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Surface codes, the 2D classical Ising model, and non-interacting fermions Goff, Leonard Thomas 2011

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Surface Codes, the 2D Classical Ising Model, and Non-Interacting Fermions by Leonard Goff  B.S. Physics, University of Maryland, College Park, 2008 B.A. Philosophy, University of Maryland, College Park, 2008  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Physics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) July 2011 © Leonard Goff 2011  Abstract In this thesis, we consider the task of simulating measurement-based quantum computation (MBQC) on surface code states: the generalization of Kitaev’s toric code to graphs embedded on a surface of higher genus. We define a family of higher genus graphs and a simple ordering of single qubit measurements, and find that simulating MBQC on any of the associated surface code states is equivalent to evaluating the inner-product between a product state and a surface code state on another graph. We further find that such an inner-product can always be written as a sum of one or more 2D classical Ising model partition functions, with appropriate couplings. For certain higher genus square lattices, we develop a means to evaluate this partition function in a number of steps that scales polynomially in the number of qubits, but exponentially in the genus of the embedded graph. The method makes use of the transfer matrix formalism for the Ising partition function, and a subsequent mapping to fermion operators. We synthesize these results to relate the simulation of MBQC on certain surface code states to a system of fermions interacting with the encoded qubits of the surface code. We identify a family of states in the code space of the surface code on our higher genus graphs for which MBQC can be simulated efficiently in all parameters, including the genus of the embedded graph. Finally, we identify two connections between the complexity of this task and entanglement.  ii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 1  I Measurement-Based Quantum Computation with Surface Code States . . . . . . . . . . . . . . . . . . . . . . . .  4  2 The 2.1 2.2 2.3 2.4 2.5 2.6  Surface Code . . . . . . . . . . . . . . . . . . . . Stabilizer Codes . . . . . . . . . . . . . . . . . . . . The Surface Code Stabilizers . . . . . . . . . . . . . Encoded Operators . . . . . . . . . . . . . . . . . . . The Code Space Explicitly . . . . . . . . . . . . . . . Overlaps Between Product States and |K(G)⟩ . . . Product State Overlaps in the Code Space . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . 5 . 5 . 6 . 8 . 10 . 13 . 16  3 MBQC with Surface Code States in General . . . . . . . . 18 3.1 Classical Simulation of MBQC . . . . . . . . . . . . . . . . . 18 3.2 Partial Measurement Probabilities . . . . . . . . . . . . . . . 20 4 MBQC on Punctured Cylinder Code States . . . . . . . . . 29 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  iii  TABLE OF CONTENTS 4.2 4.3  II  Measurements Between Holes . . . . . . . . . . . . . . . . . . Crossing Holes . . . . . . . . . . . . . . . . . . . . . . . . . .  32 36  Evaluation of the Ising Model Partition Function . . 45  5 The 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9  . . . . . . . . . .  . . . . . . . . . .  46 46 48 50 54 56 58 63 71 75  6 Efficient Simulation of Fermion Gaussian Operators . . . 6.1 Fermion Gaussian Operators . . . . . . . . . . . . . . . . . 6.2 Mapping from Coefficient Matrices to Covariance Matrices . 6.3 Lie Algebraic Multiplication . . . . . . . . . . . . . . . . . . 6.4 Synthesis: Characterizing a Product of Gaussian Operators 6.5 Application to Ising Model . . . . . . . . . . . . . . . . . .  . . . . . .  78 78 80 86 89 91  III  2D Classical Ising Model Partition Function . Introduction and History . . . . . . . . . . . . . . . Formulation of the Problem for an Arbitrary Graph Square Lattices: Transfer Matrix Formalism . . . . . Mapping to Fermions . . . . . . . . . . . . . . . . . . Partition Function on a Cylinder or Torus . . . . . . Partition Function On a Double Torus . . . . . . . . Partition Function on a Square g-Graph . . . . . . . Partition Function on a Punctured Cylinder Graph . Summary . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  Surface Code MBQC as a Fermion-Qubit Problem . 95  7 Qubits in the Ising Model Partition Function . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Ising Bits as Surface Code Qubits . . . . . . . . . 7.3 Quantum Circuit Interpretation . . . . . . . . . . . . . 7.4 Simulation Cost and Entanglement . . . . . . . . . . . 7.4.1 Entanglement Across the Fermion/Qubit Split 7.4.2 Effective Entanglement in the Output State . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  96 96 98 104 105 107 109  8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118  iv  TABLE OF CONTENTS  Appendices A Relevant Topological Graph Theory . . . . . . . . . . . . . . 123 B The Signs in Equations 2.6 and 5.8 . . . . . . . . . . . . . . 125 B.1 Sign of the Square Root in Equation 2.7 . . . . . . . . . . . . 125 B.2 Sign of the Square Root in Equation 5.8 . . . . . . . . . . . . 127 C Bipartite Entanglement of Surface Code States . . . . . . . 128 D Fermion Gaussian Operators . . . . . . . . . . . . . . . . . . 130 D.1 Relationship to Exponentials of Quadratic Fermion Operators 130 D.2 Gaussian Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 133 E Diagonalizable Complex Antisymmetric Matrices  . . . . . 135  F Lie Algebraic Time Evolution . . . . . . . . . . . . . . . . . 137 F.1 General Result . . . . . . . . . . . . . . . . . . . . . . . . . . 137 F.2 Application to Non-Interacting Fermions . . . . . . . . . . . . 140 G Supplement to MBQC on the Punctured Cylinder Code . 145  v  List of Figures 2.1  3.1  4.1 4.2  4.3 4.4 4.5  A choice of cycles and cocycles on a square toroidal graph (g=1), obeying |Cj ∩ Ck′ | = δjk .. In the figure, the free ends of each row are identified and similarly for each column (periodic boundary conditions in both directions). . . . . . . . . . . . .  10  ¯ ⊂ E such that A square lattice graph G and a choice of E ¯ ¯ S = S(E), but for which G(E) is not connected. The edges ¯ are indicated by thick lines, and the vertices in in the set E ¯ ¯ has two connected ∂ E are indicated by circles (green). G(E) ˆ components, while G(E) has only one. . . . . . . . . . . . . .  25  A three-slot punctured cylinder graph cellularly embedded on a surface of genus three. . . . . . . . . . . . . . . . . . . . . . A 4 slot punctured torus graph. The n=4 handles have positions (xj , yj ) and widths Kj for j = 1...n. Pairs of points marked by diamonds are identified within each column. . . . A set of 8 non-trivial cocycles Ck′ on a 4 slot punctured cylinder graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The lattice G at a typical stage when MBQC is “between holes”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The top half of the figure shows the two copies of the edge ¯ (bold), which are glued together along the vertices in set E ¯ (green circles) to form the graph G1 (E) ¯ ∪ G2 (E) ¯ shown in ∂E bottom left. The bottom half of the figure shows an effective ˆ corresponding to G1 (E) ¯ ∪ punctured cylinder graph G′ (E) ¯ The red horizontal edges are measured into the |+⟩ G2 (E). state, while the vertical yellow edges are measured into the |0⟩ state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  29  30 31 32  36  vi  LIST OF FIGURES 4.6  An example of the part of a punctured cylinder graph around ′ ′ are the k th hole. The non-trivial cocycles C2k−1 and C2k shown as dotted lines (orange and purple, respectively). These reflect the choice of cocycles depicted in Figure 4.3. . . . . . . 4.7 A stage of computation just before edge a of Figure 4.6 is measured. Two copies of the relevant part of G are shown on ¯ is shown on the right. the left, and the relevant part of G(E) 4.8 A stage of computation just after edge b of Figure 4.6 has been measured. Two copies of the relevant part of G are shown on ¯ is shown on the right. the left, and the relevant part of G(E) 4.9 The relevant part of G part-way through measurement of the ′ ¯ is shown in bold, and edges along C2k−1 . Again, the set E ˆ is non-bold. . . . . . . . . . . . . . . . . . . . . . . the set E 4.10 A stage of computation just after edge a of Figure 4.6 has been measured. Two copies of the relevant part of G are shown on the left, and the relevant part of the effective graph ¯ are shown on the right. . . . . . . . . . . . . . . . . . . G′ (E) ¯ during the step depicted in Figure 4.11 The relevant part of G′ (E) 4.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 5.2  5.3  5.4  A genus two square lattice . . . . . . . . . . . . . . . . . . . . A lattice composed of four toroidal graphs stitched together. The boundary conditions are given by the following rule: the two endpoints of each line defining the grid are identified with each other. The leftmost vertical lines are slightly elongated to distinguish them from the vertical lines on the fourth torus. This square g-graph lattice has a “staircase” structure. . . . . Non-trivial cycles of a square 4-graph. The dotted lines indicate cycles on the lattice itself, while the solid lines indicate cocycles (cycles on the dual lattice). . . . . . . . . . . . . . . A set of non-trivial cocycles on a square 4-graph, different from those in Figure 5.3. The binary variables α and β from Section 5.7 determine whether or not the Ising couplings along these non-trivial cocycles get multiplied by minus one. The ′ correspond to the binary variable α and the cocycles C2j j ′ cocycles C2j−1 correspond to the binary variables βj . . . . .  37  38  38  39  40 43 59  64  65  71  vii  LIST OF FIGURES 5.5  7.1 7.2 7.3  The left-most part of Figure a) shows an alternative basis for the first homology group of the dual graph of the square 4-torus. Figure a) shows the linear transformation from this basis to the one shown in Figure 5.3. Each equation, e.g. A′ = A + B, is true up to the addition of a trivial cocycle of edges. Figure b) shows the trivial cocycles on the dual lattice that must be added to demonstrate the third equation down (blue) for each step. . . . . . . . . . . . . . . . . . . . . . . .  72  Two bases Ck′ and Dk′ for the first homology group on a double torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 A quantum circuit preparing the state |C α,β ⟩ in the case of g=2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 ˜ for A more detailed quantum circuit representation of ⟨ϕ|ψ⟩ the genus four punctured cylinder graph from Figure 4.2, pictured above the dotted line. The quantum circuit depicted below the dotted line is composed of a set of N = 8 fermionic modes, and a set of 2g = 8 qubits. There is a quantum gate associated with each edge of the punctured cylinder graph. Each circle (blue) is a fermionic gate of the form ∗ e−iJ c2j c2j+1 , and each triangle (red) is a fermionic gate of the ∗ form A∗ ∗ e−iγ c2j−1 c2j . Some of these gates (hatched, green and orange) operate with couplings that are controlled by qubits from below, as shown with the vertical lines. . . . . . . 106  G.1 At certain stages in the computation, more columns must be ¯ The edge set E ¯ is shown in bold. added to the graph G′ (E). Compare with Figure 4.10. Two copies of the relevant part ¯ is of G are shown on the left, and the relevant part of G(E) shown on the right. . . . . . . . . . . . . . . . . . . . . . . . . 147 G.2 A stage of computation just before edge b of Figure 4.6 is ¯ is shown in bold. Two copies of measured. The edge set E the relevant part of G are shown on the left, and the relevant ¯ is shown on the right. . . . . . . . . . . . . . . . 147 part of G(E)  viii  Acknowledgements My sincerest gratitude goes to my supervisor Robert Raussendorf, with whom it has been my privilege and pleasure to work. I also want to thank other members of the UBC Quantum Information group: Pradeep Sarvepelli, Tzu-Chieh Wei, Matthew Sholte, and Poya Haghnegahdar, for their helpful discussions throughout this research.  ix  To Mom, Dad and my brother Tim, with love.  x  Chapter 1  Introduction 1.1  Introduction  Measurement-based quantum computation (MBQC) is a scheme of quantum computation in which computation begins with an initial quantum state and proceeds via successive single qubit measurements, rather than by unitary evolution as in the original circuit model of quantum computation. MBQC was originally proposed by Raussendorf and Briegel [53]. The initial quantum state in MBQC is called the resource state. It is known that with the 2D cluster state [5] as a resource state, universal quantum computation can be performed with polynomial overhead as compared with the circuit model [6]. Despite the inherent indeterminism of each quantum measurement outcome, MBQC mimics the deterministic evolution of the circuit model by adaptive choice of future single qubit measurement bases depending on past outcomes. These bases are computed via polynomial sized classical side processing. After performing measurements to process quantum information, one is left with a pure quantum state defined on the unmeasured qubits (the measured qubits have been disentangled from the rest and can be ignored). Universal MBQC is defined as the ability to prepare an arbitrary quantum state in this way, up to known single qubit rotations [11]. Classical output can then be obtained from this state by measuring the rest of the qubits in appropriate bases. While 2D cluster states were the first proposed resource state for MBQC, other important resource states have been identified in subsequent investigations. For instance, it has been shown that with a 3D cluster state, one can perform MBQC with fault-tolerance [51]. Systematic studies of other universal resource states and their properties has been initiated in [22], [23], [13], resulting in the identification of several such families of states. It has recently been shown that the 2D AKLT state is also a universal resource state for MBQC [62],[61],[45]. An interesting class of states to study in this context are stabilizer states. Stabilizer states are central to the theory of quantum error correction and 1  1.1. Introduction the Gottesman-Knill theorem [48], and are also important to MBQC. In particular, the class of stabilizer states includes that of graph states, of which the cluster state is an example [28]. Of course, not all stabilizer states will be universal resource states for MBQC. Ref [13] discusses two entanglement monotones: the entropic entanglement width and the Schmidt rank width, which must diverge with the number of qubits N for any family of universal resource states. However, this result is probably too weak to establish the non-universality of resource states in general. For a resource state |R⟩ could have diverging entanglement, but if its Schmidt-rank width diverges only logarithmically with N, it has been shown that MBQC with |R⟩ can be simulated on a classical computer with resources that are polynomial in N [11]. Under the assumption that there exist quantum computational algorithms that cannot be simulated classically with poly(N ) (time and space) resources, such states |R⟩ must not be universal resource states. Raussendorf and Briegel identified a stabilizer state: the planar code state, for which MBQC can be simulated on a classical computer with resources polynomial in the number of qubits, under the weak restriction that the sets of measured and unmeasured qubits are connected at each stage of the computation [52]. But the authors also show that the Schmidt-rank width for the planar code state grows at least linearly with N, so the simulation result cannot be subsumed under the one based on this entanglement measure in Ref [11]. In this thesis, we will study a generalization of the planar code state: surface code stabilizer states, and consider the task of classically simulating MBQC with such states as resource states. In particular, we will ask the question: In what parameter(s) does the cost of simulation grow faster than polynomially? The aim of the investigation is to contribute to our understanding of what it is about quantum computation that makes it more powerful than classical computation. Efficient quantum algorithms such as Shor’s factoring algorithm suggest that for certain computational tasks, a quantum computer offers an exponential speedup over any classical computer, implying that no algorithm on a classical computation could simulate the output of a universal quantum computer with sub-exponential overhead. But at this point, this is merely a conjecture. In the absence of a more fundamental physical understanding of what the power of quantum computation arises from, it is likely to remain so. One candidate for explaining the quantum computational speedup is entanglement, which is a purely quantum physical phenomenon with no classical counterpart. Useful quantum computational tools such as the quantum Fourier transform and quantum teleportation exploit entanglement, 2  1.1. Introduction and these operations are costly to simulate on a classical computer. An important no-go theorem regarding entanglement was obtained by Vidal [59], which says that any circuit model quantum computation can be classically simulated with a number of steps that is polynomial in the entanglement monotone χ, where χ is the maximum value throughout the computation of the maximal Schmidt rank across all bipartitions of the qubits. A simulation scheme based on the Schmidt-rank width entanglement measure for the resource state provides an analogous simulation result for MBQC [11]. These results establish entanglement as a necessary condition for a quantum computational speedup. However, the presence of entanglement in a quantum computational process is far from being a sufficient condition for a quantum computational speedup. The Gottesman-Knill Theorem shows a class of highly entangling qubit operations that can be efficiently classically simulated [48], as does the analysis of matchgate quantum circuits [33]. A more recent result by Gross, Flammia, and Eisert [24] turns the relationship between quantum computational power and entanglement on it’s head: quantum computations that produce states with too much entanglement can be efficiently simulated by a classical fair coin, and the simulation cost actually decreases the larger the entanglement. Thus, the role of entanglement in the hypothesized quantum computational speedup is somewhere between being a necessary and a sufficient condition. Making this connection more precise is an active area of research. In Part I of this thesis, we define the surface code states and consider the task of classically simulating MBQC with surface code states as resource states. We will introduce cases where simulation is equivalent to the evaluation of a 2D classical Ising model partition function with complex inhomogeneous couplings. The relationship between the surface code and the 2D classical Ising model has previously been used to compute its error threshold for quantum error correction [14], and was identified in the case of planar lattices by Brayvi and Raussendorf [52]. In Part II, we take on the task of evaluating the Ising partition function for two families of higher genus graphs. We make a novel application of the so-called transfer matrix formalism, and map the problem onto one involving non-interacting fermions. We develop an algorithm to compute the partition function in a number of steps that is polynomial in the size of the lattice, but exponential in its genus. In Part III we bring the surface code and the Ising model fermions together into one picture, and discuss how the complexity of simulating MBQC on surface code states can be related to entanglement.  3  Part I Measurement-Based Quantum Computation with Surface Code States  4  Chapter 2  The Surface Code 2.1  Stabilizer Codes  The foremost practical challenge of building a quantum computer with a useful number of qubits is battling decoherence, which inevitably arises from interaction between a quantum information processing device and its environment. Decoherence must be dealt with in some way in order to protect the coherent superposition of states which is so special to quantum physics. A major triumph of the quantum information field was the discovery that quantum computation can be performed fault-tolerantly, such that errors due to decoherence can be corrected with arbitrary reliability, if they occur with probability below some non-zero threshold. A central scheme in fault-tolerant quantum computation is the use of a class of quantum error correcting codes called stabilizer codes, which we will briefly introduce here in order to define the surface code. For a more detailed introduction to stabilizer codes, see [21] and [48]. Stabilizer codes are a means of performing quantum error correction by using extra qubits to encode quantum information non-locally in an entangled state, protecting it from suitably local errors. Formally, consider a system of n physical qubits and its associated Hilbert Space Hn = (C2 )⊗n , and k independent, commuting stabilizer generators s1 ...sk . Each stabilizer generator is a member of the Pauli group, which is the group generated by tensor products of the Pauli matrices for each qubit ( ) ( ) ( ) ( ) 0 1 0 −i 1 0 1 0 X= , Y = , Z= , X= 1 0 i 0 0 −1 0 1 along with the multiplicative factors ±1, ±i. We define a code space CS ⊂ Hn as the mutual +1 eigenspace of all of the stabilizer generators CS := {|ψ⟩ ∈ Hn : sj |ψ⟩ = |ψ⟩ ∀j} The dimensionality of the code space is 2n−k , and it can encode an n − k qubit state in a manner that is protected against local errors [48]. The 5  2.2. The Surface Code Stabilizers stabilizer generators themselves generate a group, members of which we call stabilizer operators. Any error which takes a state out of the code space can be detected by measuring stabilizer operators, and then corrected. The only errors which a stabilizer code space is not protected from are errors that commute with all of the stabilizer operators, yet act non-trivially on the code space. A good stabilizer code makes such errors highly improbable by requiring many qubits to go foul simultaneously to implement an nondetectable error. The minimum such number of qubits is called the distance of the stabilizer code.  2.2  The Surface Code Stabilizers  Now we introduce the surface code, which is a stabilizer code that has a topological sort of defense against non-detectable errors. Consider a connected graph G with no self-loops, and a cellular embedding of it on a closed, orientable surface S (see Appendix A for definitions). Denote the sets of vertices, edges and faces by V , E and F , respectively. We associate a qubit with each of the edges e ∈ E of the graph G. A surface code is a stabilizer code with the stabilizer generators1 ∏ As := Ze ∀s ∈ V e∈δs  Bp :=  ∏  Xe  ∀p ∈ F  e∈∂p  where δs is the set of all edges incident on vertex s, and ∂p is the set of edges on the boundary of face p (we use the letter p for plaquette). We again refer to Appendix A for definitions. We will sometimes refer to the As as site operators and the Bp as face operators. The code space CS of a surface code is CS = {|ψ⟩ : As |ψ⟩ = Bp |ψ⟩ = |ψ⟩  ∀s ∈ V, p ∈ F }  We note that the code space is also the ground-space of the following exactly-solvable Hamiltonian, with some arbitrary energy scale E: ∑ ∑ H = −E As − E Bp s∈V  p∈F  1  Here we use X Pauli operators for the faces and Z for the vertices (as in [52]), rather than Z operators for the faces and X for the vertices as with Kitaev’s toric code([40]) and most other treatments of the surface code. This simplifies some of the later discussion. The two code spaces are equivalent up to a global Hadamard transformation.  6  2.2. The Surface Code Stabilizers We now consider the independence and commutation relations of the surface code stabilizer generators. As products of Pauli X operators: [Bp , Bq ] = 0 for all p, q ∈ F , and as products of Z operators: [Bs , Bt ] = 0 for all s, t ∈ V . The following condition holds generally for cellular embeddings: Even Sharing: the boundary of any face p shares an even number of edges (or zero) with the set of edges incident on any vertex s It follows from Even Sharing that [As , Bp ] = 0 for all s ∈ V and p ∈ F , thus all of the stabilizer generators commute. Since every edge is incident on exactly two vertices, the product of all of the vertex stabilizers is equal to the identity ∏ ∏ ∏ ∏ As = Ze = Ze2 = I s∈V  s∈V e∈δs  e∈E  So, only |V | − 1 of the |V | vertex generators are independent (the rest of the vertex generators are mutually independent given that G is connected). Similarly, one of the face stabilizers Bp is generally not independent of the rest. Consider a particular face q. Since the faces partition S, the region remaining when we combine the |F | − 1 faces in the set F/q will be precisely the chosen face q. Every edge either on the boundary of no faces or it is on the boundary of exactly two faces, so:     ∏ ∏ ∏ ∏ ∏  ∏  2   Xe = B q Xe =  X = Bp = X e e  p∈F/q  p∈F/q e∈∂p  e∈E:(e∈∂q, / ∃p∈F :e∈∂p)  e∈∂q  e∈∂q  So there are only |F | − 1 independent Bq operators. A quantum stabilizer code on n qubits with k independent, commuting stabilizer generators encodes n − k qubits. In our case, n = |E| and k = |F | + |V | − 2, thus the number of encoded qubits m is 2 − χ, where χ is the Euler characteristic: χ := |V | − |E| + |F |. For a closed, orientable surface S, the Euler characteristic is related to the genus of S as χ = 2 − 2g. So: m = 2g; the number of encoded qubits is twice the genus of S. The striking result here is that m depends only on a property of the graph’s topological properties, and not upon the size or local structure of the graph G. We mention here two special cases of surface codes that are familiar in the quantum information literature. In the case that the graph G is a rectangular lattice with ordinary boundary conditions, then the genus is zero and the code space contains only one state, which is the planar code state 7  2.3. Encoded Operators studied in [52]. If the graph G is a rectangular lattice with periodic boundary conditions in both directions (wrapped on a torus), then the resulting quantum stabilizer code is the well-studied toric code introduced by Kitaev [40].  2.3  Encoded Operators  To encode a 2g qubit quantum state into the code space of the surface code, we need to define operators acting as logical Pauli operators within the code space. We adopt a notation where all encoded operators and states acting in the physical Hilbert space are marked with a tilde. For instance, Xj will denote the operator in the Hilbert space of the |E| physical qubits, but acting in the code space as a Pauli X matrix for the j th encoded qubit. A quantum state |ψ⟩ will denote the encoding of the 2g qubit state |ψ⟩ into the surface code. Consider a set of edges C. Then define string operators XC and ZC ∏ ∏ XC := Xe and ZC := Ze e∈C  e∈C  If for any vertex v ∈ V , C contains an even number of edges incident on v (i.e. C comprises a set of cycles on G), then XC commutes with all of the stabilizer generators, and thus maps a state in CS into a state in CS. The same is true of ZC if for any face p, C contains an even number of edges on the boundary of p. This is true if C comprises a set of cocycles on G (i.e. ˜ see Appendix A). cycles on the dual graph G, For any set of edges C which forms the boundary of a region of S, XC is a product of face operators: ∏ XC = Bp p∈F (C)  where F (C) is a set of faces who’s union is equal to the region bounded by C. Such an operator will act trivially on the code space: XC |ψ⟩ = |ψ⟩  ∀|ψ⟩ ∈ CS  Consequently, for any two edge sets C1 and C2 who differ from one another only by the boundary C of some region, XC1 and XC2 will act identically on the code space. Put differently, the action of the operator XC for any cycle C depends only on the homology class of C. The same is true 8  2.3. Encoded Operators ˜ if C is a cocycle on of the ZC operators with respect to the dual lattice G: G, then the action of the operator ZC depends only on the homology class ˜ Thus if we define the encoded Pauli operators to of C with respect to G. be string operators corresponding to cycles or cocycles of the graph G, they will really only need to be defined up to a homology class. We know from Section 2.2 that the codespace CS has dimension 22g , where g is the genus of the surface S that G is embedded on. We wish now to find a set of independent logical Pauli operators Xj and Zj for j = 1...2g acting non-trivially on this space, but commuting with the stabilizers, with the properties: 1. [Xj , Xk ] = [Zj , Zk ] = 0 2. [Xj , Zk ] = 0 if j ̸= k 3. Xj Zj = −Zj Xj Note that for any two edge sets C1 and C2 : [XC1 , XC2 ] = [ZC1 , ZC2 ] = 0, so the first property will be trivially satisfied if we choose Xj and Zj to be X and Z type string operators, respectively. For any two edge sets such that C1 and C2 that share an even number of edges: [XC1 , ZC2 ] = 0, while if the two sets share an odd number of edges: XC1 ZC2 = −ZC2 XC1 . So, the second and third properties could be satisfied by setting Xj := XCj and Zj := ZCj′ , with the Cj cycles and Cj′ cocycles, and |Cj ∩ Ck′ | = δjk (mod 2) - i.e. Cj and Ck′ have an odd number of edges in common if j = k, and share an even number of edges otherwise. We also want each of the logical operators to be independent, and to act non-trivially on the code space. To achieve this, we need to choose the C1 ...C2g to be a set of homologically independent non-trivial cycles, and the ′ to be a set of homologically independent non-trivial cocycles. In C1′ ...C2g this way none of the encoded operators act trivially on the codespace (none of the Cj′ or Cj correspond to the boundary of a region of S), each of the Xj are mutually independent, each of the Zj are mutually independent, and all of the logical Pauli operators commute with the stabilizers. All that remains is to choose the cycles and cocycles such that |Cj ∩Ck′ | = δjk (mod 2). Any closed, orientable surface of genus g is homeomorphic to a connected sum of g tori [2]. Each torus contributes two homologically distinct non-trivial cycles, which can be chosen in a natural way: one going through the hole and one circumnavigating it. These two cycles cross at exactly one point on S, so we can generally satisfy |Cj ∩ Ck′ | = δjk (mod 9  2.4. The Code Space Explicitly 2) by choosing C2j−1 to be a cycle of G going through the hole of the j th ′ ˜ circumnavigating the hole of the j th torus, torus, C2j−1 to be a cycle of G ′ C2j to be a cycle of G circumnavigating the hole of the j th torus, and C2j ˜ going through the hole of the j th torus. Note that the to be a cycle of G cycles Cj and Cj′ need not be chosen in this way. The only requirements are that the Cj are non-trivial cycles on G that each belong in a different homology class, that the Cj′ are non-trivial and mutually non-homologous cocycles, and that |Cj ∩ Ck′ | = δjk . A suitable choice in the simple case of a torus is shown in Figure 2.1.  Figure 2.1: A choice of cycles and cocycles on a square toroidal graph (g=1), obeying |Cj ∩ Ck′ | = δjk .. In the figure, the free ends of each row are identified and similarly for each column (periodic boundary conditions in both directions).  2.4  The Code Space Explicitly  We now construct the code space explicitly for a given graph G, in terms of the eigenstates of the physical Pauli operators. Consider a joint-eigenstate of the Pauli Z operators for each of the |E| physical qubits: |x⟩ := |x1 ....x|E| ⟩ where each xj is a binary variable xj ∈ {0, 1}, and Ze |x⟩ = (−1)xe |x⟩ for any e ∈ E. The action of a vertex stabilizer As on |x⟩ is ( ) ∑ ∏ As |x⟩ = Ze |x⟩ = (−1) e∈δs xe |x⟩ e∈δs  10  2.4. The Code Space Explicitly ∑ Thus any Pauli Z basis state for which e∈δs xe = 0 (mod 2) will be an ∑ +1 eigenstate of the As stabilizer. If the bitstring x is chosen such that e∈δs xe = 0 for all vertices s, then |x⟩ is in the joint +1 eigenspace of all of the As operators. Denote the set of bitstrings x for which this is true by E0 (G). A bitstring over the edges can be though of as a subset of the edges, the subset being those edges e for which xe = 1. Then we see that E0 (G) is the set of all closed subgraphs of G, where a closed subgraph is a subset of the edges such that every vertex has an even number of edges incident upon it from the set [37]. A closed subgraph is equivalent to a set of one or more disjoint cycles. The edges in a closed subgraph consist only of closed paths with no endpoints. From here on, we will interchangeably talk about bitstrings in {0, 1}⊗|E| and subsets of edges with the obvious equivalence in mind. In this language, the bitwise addition x ⊕ y of two bitstrings x and y is identical to the symmetric difference of the edge sets x and y. The action of the a face stabilizer generator on a Pauli Z eigenstate is   ∏ Bp |x⟩ =  Xe  |x⟩ = |x ⊕ ∂p⟩ e∈∂p  where the bitstring x ⊕ ∂p indicates flipping the bit xe for all e ∈ ∂p. With the Even Sharing condition from Section 2.2 (|∂p ∩ δs| = 0 (mod 2) ∀s ∈ V ), we have that for any bitstring x: x ∈ E0 (G) iff x ⊕ ∂p ∈ E0 . It is now easy to see that the following state is in the code space of the surface code: ∑ 1 |K(G)⟩ = √ |x⟩ (2.1) |E0 (G)| x∈E (G) 0  Since Bp is unitary and hence invertible, it simply permutes the order of terms in the symmetric summation over E0 (G). For a planar lattice, Equation 2.1 is the unique solution to the stabilizer equations, since in this case the dimension of the code space is 20 = 1. The normalization of |K(G)⟩ dictates an overall coefficient of |E0 |−1/2 , which we would like to be able to compute. E0 (G) is defined by |V | equations ∑ relating |E| binary variables: x ∈ E0 (G) iff ∀s ∈ V : e∈δs xe = 0 (mod 2). Since every edge is incident on exactly two vertices, we can add the equations for all vertices s and obtain 0 = 0. So long as G is connected, the equations are otherwise linearly independent so we have a total of |V | − 1 independent equations. So E0 contains |E| − |V | + 1 binary degrees of freedom, and |E0 (G)| = 2|E|−|V |+1 = 22g ∗ 2|F |−1 . |E0 (G)| is the size of the so-called cycle-space of G. 11  2.4. The Code Space Explicitly The size of the set E0 (G) allows us to provide a complete characterization of it. Consider the following fact: any cycle on the graph G is a member of the set E0 (G), since it has an even number of edges incident on any vertex. For any set of faces, the superposition cycle f of the boundaries of the faces in the set is a cycle on G and so f ∈ E0 (G). As we saw in Section 2.2, every product of face operators except one is unique, so 2|F |−1 members of E0 (G) correspond to the various subsets of F with unique products of face operators. These comprise the set of all trivial cycles on G. For each face p except the last, there is a binary degree of freedom associated with the choice of whether or not to include p in the product. Any non-trivial cycle Ck is also a member of E0 (G). Pick a set C of 2g non-trivial cycles, each belonging to a different homology class. Now consider the superposition cycle generated by repeated superposition over any of the 22g subsets of C. For each such superposition cycle x, x ∈ E0 (G). We may deform x by taking its superposition with a product of face operators f . Each resulting set of edges is unique, giving us the full set E0 of size 22g ∗ 2|F |−1 . It turns out that the state |K(G)⟩ is always the +1 eigenstate for all of the encoded Pauli X operators, provided that they are defined as Xj = XCj for each j = 1...2g, where Cj is some cycle on the lattice G:   ∏ ∑ 1 Xe  |K(G)⟩ = √ Xj |K(G)⟩ =  |x ⊕ Cj ⟩ |E (G)| 0 e∈C x∈E0 j  Since Cj is a cycle, every vertex s ∈ V has either zero or two edges incident upon it in the set Cj . Thus x ∈ E0 (G) iff x⊕Cj ∈ E0 (G). So as with Bp , the unitary operator Xj simply permutes the order of the symmetric summation over E0 (G), and Xj |K(G)⟩ = |K(G)⟩. Thus, |K(G)⟩ is the encoded |+⟩ state |+⟩ of the surface code. We can now construct an orthogonal basis for all of the code space by applying encoded Z operators to |K(G)⟩. Let α = α1 ...αg and β = β1 ...βg be bitstrings, and define   g ∏ |Xα,β ⟩ =  (Z2j−1 )αj (Z2j )βj  |K(G)⟩ (2.2) j=1  Here we keep track of the even and odd qubits separately for later convenience. The state |Xα,β ⟩ is a mutual eigenstate of all of the encoded Pauli X operators Xj , with eigenvalue (−1)αj for encoded qubit 2j − 1 and eigenvalue (−1)βj for encoded qubit 2j. Furthermore, the 22g different |Xα,β ⟩ 12  2.5. Overlaps Between Product States and |K(G)⟩ form an orthonormal basis of the code space, because   g ∏ ⟨Xγ,ρ |Xα,β ⟩ = ⟨K(G)|  (Z2j−1 )αj +γj (Z2j )βj +ρj  |K(G)⟩ (2.3) j=1  If (α, β) = (γ, ρ), then ⟨Xγ,ρ |Xα,β ⟩ = ⟨K(G)|K(G)⟩ = 1 as expected. If on the other hand α, β ̸= γ, ρ, then the above summation vanishes. To see this, suppose that the operator Z˜k occurs in Equation 2.3 (i.e. its exponent is one, not zero). Since the state |K(G)⟩ is a +1 eigenstate of the corresponding ˜ k , we may insert it into Equation 2.3 as X ˜ k |K(G)⟩. encoded X operator X ˜ ˜ ˜ Xk commutes with all of the Z operators except Zk , which it anticommutes ˜ k all the way to the left and absorb it with. We can use this to commute X into the bra ⟨K(G)|, incurring a minus sign along the way. We have then that ⟨Xγ,ρ |Xα,β ⟩ = −⟨Xγ,ρ |Xα,β ⟩, so ⟨Xγ,ρ |Xα,β ⟩ = 0. Thus: ⟨Xγ,ρ |Xα,β ⟩ = δγ,α δβ,ρ  2.5  (2.4)  Overlaps Between Product States and |K(G)⟩  Consider an arbitrary product state in the Hilbert space of the physical qubits ⊗ |ϕ⟩ = |ϕe ⟩e e∈E  where |ϕe ⟩e := ae |0⟩e + be |1⟩e . To simplify matters, we take each ae to be real positive or zero. This simply reflects a choice of overall phase of |ϕ⟩, which is unimportant if we consider a single state |ϕ⟩. When extending the results of this section to a coherent superposition over product states |ϕ⟩ (as we will have occasion to in Sections 2.12 and 7.4), extra care must be taken to keep track of any phase that must be factored out for each term. For a planar lattice G, it was shown in [52] that ⟨K(G)|ϕ⟩ is proportional to a 2D classical Ising model partition function with appropriate couplings. Here we will generalize this result to the state |K(G)⟩ corresponding to  13  2.5. Overlaps Between Product States and |K(G)⟩ arbitrary graph G: ⟨K(G)|ϕ⟩ =  =  =  √  (  ∑  1  )  ⊗  ⟨x| ae |0⟩e + be |1⟩e |E0 (G)| x∈E (G) e∈E 0 ( ) ( ) ( ) ⊗ ∏ ∑ 1 be √ ⟨x| |0⟩e + |1⟩e ae ae |E0 (G)| e∈E e∈E x∈E0 (G) ( ) ( ∏ ∑ ∏ be )xe 1 √ ae ae |E0 (G)| e∈E x∈E (G) e∈E 0  If ae = 0 for any edge e, then one can take a limit of ae + δ as δ → 0 from the right, and use continuity of ⟨K(G)|ϕ⟩ as a function of ae . The above equation relates ⟨K(G)|ϕ⟩ to the partition function of a 2D classical Ising Model on the graph G, to be introduced in Chapter 5. We will see in Equation 5.2 that for an Ising model defined on the graph G with coupling strengths Je ∗(−kB T ) associated with each edge e ∈ E, the partition function at temperature T is ( ) ( ) ∏ ∑ ∏ Z({Je }) = 2|V | cosh(Je ) tanh(Je ) (2.5) e∈E  e∈E0 (G)  e∈e  Comparing ( ) with the expression for ⟨K|ϕ⟩, notice that if we set Je = be −1 tanh ae , then ⟨K(G)|ϕ⟩ =  2|V |  √  (  1 |E0 (G)|  ∏  e∈E  ae cosh(Je )  ) Z({Je })  (2.6)  Using the identity sech2 (z) = 1 − tanh2 (z), we can rewrite Equation 2.6 as ⟨K(G)|ϕ⟩ =  2|V |  √  (  1 |E0 (G)|  ∏√  e∈E  ) a2e − b2e  ( ( ) ) be −1 } (2.7) Z {tanh ae  Despite the presence of the expression abee in Equation 2.7, the situation where ae = 0, be = 1 does not result in any infinite quantities appearing in the partition function. This can be seen by writing tanh−1 (z) = 1 ln() denotes the multivalued complex log2 (ln(1 + z) − ln(1 − z)),( where ) arithm. Then: tanh−1  be ae  =  1 2  (ln(ae + be ) − ln(ae − be )), which is well 14  2.5. Overlaps Between Product States and |K(G)⟩ behaved when ae = 0, be = 1, i.e. the qubit e is in a Pauli Z eigenstate. However, singular quantities do arise ( )if ae = ±be , i.e. the qubit e is in a Pauli X eigenstate. Then tanh−1 abee becomes infinite, and the coefficient √ a2e − b2e goes to zero. In this case, one can use the continuity of ⟨K(G)|ϕ⟩ as a function of ae and be and express ⟨K(G)|ϕ⟩ as a limit of a sequence of well behaved coefficients such that ae → ±be . In Section 4.2, we explain how measuring a qubit in the X basis has the same effect as performing graph minor operations on the graph G. This fact will be useful when simulating intermediate steps of MBQC on the surface code. √ There is a sign ambiguity in Equation 2.7 arising from a2e − b2e , where a2e − b2e is generally a complex number. For practical calculations, we would like to specify which of the two choices of square root makes Equation 2.7 true. The answer depends a choice one makes when computing the couplings Je , which arises from its definition as an inverse hyperbolic tangent. The inverse hyperbolic tangent is a multivalued function for complex argument, because tanh(z) = tanh(z + iπ). So, we may choose any branch of π radians in the complex plane to fix the imaginary ( )part of each coupling Je . A be −1 −1 suitable choice would be to use T anh ae , where T anh (z) is defined by insisting that π π (2.8) − < I(T anh−1 (z)) ≤ 2 2 where I(z) denotes the imaginary part of the complex number z. Note however that cosh(z) = − cosh(z + iπ). We will find it convenient for later analysis to choose the rule for calculating the Je from the coefficients ae , be such that it has the property that Je → −Je when abee → − abee . In this case, cosh(Je ) will be invariant under the switch abee → − abee . The choice of branch in Equation 2.8 does not have this property if the imaginary part of Je equals π2 , because then under abee → − abee the required transformation of Je would be Je → −Je + π. Under this transformation, cosh(Je ) would flip sign (these properties can be verified by complex hyperbolic identities provided in Appendix B). So, ( we ) will instead use the slightly more complicated rule be −1 −1 that Je = T anh ae , where T anh (z) is instead defined by −  π π ≤ I(T anh−1 (z)) ≤ 2 2  (2.9)  subject to sgn(I(T anh−1 (z))) = sgn(z), if z is real with absolute value greater than one. Here, sgn(x) denotes the sign of the real nonzero number x. This condition eliminates any choice between I(T anh−1 (z)) = ± π2 , while 15  2.6. Product State Overlaps in the Code Space ensuring that Je → −Je when ⟨K(G)|ϕ⟩ =  → − abee , as desired. If we do this then ( ) ( ( ) ) ∏ ( 2 ) 1 be 2 −1 √ Sqrt ae − be Z {T anh } |V | ae 2 |E0 (G)| be ae  e∈E  (2.10) with no sign ambiguity, where Sqrt(z) denotes the principal square root of z. See Appendix B for a proof.  2.6  Product State Overlaps in the Code Space  Consider the overlap of the state |Xγ,ρ ⟩ with an arbitrary product state |ϕ⟩ as given in Section 2.5:   g ∏ ⟨Xγ,ρ |ϕ⟩ = ⟨K(G)|  (Z2j−1 )γj (Z2j )ρj  |ϕ⟩ j=1  = ⟨K(G)|ϕγ,ρ ⟩ )) ( (∏ ( 2 ( γ,ρ ) ) 2 e∈E Sqrt ae − be −1 b √ } (2.11) Z {T anh = a 2|V | |E0 (G)| (∏ ) g γj (Z )ρj |ϕ⟩. To move where the product state |ϕγ,ρ ⟩ := ( Z ) 2j−1 2j j=1 from the first line of Equation 2.11 to the second we take the encoded Pauli Z operators to operate to the right, rather then to the left. In the third line the product state |ϕγ,ρ ⟩ is characterized by the coefficients ae and bγ,ρ e in the same way that |ϕ⟩ is characterized by ae and be . The modified coefficients ′ bγ,ρ e are obtained by multiplying be by −1 for any edge e ∈ C2j−1 such that ′ such that ρ = 1. If an edge e occurs on more γj = 1 and for every e ∈ C2j j ′ than one such cocycle Ck , be needs to be multiplied by −1 once for each. For the prefactor, we have also used that (ae )2 − (bγ,ρ )e 2 = ae 2 − be 2 . Note that the assumption of each ae being real positive or zero is still valid for |ϕγ,ρ ⟩, provided that the condition on the ae was satisfied initially for |ϕ⟩. From the above it is clear that ⟨Xγ,ρ |ϕ⟩ is also proportional to an Ising Model partition function, just with modified couplings. In the last Section, we defined the Ising couplings by a single-valued inverse hyperbolic tangent of abee that is an odd function of its argument. Thus, application of a Pauli Z operator to any edge has the effect of multiplying its Ising coupling by −1. Now consider any∑ 2g qubit state |ψ⟩, with coefficients cγ,ρ in the Pauli X eigenbasis: |ψ⟩ = γ,ρ cγ,ρ |Xγ,ρ ⟩. Here |Xγ,ρ ⟩ is the X eigenstate with 16  2.6. Product State Overlaps in the Code Space eigenvalue (−1)γj for the odd qubit number 2j − 1 and (−1)ρj for the even qubit number 2j. ∑ Then the encoding of |ψ⟩ into the code space of the surface code is |ψ⟩ = γ,ρ cγ,ρ |Xγ,ρ ⟩. For any such |ψ⟩ ∈ CS we repeat the above maneuver to obtain ∑ ⟨ψ|ϕ⟩ = c∗γ,ρ ⟨K(G)|ϕγ,ρ ⟩ γ,ρ  ∏ =  (  2 2 e∈E Sqrt ae − be √ 2|V | |E0 (G)|  )  ∑ γ,ρ  c∗γ,ρ Z  ( {T anh  −1  (  ) ) bγ,ρ } (2.12) a  We see then that ⟨ψ|ϕ⟩ is always proportional to a linear combination over a maximum of 22g Ising model partition functions. We will consider the explicit evaluation of these Ising model partition function in Chapter 5.  17  Chapter 3  MBQC with Surface Code States in General 3.1  Classical Simulation of MBQC  Here we briefly discuss the simulation of measurement-based quantum computation using a classical computer. A scalable scheme of MBQC can be thought of as a pair {Ψ, A}, where Ψ is a family of resource states (e.g. Ψ = {|ψ(N )⟩, N = 1...∞} where |ψ(N )⟩ is an N qubit state), and A is a classical algorithm for determining the basis to perform each single qubit measurement in. Let {|ϕ0j ⟩, |ϕ1j ⟩} denote an orthonormal basis for the local Hilbert space of the j th qubit. Then A is a function taking all past measurement outcomes as an input, and outputting a suitable basis for the next measurement: ( ) s A |ϕs11 ⟩, |ϕs22 ⟩, ...|ϕj j ⟩ = {|ϕ0j+1 ⟩, |ϕ1j+1 ⟩} where the sl ∈ {0, 1} are the random measurement outcomes obtained up to step j. A run of MBQC begins with an N-qubit resource state |R⟩ ∈ Φ. The output is some desired M-qubit state   N ⊗ ( s −M )  | |R⟩ Uj  |ϕout ⟩ = ⟨ϕs11 |⟨ϕs22 |...⟨ϕNN−M j=N −M +1  where M < N and Uj is a possible local “correction” unitary for the j(th output qubit, ) which is some known function of s1 ...sN −M . The state ⊗N out ⟩ has been called the quantum output of the MBQC j=N −M +1 Uj |ϕ run [10]. At the end of a computation, however, one is eventually interested in some classical output which we assume to be given by measuring |ϕout ⟩ in some local computational basis. The final output of MBQC is then a full sequence of N measurement outcomes |ϕs11 ⟩, |ϕs22 ⟩, ...|ϕsNN ⟩, which is given 18  3.1. Classical Simulation of MBQC with probability ( ) p |ϕs11 ⟩, |ϕs22 ⟩, ...|ϕsNN ⟩ = ⟨R|ϕs11 ⟩|ϕs22 ⟩...|ϕsNN ⟩  2  It follows then that knowledge of the inner-product between the resource state and an arbitrary product state is sufficient to simulate MBQC on any classical computer that is capable of running A and(equipped with a random ) number generator. The probability distribution p |ϕs11 ⟩, |ϕs22 ⟩, ...|ϕsNN ⟩ can be computed from the overlaps and the distribution can be sampled from using the random number generator. We say that MBQC can be efficiently on a classical computer ( s1 simulated sN ) s2 if it is possible to sample from p |ϕ1 ⟩, |ϕ2 ⟩, ...|ϕN ⟩ with resources that are sub-exponential in N. For efficient simulation, the ability to efficiently compute local overlaps is neither necessary nor sufficient. It is not necessary because there may be means of sampling from the probability distribution without explicit knowledge of the overlaps, see for example Ref [12]. Knowledge of the overlaps is not sufficient because even if the distribution can be computed efficiently, it contains 2N possible outcomes. Thus, sampling from the distribution will generally require a number of steps that is exponential in N. However, under the stronger condition that one can efficiently compute the probability of obtaining an arbitrary set of measurement outcomes on subsets of the ( ) qubits, it is possible to not only efficiently compute p |ϕs11 ⟩, |ϕs22 ⟩, ...|ϕsNN ⟩ , but also sample from it. This was pointed out in [52] and we present the idea here. If one has an efficient means of computing ( ) ( ) s s s p |ϕs11 ⟩, ...|ϕj j ⟩ = tr{j+1...N } ⟨ϕs11 |...⟨ϕj j |R⟩⟨R|ϕs11 ⟩...|ϕj j ⟩ (3.1) for any j, then at any step of MBQC one can use Bayes’ formula to compute the probability of obtaining either of the two possible outcomes for the next qubit, conditioned on the outcomes already obtained. At step j: ( ) j+1 ( ) p |ϕs11 ⟩, ...|ϕsj+1 ⟩ sj+1 s ) p |ϕj+1 ⟩ |ϕs11 ⟩, ...|ϕj j ⟩ = ( s p |ϕs11 ⟩, ...|ϕj j ⟩ This process can be iterated at each of the N steps of simulation, such that one is only ever sampling from a two-outcome probability distribution. We assume that A is a efficient classical algorithm, that is: A requires resources that are sub-exponential in N. Then, the ability to evaluate Equation 3.1 for any |R⟩ ∈ Ψ efficiently in N is sufficient to efficiently simulate the MBQC scheme {Ψ, A} on a classical computer. 19  3.2. Partial Measurement Probabilities  3.2  Partial Measurement Probabilities  To consider the prospect of efficiently classically simulating MBQC on a state in the code space of the surface code, we seek an efficient means of computing the probabilities of local measurement outcomes on a subset of the qubits, since the ability to compute the product state overlaps we considered in Section 2.5 efficiently is not sufficient. However, we will see that in at least some cases, computing the probabilities maps back onto the problem of evaluating inner products between a surface code state and a product state. To obtain such probabilities of partial measurements, we will need to perform a partial trace over the qubits not being measured. Here we will follow the approach taken by Raussendorf and Brayvi in [52] when they considered the planar code-state. Most of the analysis turns out to be the same, but here we pursue it for higher genus graphs and with slightly greater generality regarding the sets of measured and unmeasured qubits. We again assume that the graph G in question is connected and contains ¯ ⊆ E, and denote the compleno self-loops. Consider a subset of the edges E ¯ ˆ ment of E in E by E. We will begin by obtaining a Schmidt decomposition ¯ ∪ E. ˆ First of the state |K⟩ with respect to the bipartition of qubits: E = E ¯ we make some definitions. Let G(E) denote the subgraph of G that con¯ as well as all vertices V¯ which have at least one tains only the edges E, ¯ Define Vˆ and G(E) ˆ similarly, and edge incident on them from the set E. ¯ ¯ ˆ let ∂ E ⊆ V := V ∩ V be the set of vertices containing at least one edge ¯ and E. ˆ (Since the definition of ∂ E ¯ incident upon it from both of the sets E ¯ ˆ ˆ is symmetric between E and E, we could just as well have called it ∂ E. In ¯ for simplicity.) We can think of ∂ E ¯ as the the following, we will just use ∂ E ¯ and E. ˆ boundary between the sets E ¯ ¯ and Let E0 (E) denote the set of closed subgraphs on the graph G(E), ˆ analogously. In Section 2.4 we found that the size of the set define E0 (E) E0 for the full graph G was e|E|−|V |+1 , under the assumption that G was ¯ is not necessarily connected: connected. To accommodate the case where E ¯ and let CC j ⊆ let n ¯ be the number of connected components of G(E) th ¯ V , j = 1...¯ n denote the vertex set of the j connected component. Make ˆ Then we have the following Lemma: the analogous definitions for E. ¯ V¯ |+¯ n (and similarly for E) ¯ = 2|E|−| ˆ Lemma 3.2.1 |E0 (E)|  ¯ ¯ Proof E0 (E) {xe } satisfying the |V¯ | binary ∑ is a set of |E| binary variables ¯ equations: e∈δs xe = 0 for all s ∈ V . A connected component CC j of ¯ ¯ e G(E) is a minimal set of vertices in V¯ such that for any edge e ∈ E: is either incident upon two vertices in CC j or upon no vertices in CC j . 20  3.2. Partial Measurement Probabilities ∑ Thus, if we add together the equation e∈δs xe = 0 over all s ∈ CC j for a particular value of j, then each binary variable xe appears either twice or not at all, and we obtain 0 = 0. The equations are otherwise bitwise linearly independent so the total number of independent binary equations is |V¯ | − n ¯. ¯ ∂ E) ¯ denote the set of binary strings over the edges such Now let E0 (E, that the closed subgraph condition holds everywhere except possibly on the ¯ Define E0 (E, ˆ ∂ E) ˆ similarly for E. ˆ vertices in ∂ E. ¯ V¯ |+|∂ E| ¯ ¯ ∂ E)| ¯ = 2|E|−| ˆ Lemma 3.2.2 |E0 (E, (and similarly for E)  ¯ is a set of |E| ¯ binary variables {xe } satisfying the |V¯ | − |∂ E| ¯ Proof E0 (E) ∑ ¯ ¯ binary equations: e∈δs xe = 0 for all s ∈ V /∂ E. The exclusion of the ¯ vertices in ∂ E removes any linearly dependence amoung these equations, ¯ because by the connectedness of G, each connected component of G(E) ¯ ¯ contains at least one vertex in ∂ E: i.e. CC j ∩ ∂ E ̸= ∅ for all j. ¯ ∂ E). ¯ For any x ¯ ∂ E), ¯ We now study the structure of the set E0 (E, ¯ ∈ E0 (E, let ∂ x ¯ be a bitstring encoding x ¯ incident on the ∑ the parity of edges from ¯ ¯ vertices s ∈ ∂ E, i.e. ∂ x ¯s = e∈δs x ¯e for each s ∈ ∂ E. Following Ref [52], ¯ u) ⊂ E0 (E, ¯ ∂ E) ¯ to be the we call ∂ x ¯ the syndrome of x ¯. Then define E0 (E, ¯ ¯ ¯ ¯ set E0 (E, u) := {¯ x ∈ E0 (E, ∂ E) : ∂ x ¯s = us ∀s ∈ ∂ E}, where u = u1 ...u|∂ E| ¯ ˆ ˆ ¯ is a given syndrome. Define E0 (E, u) and ∂ x ˆ for any x ˆ ∈ E0 (E, ∂ E) in the same way. ¯ ∂ E) ¯ = ∪ ¯ u) where S(E) ¯ is the set of Lemma 3.2.3 E0 (E, ¯ E0 (E, u∈S(E) ¯ all bitstrings over the vertices in ∂ E that are even within ∑ each connected ¯ ⊗|∂ E| ¯ ¯ component of G(E), i.e. S(E) := {u ∈ {0, 1} : us = ¯ s∈∂ E∩CC j ˆ 0 for each j = 1...¯ n} (and similarly for E). ¯ ∂ E) ¯ has some parity ∂ x ¯ Proof Since every x ¯ ∈ E0 (E, ¯ on the vertices in ∂ E, ∪ ¯ ¯ ¯ it is immediate that E0 (E, ∂ E) = u∈u E0 (E, u) for some set u of bitstrings ¯ We only need to show that u = S(E). ¯ As a first over the vertices in ∂ E. ¯ u) = ∅ if u ∈ ¯ which establishes that u ⊆ step, we will see that E0 (E, / S(E), ∑ ¯ ¯ ¯ S(E). Indeed, E∑ ¯e = 0 for all 0 (E, u) is defined by the |V | equations: e∈δs x ¯ and ¯ s ∈ V¯ /∂ E, x ¯ = u for all s ∈ ∂ E. If we add together all of these s e∈δs e equations within a single connected component CC j , each binary variable x and we obtain: 0 = ∑e appears either twice or not at all in the summation, ¯ u . Thus the equations defining E ( E, u) are inconsistent if u ∈ / ¯ s 0 s∈∂ E∩CC j ¯ On the other hand, there are no further linear dependencies amoung S(E). 21  3.2. Partial Measurement Probabilities ¯ u) ̸= ∅ if u ∈ S(E). ¯ Since E0 (E, ¯ u) ∩ E0 (E, ¯ u′ ) = ∅ the equations, so E0 (E, ¯ such that u ̸= u′ , it follows that u cannot be a proper for any u, u′ ∈ S(E) ¯ subset of S(E). This completes the proof. ¯ u)| = 2|E|−|V |+¯n for all u ∈ S(E) ¯ (and similarly Corollary 3.2.4 |E0 (E, ˆ for E) ¯  ¯  ¯ u) has the same size for Proof The above considerations imply that E0 (E, ¯ and so |E0 (E, ¯ ∂ E)| ¯ = |S(E)| ¯ ∗ |E0 (E, ¯ u)|. From its definition, each u ∈ S(E) ¯ n| |∂ E|−|¯ ¯ ¯ ¯ |S(E)| = 2 , while |E0 (E, ∂ E)| is given by Lemma 3.2.2. ¯ E0 (E, ¯ u) = z¯(u) ⊕ E0 (E) ¯ where z¯(u) Corollary 3.2.5 For any u ∈ S(E), ¯ u) (and similarly for E). ˆ is any fixed member of the set E0 (E, ¯ and z¯(u) ∈ E0 (E, ¯ u), x ¯ since Proof For any x ¯ ∈ E0 (E) ¯ ⊕ z¯(u) ∈ E0 (E), ¯ ¯ ∂(¯ x ⊕ z¯(u)) = ∂ x ¯ + ∂ z¯(u) = 0 ⊕ u = u. Thus, z¯(u) ⊕ E0 (E) ⊆ E0 (E, u). Fur¯ V¯ |+¯ n , so E (E, ¯ = |E0 (E)| ¯ = |E0 (E, ¯ u)| = 2|E|−| ¯ u) = thermore, |¯ z (u)⊕E0 (E)| 0 ¯ z¯(u) ⊕ E0 (E). We are now in a position to construct a Schmidt decomposition of the ¯ E) ˆ bipartition of qubits. With our state |K(G)⟩ with respect to the (E, definitions, first note that the state |K(G)⟩ can be written |K(G)⟩ := √  1  ∑  |E0 | x∈E  |x⟩ = √  ∑  1  ∑  |E0 (G)| x¯∈E  δ∂ x¯,∂ xˆ |¯ x⟩|ˆ x⟩  ¯ ¯ ˆ∈E (E,∂ ˆ E) ¯ 0 (E,∂ E) x 0  0 (G)  (3.2) Equation 3.2 is simply a restatement of the fact that ¯ ∂ E) ¯ ⊗ E0 (E, ˆ ∂ E) ¯ : ∂x E0 (G) = {x = (¯ x, x ˆ) ∈ E0 (E, ¯ = ∂x ˆ} which follows immediately from the equations defining the sets E0 (G), ¯ ∂ E) ¯ and E0 (E, ˆ ∂ E). ¯ Now, by Lemma 3.2.3: E0 (E, |K(G)⟩ = √  1  ∑  |E0 (G)| u¯∈S(E) ¯  ∑  ∑  ∑  δu¯,ˆu |¯ x⟩|ˆ x⟩  (3.3)  ¯ u) x ˆ x ˆ u) ¯∈E0 (E,¯ u ˆ∈S(E) ˆ∈E0 (E,ˆ  ¯ ∩ S(E), ˆ then we can rewrite Equation 3.3 as a If we define S = S(E) Schmidt decomposition of |K(G)⟩: ∑ 1 |K(G)⟩ = √ ¯ |KE¯ (u)⟩ ⊗ |KEˆ (u)⟩ 2|∂ E|−(¯n+ˆn−1) u∈S  (3.4) 22  3.2. Partial Measurement Probabilities where |KE¯ (u)⟩ := √ |KEˆ (u)⟩ := √  ∑  1  ¯ u)| |E0 (E, x ¯∈E  |¯ x⟩  ¯ 0 (E,u)  ∑  1  ˆ u)| xˆ∈E (E,u) ˆ |E0 (E, 0  |ˆ x⟩  For the normalization of Equation 3.4, we have used that |E0 | = 2|E|−|V |+1 , ¯ u)| and |E0 (E, ˆ u)| from Corollary 3.2.4, the explicit expressions for |E0 (E, ¯ ˆ ¯ and the fact that |V | = |V | + |V | + |∂ E|. To verify that Equation 3.4 yields a Schmidt decomposition of |K(G)⟩, note that the coefficient of each term is real and the |KE¯ (u)⟩ and |KEˆ (u)⟩ are each orthonormal sets of vectors: e.g. ⟨KE¯ (u′ )|KE¯ (u)⟩ = δu,u′ . The reduced density matrix on the subsystem ¯ is, by Equation 3.4: of qubits corresponding to the edges in E ρE¯ = tre∈Eˆ (|K(G)⟩⟨K(G)|) =  1 ¯ n+ˆ n−1) 2|∂ E|−(¯  ∑  |KE¯ (u)⟩⟨KE¯ (u)|  (3.5)  u∈S  From Equation 3.5 it is clear that the Schmidt rank of |K(G)⟩ with ¯ E ˆ is |S|. Since the superposition is uniform we respect to the bipartition E, ¯ can read off |S| from the normalization: |S| = 2|∂ E|−(¯n+ˆn−1) , and the entropy ¯ − (¯ of entanglement of the state ρE¯ is log2 |S| = |∂ E| n+n ˆ − 1). Thus |K(G)⟩ obeys the so-called entanglement area law: the entropy of entanglement of a block of spins grows linearly with the size of its perimeter. We note that this result has also been obtained for surface code states in Ref [27]. In the case ¯ and E ˆ are each connected sets (¯ that E n=n ˆ = 1), we reproduce the result ¯ from Ref [52] that |S| = e|∂ E|−1 . In Appendix C, we show that Equation 3.4 allows us to also consider the bipartite entanglement of arbitrary surface code states, for certain bipartitions. Now consider any product state in the Hilbert space of the qubits asso¯ ciated with the edges in E: ⊗ |ϕe ⟩e |ϕE¯ ⟩ = ¯ e∈E  where |ϕe ⟩e := ae |0⟩e +be |1⟩e . Since the overall phase of |ϕe ⟩e is not physically observable, we take ae to be real positive or zero. Now from Equations 3.1 and 3.5, the probability of obtaining the sequence of measurement out-  23  3.2. Partial Measurement Probabilities comes {|ϕe ⟩}e∈E¯ from the state |K(G)⟩ is ( ) p {|ϕe ⟩}e∈E¯ = ⟨ϕE¯ |ρE¯ |ϕE¯ ⟩ ∑ 1 ⟨ϕE¯ |KE¯ (u)⟩⟨KE¯ (u)|ϕE¯ ⟩ = ¯ |∂ E|−(¯ n +ˆ n −1) 2 u∈S ∑ 1 ∗ = ⟨ϕ | |KE¯ (u) ⊗ KE¯ (u)⟩ (3.6) ¯ ⊗ ϕE ¯ E ¯ 2|∂ E|−(¯n+ˆn−1) u∈S where |KE¯1 (u) ⊗ KE¯2 (u)⟩ is a tensor product of two copies of the state ¯ qubits, and |ϕ ¯ ⊗ ϕ∗¯ ⟩ is a tensor |KE¯ (u)⟩ in the Hilbert space of 2|E| E1 E2 product of the state |ϕE¯ ⟩ and |ϕ∗E¯ ⟩, where |ϕ∗E¯ ⟩ is obtained from |ϕE¯ ⟩ by complex conjugating all of the coefficients in the Z basis. Here, we will follow the approach of Ref [52] and consider the graph ¯ with the vertices ∂ E ¯ in each copy obtained by taking two copies of G(E), identified with each other (see Figure 4.5 for an example). Denote this ¯ ¯ graph ( by G1 (E) ∪ G ) 2 (E), and denote the set of closed subgraphs on it by ¯ ¯ ¯ E) ˆ bipartition of E0 G1 (E) ∪ G2 (E) . Then, in direct analogy to( the (E, ) ¯ ∪ G2 (E) ¯ as edges considered before, we can write the set E0 G1 (E) ( ) ¯ ∪ G2 (E) ¯ = {x = (¯ ¯ ∂ E) ¯ ⊗ E0 (E, ¯ ∂ E) ¯ : ∂x x, y¯) ∈ E0 (E, ¯ = ∂ y¯} E0 G1 (E) (3.7) And therefore ∑ 1 ¯ ∪ G2 (E)⟩ ¯ |x⟩ |K(G1 (E) := √ ¯ ∪ G2 (E))| ¯ |E0 (G1 (E) ¯ ¯ x∈E0 (G1 (E)∪G2 (E)) ∑ ∑ 1 = √ |¯ x⟩|¯ y⟩ ¯ ∪ G2 (E))| ¯ |E0 (G1 (E) ¯ ¯ u∈S(E) x ¯,¯ y ∈E (E,u) 0  =  ∑ ¯ u)| |E0 (E, √ |KE¯1 (u) ⊗ KE¯2 (u)⟩ ¯ ∪ G2 (E))| ¯ |E0 (G1 (E) ¯ u∈S(E)  The coefficient works out to: ¯ ¯ ¯ u)| |E0 (E, 2|E|−|V |+¯n 1 √ =√ ¯ =√ ¯ ¯ ¯ ¯ ∪ G2 (E))| ¯ |E0 (G1 (E) 22|E|−(2V −|∂ E|)+¯n 2|∂ E|−¯n ( ) ¯ then p {|ϕe ⟩} ¯ Comparing with Equation 3.6, we see that if S = S(E), e∈E can be written as a product state overlap with the |K⟩ state for the surface ¯ ∪ G2 (E)): ¯ code of the graph G1 (E)  ( ) p {|ϕe ⟩}e∈E¯ =  1 ¯ ∪ G2 (E)⟩ ¯ √ ¯ ⟨ϕE¯ ⊗ ϕ∗E¯ |K(G1 (E) |∂ E|−¯ n n ˆ −1 2 2 24  3.2. Partial Measurement Probabilities ¯ is such that S = S(E), ¯ partial Thus, if the set of measured edges E measurement probabilities take the form of the inner product between a surface code state (on a different graph) and a product state. In Section 2.5, we showed that such a quantity is proportional to the partition function of an Ising model. In this case, the Ising model will be defined on the graph ¯ ∪ G2 (E)), ¯ with Ising couplings Je = T anh−1 ( be ) for the edges of the G1 (E) ae ¯ and couplings Je∗ = T anh−1 ( b∗e∗ ) for the second copy of first copy of G(E), ae ¯ Note that our assumption that ae is real positive or zero is still valid G(E). for the product state |ϕE¯ ⊗ ϕ∗E¯ ⟩. ¯ Under what conditions is it the case that S = S(E)? Recall that the ¯ and S(E). ˆ Thus, set S is defined as the intersection of the two sets S(E) ¯ ¯ ˆ ¯ ˆ are S = S(E) iff S(E) ⊆ S(E). This is the case if both G(E) and G(E) connected graphs, which is the restriction that is assumed in [52] regarding MBQC on the planar code state. In this case we also have that n ¯=n ˆ = 1. ¯ that satisfying S(E) ¯ ⊆ S(E) ˆ But there are other interesting choices of E but not n ¯ = n ˆ = 1. For instance, under the weaker condition that if for ¯ (CCj ∩ ∂ E) ¯ ⊆ (CC ˆ k ∩ ∂ E) ¯ for each connected component CCj of G(E), ˆ ˆ ¯ ˆ some connected component CCk of G(E), then S(E) ⊆ S(E). Then, any ¯ that has even parity within each connected bitstring u over the vertices in ∂ E ¯ ¯ in Lemma 3.2.3) also has even component of G(E) (see the definition of S(E) ˆ but not vice-versa. An parity within each connected component of G(E), ¯ example of a new E satisfying this weaker condition, but not the one from [52], is depicted in Figure 3.1.  ¯ ⊂ E such that S = S(E), ¯ Figure 3.1: A square lattice graph G and a choice of E ¯ ¯ but for which G(E) is not connected. The edges in the set E are indicated by ¯ are indicated by circles (green). G(E) ¯ has two thick lines, and the vertices in ∂ E ˆ connected components, while G(E) has only one.  Now we consider ∑ partial probabilities for an arbitrary resource state |ψ⟩ ∈ CS where |ψ⟩ = γ,ρ cγ,ρ |Xγ,ρ ⟩, where the summations over γ and ρ are over all bitstrings with g components. Using the Schmidt decomposition in  25  3.2. Partial Measurement Probabilities Equation 3.4, we have ( ) ρE¯ (|ψ⟩) := tre∈Eˆ |ψ⟩⟨ψ| = ∗  ∑ ∑  1 ¯ n+ˆ n−1) 2|∂ E|−(¯  cγ,ρ c∗δ,ϵ  u,u′ ∈S γ,ρ δ,ϵ  ( ) γ,ρ δ,ϵ δ,ϵ ′ ′ ZEγ,ρ ¯ (u)⟩⟨KE ¯ (u )|ZE ˆ Z ˆ |KE ˆ (u)⟩⟨KE ˆ (u )|Z ˆ ¯ |KE ¯ ∗ tre∈E  where ZEγ,ρ ¯ :=  E  ∏g  ∏  j=1  ′ ¯ ∩E e∈C2j−1 ′ ′ ¯ e ∈C2j ∩E  E  γ ρ ˆ Recall that Ze j Ze′j and similarly for E.  the cocycles Ck′ need only be defined up to a homology class. Nevertheless, we take a given choice as fixed in order to take the above partial trace. Using the cyclic property of the trace: ( ) δ,ϵ ′ tre∈Eˆ Z γ,ρ |K (u)⟩⟨K (u )| Z = ⟨KEˆ (u′ )|Z γ⊕δ,ρ⊕ϵ |KEˆ (u)⟩ ˆ ˆ ˆ ˆ ˆ E E E E E ∑ 1 |ˆ x⟩ = ⟨ˆ y |Z γ⊕δ,ρ⊕ϵ ˆ E ˆ |E0 (E, u)| ˆ x ˆ∈E0 (E,u) ˆ ′) yˆ∈E0 (E,u  = ∗ where x) := fEkˆ (ˆ  ∑  δu,u′ ˆ u)| |E0 (E, (  g ( ∏  (ˆ x) f 2j−1 ˆ  (−1)  E  j=1 ˆ x ˆ∈E0 (E,u)  x) f 2j ˆ (ˆ  (−1)  x ˆe  ∑ )(ρ⊕ϵ)j  E  (mod 2)  ˆ e∈Ck′ ∩E  The function fEkˆ (ˆ x) counts the parity of the overlap between the edge ˆ If C ′ can be chosen to consist of set x ˆ and the cocycle Ck′ , restricted to E. k ¯ then f k (ˆ edges only in E, x ) = 0 for any x ˆ . In the above, we have used that ˆ E |ˆ x⟩ = δxˆ,ˆy ∗ (−1)f (ˆx) , and the fact that x ˆ = yˆ implies that u = u′ . ⟨ˆ y |Z γ⊕δ,ρ⊕ϵ ˆ E Consider any subset n ⊆ {1...2k}. Then we have the following Lemma:  ∑ ∏ f kˆ (ˆ x) E Lemma 3.2.6 The quantity xˆ∈E0 (E,u) = 0 iff there exists ˆ k∈n (−1) k (ˆ ′) ∏ f x ˆ such that ˆ E an x ˆ′ ∈ E0 (E) = −1. k∈n (−1) ˆ and x ˆ u) iff x ˆ′ ∈ E0 (E) ˆ: x ˆ ∈ E0 (E, ˆ⊕x ˆ′ ∈ Proof Sufficiency: For any x ˆ u). Bitwise addition is invertible, so with respect to any x ˆ E0 (E, ˆ′ ∈ E0 (E) ′ ˆ u) into pairs: (ˆ we can partition the set E0 (E, x, x ˆ⊕x ˆ ). Then, for each x ˆ 26  )(γ⊕δ)j  3.2. Partial Measurement Probabilities the corresponding term in the summation is canceled by the term of its pair, because ∏ ∏ ∏ f k (ˆ x⊕ˆ x′ ) f k (ˆ x′ ) f k (ˆ x) f k (ˆ x) (−1) Eˆ = (−1) Eˆ (−1) Eˆ = − (−1) Eˆ k∈n  k∈n  k∈n  Necessity: By Corollary 3.2.5 we may replace the summation over the ˆ u) by one over E0 (E): ˆ set E0 (E, ∑  ∏  (−1)  f kˆ (ˆ x) E  =  ∑  ∏  f kˆ (ˆ x+ˆ z (u))  (−1)  E  ˆ k∈n x ˆ∈E0 (E)  ˆ k∈n x ˆ∈E0 (E,u)  ˆ u). If there exists no where zˆ(u) is any fixed element in the set E0 (E, k (ˆ ′) ∏ f x ˆ such that ˆ E x ˆ′ ∈ E0 (E) = −1, then k∈n (−1) ∏  k  (−1)fEˆ (ˆx+ˆz (u)) =  k∈n  ∏  k  k  (−1)fEˆ (ˆx) (−1)fEˆ (ˆz (u)) =  k∈n  ∏  k  (−1)fEˆ (ˆz (u))  k∈n  ˆ so all terms in the sum have the same sign. for each x ˆ ∈ E0 (E), ′ k ˆ such that ∏ (−1)fEˆ (ˆx ) = Corollary 3.2.7 If there exists no x ˆ′ ∈ E0 (E) k∈n −1, then ∏ ∑ ∏ f k (ˆ z (u)) f k (ˆ x) ˆ (−1) Eˆ (−1) Eˆ = |E0 (E)|  k∈n ˆ x ˆ∈E0 (E,u)  k∈n  ˆ u). where zˆ(u) is any fixed element in the set E0 (E, We note that a consequence of Lemma 3.2.6 is that if for any value k ∈ n, there exists a cycle x ˆ′ on G which is: a) homologous to the non-trivial cycle ∏ f kˆ (ˆ x) ˆ then: ∑ E Ck , and b) contained entirely within E, = ˆ k∈n (−1) x ˆ∈E0 (E,u) 0. This condition is equivalent to there being an string operator on the ˆ that acts on the surface code of G as an X operator X ˜k subgraph G(E) k ′ for the logical qubit k. To verify the claim, observe that fEˆ (ˆ x ) = 1 and  f jˆ (ˆ x′ ) = 0 (mod 2), for any j ̸= k. These two statements follow from E |Cj ∩ Ck′ | = δjk (mod 2) and the fact that the boundary of any face has an even number of edges in common with any cocycle Ck′ ). Then use Lemma 3.2.6. Motivated by this result, let Aodd denote the set of values j such that ˆ and let Aeven denote the set of j there exists a logical X2j−1 in the set E, ˆ Let Bodd ⊆ Ac denote such that there exists a logical X2j in the set E. odd 27  3.2. Partial Measurement Probabilities the set of values j which do not belong to Aodd and furthermore for which ˆ is not empty. Define Beven analogously. We may then write the Cj′ ∩ E probability of obtaining a sequence of partial measurement outcomes |ϕE¯ ⟩ as ( ) p|ψ⟩ {|ϕe ⟩}e∈E¯ = ⟨ϕE¯ |ρE¯ (|ψ⟩)|ϕE¯ ⟩ ∑ ∑ ∏ ∏ 1 ∗ = c c δ δρj ,ϵj γ,ρ γ ,δ δ,ϵ j j ¯ 2|∂ E|−(¯n+ˆn−1) u∈S γ,ρ,δ,ϵ j∈Aodd j∈Aeven )(ρ⊕ϵ)j )(γ⊕δ)j ∏ ( ∏ ( f 2j−1 (ˆ z (u)) f 2j (ˆ z (u)) (−1) Eˆ (−1) Eˆ ∗ j∈Beven  j∈Bodd  ∗  δ,ϵ ⟨ϕE¯ |ZEγ,ρ ¯ (u)⟩⟨KE ¯ (u)|ZE ¯⟩ ¯ |KE ¯ |ϕE  (3.8)  ¯ ∪ G2 (E) ¯ obtained by taking two We again consider the graph G1 (E) ¯ ¯ copies of G(E), with the vertices ∂ E in each copy identified with each other. Then we can rewrite Equation 3.8 as a state overlap in the Hilbert-space ¯ qubits: corresponding to 2|E| ) ( p|ψ⟩ {|ϕe ⟩}e∈E¯ = ∗  1  ∑  cγ,ρ c∗δ,ϵ  ¯ n+ˆ n−1) 2|∂ E|−(¯ j∈Aodd γ,ρ,δ,ϵ ) ∑ ∏ ( (γ⊕δ)j f 2j−1 (ˆ z (u)) (−1) Eˆ u∈S j∈Bodd  ∗  ∏  ⟨ϕE¯1 ⊗  δ,ϵ ϕ∗E¯2 |ZEγ,ρ ¯1 (u) ¯1 ZE ¯2 |KE  ∏  δγj ,δj  δρj ,ϵj  j∈Aeven  ∏ (  f 2j z (u)) ˆ (ˆ  (−1)  )(ρ⊕ϵ)j  E  j∈Beven  ⊗ KE¯2 (u)⟩  (3.9)  ¯1 indicates the copy of the edges E ¯ that are in the subgraph Here E ¯ ¯ ¯ in the subgraph G2 (E). ¯ G1 (E), and E2 indicates the copy of the edges E ∗ ¯ The state |ϕE¯1 ⊗ ϕE¯ ⟩ is a tensor product of the state |ϕE¯ ⟩ on the edges E1 , 2 and the same state but with complex conjugated coefficients (in the Z basis) γ,ρ ¯2 . Z γ,ρ on the edges E ¯m for m = 1, 2 is the operator ZE ¯ applied to the edge E ¯ set Em . In the next Chapter we will consider Equation 3.9 for a specific family of lattices. For lattices in this family, we will consider a measurement scheme in which Equation 3.9 is proportional to a summation over Ising ¯ ∪ G2 (E), ¯ at each step of the model partition functions on the graph G1 (E) ( ) computation. In this case, we will see that p|ψ⟩ {|ϕe ⟩}e∈E¯ is proportional to the inner product between |ϕE¯1 ⊗ ϕ∗E¯ ⟩ and a state in the code space of a 2 ¯ ∪ G2 (E). ¯ surface code defined on the graph G1 (E)  28  Chapter 4  MBQC on Punctured Cylinder Code States 4.1  Introduction  In this Chapter we will apply the considerations from Section 3.2 regarding simulating MBQC on surface code states to a specific class of higher genus graphs, which we will call punctured cylinder graphs. To construct a punctured cylinder graph, consider an N × M square lattice with periodic boundary conditions in the vertical direction, embedded on the surface of a solid disk. Then, imagine drilling n thin slots through the disk, each one in between two rows of vertices on the graph. Finally, vertical edges are extended through each slot, as in Figure 4.1 below.  Figure 4.1: A three-slot punctured cylinder graph cellularly embedded on a surface of genus three.  A family of such punctured cylinder graphs is parameterized by the dimensions of the lattice along with the position and width of each slot: {N, M, {x1 , y1 , K1 }, ...{xn , yn , Kn }}. We will take the slots to be ordered from left to right (xj+1 > xj ), and assume that no two slots are above one another (xj+1 ≥ xj + Kj ). Both of these assumptions simplify notation and allow for a clearer discussion, but should not effect the overall conclusions 29  4.1. Introduction of the following analysis. A cellular embedding like that shown in Figure 4.1 of a punctured cylinder graph has N M vertices, 2N M − N edges, and N M − 2n − N + 2 faces, so the Euler characteristic is χ = |F | + |V | − |E| = 2 − 2n. We can see then that the genus of such an embedding of a punctured cylinder graph is equal to the number of slots: g = n. We will interchangeably refer to the “slots” we have been describing as “holes” or “handles”. A flattened representation of a punctured cylinder graph is shown in Figure 4.2 below:  Figure 4.2: A 4 slot punctured torus graph. The n=4 handles have positions (xj , yj ) and widths Kj for j = 1...n. Pairs of points marked by diamonds are identified within each column.  We will assume a particular order in which to make the single qubit measurements on the punctured cylinder lattice. Since the punctured cylinder graph has a left and right boundary, we may unambiguously start at the leftmost column, and measure the qubits column by column proceeding to the right. That is: first we measure all qubits on the vertical edges in column 1, then all of the qubits on horizontal edges between columns 1 and 2, then the vertical edge qubits in column 2, and so on. The order in which qubits are measured within a column is not important, but for simplicity we will show them as occurring row by row as one moves down a column of horizontal or vertical edges. Note that this measurement scheme implies that ¯ and E ˆ are connected at all steps, so n ¯ =S the sets E ¯ =n ˆ = 1 and S(E) in the notation of Section 3.2. Since the holes in the punctured cylinder 30  4.1. Introduction  Figure 4.3: A set of 8 non-trivial cocycles Ck′ on a 4 slot punctured cylinder graph.  are arranged from left to right, having MBQC proceed from left to right ˆ means that the boundary between the measured and unmeasured qubits ∂ E touches at most one hole at a time. In Section 3.2 we derived Equation 3.9 for the probability of obtaining ¯ measurement outcomes corresponding to the state |ϕE¯ ⟩ on the edge set E, ∑ with the surface code state |ψ⟩ = γ,ρ cγ,ρ |Xγ,ρ ⟩ as the initial resource state. We copy Equation 3.9 here for convenience: ) ( p|ψ⟩ {|ϕe ⟩}e∈E¯ = ∗  ∑  1 ¯ n+ˆ n−1) 2|∂ E|−(¯  ∑ ∏ (  γ,ρ,δ,ϵ  (−1)  ∏  cγ,ρ c∗δ,ϵ  (ˆ z (u)) f 2j−1 ˆ  j∈Aodd  )(γ⊕δ)j  where ZEγ,ρ := ¯ m  ⟨ϕE¯1 ⊗  ∏g j=1  ∏  ∏ (  z (u)) f 2j ˆ (ˆ  (−1)  )(ρ⊕ϵ)j  E  j∈Beven  δ,ϵ ϕ∗E¯2 |ZEγ,ρ ¯1 (u) ¯1 ZE ¯2 |KE  ′ ¯ e∈C2j−1 ∩E ′ ∩E ¯ e′ ∈C2j  δρj ,ϵj  j∈Aeven  E  u∈S j∈Bodd  ∗  ∏  δγj ,δj  ⊗ KE¯2 (u)⟩  (4.1)  γ ρ ¯ subZe j Ze′j operating on the Gm (E)  ¯ ∪ G2 (E), ¯ for m = 1, 2. Aodd denotes the set of values j such graph of G1 (E) ˆ and Aeven the set of that there exists logical X2j−1 exclusively on edges E, ˆ Bodd ⊆ Ac denotes j such that there exists a X2j exclusively on edges E. odd the set of values j which do not belong to Aodd and furthermore for which ˆ is not empty, and Beven is the analogous subset of Ac . We will Cj′ ∩ E even take some choice of the non-trivial cocycles Ck′ defining the logical Pauli-Z operators as fixed, and as given in Figure 4.3. In this Chapter, we will show that in every step( of MBQC)on a punctured cylinder surface code state G, the probability p|ψ⟩ {|ϕe ⟩}e∈E¯ can be written 31  4.2. Measurements Between Holes as an inner product between a product state and a surface code state for a larger punctured cylinder graph G′ . As we saw in Section 2.6,such inner products are proportional to an Ising model partition function. Along with the conclusions of Parts II and III, this will allow us to conclude that MBQC on states in the code space of the punctured cylinder code can be simulated in a number of steps that is polynomial in all parameters except for the genus g of G. For certain states in the code space, the simulation cost will furthermore be independent of g.  4.2  Measurements Between Holes  The simplest situation for Equation 4.1 is when all of the edges in column ¯ while all edges in column xk+1 are still in the set E. ˆ xk +Kk are in the set E, We say that during such a step, computation is “between” two holes. When computation is between holes, the subgraph of G in which the boundary ¯ resides is a simple rectangular lattice, as shown in Figure 4.4. In Figure ∂E 4.4, the two points marked by diamonds in each column are identified with ¯ while the thin lines each other. The thick lines indicate qubits in the set E, ˆ Here we show only the subgraph around the indicate qubits in the set E. ¯ ˆ boundary of E and E, while GL and GR indicate the rest of the lattice. All of the slots of the punctured cylinder graph are contained in the subgraphs ¯ and all of edges in GR GL and GR . All of edges in GL belong to the set E, ˆ ¯ belong to the set E. The vertices in ∂ E are marked with circles (green). ¯ contains either all or none of the holes. This situation also arises if E  Figure 4.4: The lattice G at a typical stage when MBQC is “between holes”.  In this case, the sets Bodd and Beven are empty. This is because for any ˆ ̸= ∅, then C ′ is entirely in GR and so is a possible encoded Ck′ , if Ck′ ∩ E k Xk . The sets Aodd and Aeven both contain all of the integers from k + 1 to g. 32  4.2. Measurements Between Holes ′ ...Cg′ are entirely in GR : Since the cycles C1′ ...Ck′ are entirely in GL and Ck+1 ∏ ¯m γ ¯m ρ ¯m k E E E j j ZEγ,ρ is the logical Pauli Z operator ¯m = j=1 (Z2j−1 ) (Z2j ) , where Zl on encoded qubit l in the surface code on graph G, applied to the subgraph ¯ of G1 (E) ¯ ∪ G2 (E). ¯ So, we may write Equation 4.1 as Gm (E)  ( ) p|ψ⟩ {|ϕe ⟩}e∈E¯ = ∗ ∗  ∑  1 ¯ 2|∂ E|−1  γ,ρ,δ,ϵ  ⟨ϕE¯1 ⊗ ϕ∗E¯2 | ∑  cγ,ρ c∗δ,ϵ k ∏  g ∏  δγj ,δj δρj ,ϵj  j=k+1 ¯  ¯  ¯  ¯  E1 E1 ρj E2 E2 ϵj (Z2j−1 )γj (Z2j ) (Z2j−1 )δj (Z2j )  j=1  |KE¯1 (u) ⊗ KE¯2 (u)⟩  (4.2)  u∈S  ¯ the state ∑ Since S = S(E), ¯1 (u) ⊗ KE ¯2 (u)⟩ is proportional to u∈S |KE ¯ ¯ the |K⟩ state corresponding to the ∑ graph G1 (E) ∪ G2 (E), by Equation 3.8: 1 ¯ ¯ |K(G1 (E) ∪ G2 (E))⟩ = √ ¯ ¯1 (u) ⊗ KE ¯2 (u)⟩ u∈S |KE 2|∂ E|−1  Furthermore, if we rewrite the summation operator in Equation 4.2 as ∑  cγ1 ...γg ,ρ1 ...ρg c∗δ1 ...δg ,ϵ1 ...ϵg  γ1 ...γg ρ1 ...γg δ1 ...δg ϵ1 ...ϵg  g ∏  δγj ,δj δρj ,ϵj =  j=k+1  ∑  c¯γ1 ...γk δ1 ...δk ,ρ1 ...ρk ϵ1 ...ϵk  γ1 ...γk ρ1 ...γk δ1 ...δk ϵ1 ...ϵk  where c¯γ1 ...γk δ1 ...δk ,ρ1 ...ρk ϵ1 ...ϵk :=  ∑ γk+1 ...γg ρk+1 ...γg δk+1 ...δg ϵk+1 ...ϵg  cγ1 ...γg ,ρ1 ...ρg c∗δ1 ...δg ,ϵ1 ...ϵg  g ∏  δγj ,δj δρj ,ϵj  j=k+1  (4.3) then we can see that Equation 4.2 is a summation over the codespace of ¯ ∪ G2 (E). ¯ This graph has genus a surface code defined on the graph G1 (E) 2k, and encodes 4k qubits. We associate the first 2k encoded qubits with ¯ and associate encoded qubits 2k + 1...4k the k holes in the subgraph G1 (E), ¯ with the k holes in G2 (E). Then, in Equation 4.2 the binary variables γ1 ...γk δ1 ...δk determine whether encoded Z operators act on the odd encoded qubits numbered 1, 3, 5...4k − 1, while the binary variables ρ1 ...ρk ϵ1 ...ϵk determine the application of encoded Z operators on even numbered encoded  33  4.2. Measurements Between Holes qubits 2, 4, 6...4k. We can now rename variables to write: ( ) p|ψ⟩ {|ϕe ⟩}e∈E¯ =  ∑ 1 ¯ ∪ G2 (E)) ¯ γ,ρ ⟩ √ ¯ c¯γ,ρ ⟨ϕE¯1 ⊗ ϕ∗E¯2 |K(G1 (E) 2|∂ E|−1 γ1 ...γ2k ρ1 ...γ2k  (4.4) ¯ ∪ G2 (E)) ¯ γ,ρ ⟩ is an encoded X eigenstate in the code space where |K(G1 (E) ¯ ∪ G2 (E), ¯ in the notation of Section 2.4. of a surface code defined on G1 (E) Equation 4.4 has the same form as Equation 2.12 from Section 2.6, where we considered overlaps of arbitrary surface code states with a product state on the physical qubits. We found in Section 2.6 that such an overlap is proportional to the partition function of an Ising model defined on the same graph as the surface code state. In Chapter 5, we will develop a means of evaluating the Ising model partition function on punctured cylinder graphs. Unfortunately, the graph ¯ ∪ G2 (E) ¯ is not punctured cylinder graph. However, we can transform G1 (E) ¯ ¯ into an effective punctured cylinder graph by employing two G1 (E) ∪ G2 (E) manipulations that only affect the overlap between |K(G)⟩ and a product state up to a constant of proportionality. For a connected graph G cellularly embedded on a surface S with associated code states |K(G)γ,ρ ⟩, we may perform the following operations: ˆ Edge addition: We may add an edge to G, then measure the qubit associated with the added edge to be in the |0⟩ state. The edge can be added between existing vertices on G, or by adding a new vertex and connecting it to G with the new edge. Call the new graph obtained after edge addition G′ . Then for any product state |ϕ⟩: ⟨ϕ| ⊗ ⟨0|K(G′ )γ,ρ ⟩ = √12 ⟨ϕ|K(G)γ,ρ ⟩. ˆ Vertex splitting: We can split any vertex into two, and add an edge in between the two resultant vertices. The edges incident on the vertex that is split can be divided arbitrarily between the two resultant vertices. Then measure the new qubit to be in the |+⟩ state. Call the new graph obtained by vertex splitting G′ . Then for any product state |ϕ⟩: ⟨ϕ| ⊗ ⟨+|K(G′ )γ,ρ ⟩ = √12 ⟨ϕ|K(G)γ,ρ ⟩  We note that the edge addition and vertex splitting are exactly the opposite of the graph minor operations: edge contraction, edge deletion, and deletion of isolated vertices. This implies that the class of graphs under consideration is minor closed. If we can simulate MBQC on a surface code defined on some graph G, then we can also simulate it on any surface code 34  4.2. Measurements Between Holes defined on any graph minor of G, by taking inner products with |0⟩ or |+⟩ for certain qubits. This is precisely what we will do: express Equation 4.4 as a special case of the overlap between a surface code state on some ¯ and a product state, where G1 (E) ¯ ∪ G2 (E) ¯ is a graph minor graph G′ (E) ′ ¯ of G (E). For measurements between holes, this is particularly simple to ¯ ∪ G2 (E) ¯ in the do. The bottom half of Figure 4.5 shows the graph G1 (E) ′ ¯ obtainable case depicted in Figure 4.4, and one such effective graph G (E) by the edge addition and vertex splitting operations. The effective graph ¯ is a punctured cylinder graph of genus 2k. See the footnote2 for an G′ (E) interpretation of vertex-splitting and edge addition in terms of couplings of an Ising partition function. 2  Equivalently, we can think about the effect of edge addition and vertex splitting in terms of the Ising Model partition function. See Section 5.2 for the relevant definitions. Projecting a qubit onto the state |0⟩ corresponds to taking the associated edge to have a coupling of zero in the Ising Model. Adding an edge with zero coupling leaves the Ising partition function unchanged. If a vertex was added in this operation, the partition function gets multiplied by a factor of 2, but this factor of two is then canceled in Equation 2.12 when mapping to a surface code state overlap, because √ of the addition of a vertex  E0 (G) , assuming that G is and the factor of 2−|V | . Then a factor of √12 comes from E0 (G′ ) connected. In the case of vertex splitting, measuring a qubit in the state |+⟩ corresponds to an infinite coupling along an edge in the Ising Model. This forces the two vertices that share this edge to have the same Ising spin state, thus effectively “gluing the √ two vertices back together. In the mapping to the surface code overlaps, the factor of a2e − b2e → 0 keeps all of the quantities finite. To see this, we may express the coefficients ae and be as a limit, and use the continuity of the inner product as a function of them. In particular, we may let ae (x) = √12 (1 + x) and be (x) = √12 (1 + x), and take a limit as x → 0 from √ √ the right. Then, ae (x)2 − be (x)2 = 2x and Je = ln( √1x ), using that for real y < 1, 1+y ). Now let σ1 and σ2 be the Ising spins of the vertices connected by tanh−1 (y) = 21 ln( 1−y the infinite coupling. Then the equivalent transformation on the Ising partition function due to the vertex splitting operation is:  lim  x→0  √  1 2x ∗ Z(G′ , J& ln( √ )) x  =  ∑ σ1 ,σ2  =  lim  x→∞  ∑ √  σ1 ,σ2  √  ln( √1 ))∗σ1 σ2  2x ∗ e  2 ∗ δσ1 σ2  ∑  x  ... =  √  ∑  ...  σ3 ,...  2 ∗ Z(G, J)  σ3 ,...  Since a vertex was added in this operation, the overall effect in Equation 2.12 is a factor √ of 22 = √12 .  35  4.3. Crossing Holes  ¯ Figure 4.5: The top half of the figure shows the two copies of the edge set E ¯ (green circles) to form (bold), which are glued together along the vertices in ∂ E ¯ ¯ the graph G1 (E)∪G 2 (E) shown in bottom left. The bottom half of the figure shows ˆ corresponding to G1 (E) ¯ ∪ G2 (E). ¯ The an effective punctured cylinder graph G′ (E) red horizontal edges are measured into the |+⟩ state, while the vertical yellow edges are measured into the |0⟩ state  .  4.3  Crossing Holes  ¯ contains vertices in a column between xk and xk + Kk If the boundary ∂ E for any k, then Equation 4.1 can take on a more complicated form than it does in the special case of Equation 4.2. This occurs as the computation “crosses holes” from left to right on the lattice G. For simplicity, we assume that the measurements occur from top to bottom within a given column of horizontal or vertical edges, though this assumption is not essential to the results. Figure 4.6 below shows the part of a punctured cylinder graph G around the k th hole. In this Section, we will focus on the measurement steps after edge a in Figure 4.6 has been measured, but before edge b is measured. Before edge a is measured, or after edge b is measured, the situation is the same as when computation is “between holes”. In Figures 4.7 and 4.8 we show these two ¯ who’s partition function is situations, as well as an effective graph G′ (E) 36  4.3. Crossing Holes  Figure 4.6: An example of the part of a punctured cylinder graph around the k th ′ ′ hole. The non-trivial cocycles C2k−1 and C2k are shown as dotted lines (orange and purple, respectively). These reflect the choice of cocycles depicted in Figure 4.3.  ¯ ∪ G2 (E) ¯ (which is not shown). As in Figure proportional to that of G1 (E) 4.5, the added horizontal red edges are measured to be in the |+⟩ state and the added vertical yellow lines are measured to be in the |0⟩ state. ¯ contains the edge a of Figure 4.6, there are no longer Once the edge set E ˆ This is because no cycles on G(E) ˆ can make any logical Xk in the set E. th it all the way “around” the k hole. Now the set Bodd is no longer empty, ′ ˆ ̸= ∅. So, Bodd = {k} while Aodd = {k + 1...g}. Since the because C2k−1 ∩E edge b is still unmeasured, and we are measuring from top to bottom within a given column, there exists a cycle of vertical edges in column xk + Kk − 1 ′ . So, B that has one edge (the edge b) in common with C2k even = ∅ and Aeven = {k...g}. We may then write Equation 4.1 as (  p|ψ⟩ {|ϕe ⟩}e∈E¯  )  = ∗  1 ¯ 2|∂ E|−1  ∑  cγ,ρ c∗δ,ϵ  g ∏  δγj ,δj  j=k+1  γ,ρ,δ,ϵ  δ,ϵ ⟨ϕE¯1 ⊗ ϕ∗E¯2 |ZEγ,ρ ¯ ZE ¯ 1  ∑(  2  g ∏  δρj ,ϵj  j=k f 2k−1 (ˆ z (u)) ˆ  (−1)  )(γ⊕δ)k  E  u∈S  ∗  |KE¯1 (u) ⊗ KE¯2 (u)⟩  (4.5)  37  4.3. Crossing Holes  Figure 4.7: A stage of computation just before edge a of Figure 4.6 is measured. Two copies of the relevant part of G are shown on the left, and the relevant part ¯ is shown on the right. of G(E)  Figure 4.8: A stage of computation just after edge b of Figure 4.6 has been measured. Two copies of the relevant part of G are shown on the left, and the ¯ is shown on the right. relevant part of G(E)  where in this case   k−1 ∏ ¯1 ¯ 1 ρj ¯2 ¯ 2 ϵj δ,ϵ E E E E ZEγ,ρ =  (Z2j−1 )γj (Z2j ) (Z2j−1 )δj (Z2j )  ¯ ZE ¯ 1  2   ∗     j=1  ∏    ¯ ¯  (ZeE1 )γk (ZeE2 )δk  ∗   ′ ¯ e∈C2k−1 ∩E  ∏  ρk ¯ ¯ ZeE1 ZeE2   ′ ∩E ¯ e∈C2k  ¯  and ZeEm is the Pauli Z operator acting on the edge e in the subgraph ¯ of G1 (E) ¯ ∪ G2 (E). ¯ In Figure 4.9, we show the lattice G soon after Gm (E) ′ has been measured. The edges measurement of edge a, when part of C2k−1 ′ in C2k−1 are labeled e1 to eyk (in the case depicted yk = 3), and the vertex immediately to the left of each ej is labeled vj . ˆ contains at most one edge (namely the edge ej ) incident Since the set E ˆ So, we on any of the vertices vj , it follows that: uj = zˆ(u)ej if ej ∈ E. 2k−1 ∏ (ˆ z (u)) f = ej ∈E∩C may rewrite: (−1) Eˆ (−1)uj . Furthermore, for any ′ ˆ 2k−1  38  4.3. Crossing Holes  Figure 4.9: The relevant part of G part-way through measurement of the edges ′ ¯ is shown in bold, and the set E ˆ is non-bold. along C2k−1 . Again, the set E  ∏ ¯ u): (−1)uj |¯ x ¯ ∈ E0 (E, x⟩ = e∈E∩δ x⟩, where δvj is the set of edges ¯ v Ze |¯ j  incident to the vertex vj in the graph G. Letting x ⊆ {1...yk } denote the ˆ ∩ C′ set of indices j such that ej ∈ E 2k−1 , we may then write (  ∑  f 2k−1 (ˆ z (u)) ˆ  (−1)  u∈S  E  ∑ 1 ¯ u)| |E0 (E,  =  )(γ⊕δ)k  |KE¯1 (u) ⊗ KE¯2 (u)⟩ ∑ ∏ ((−1)uj )(γ⊕δ)k |¯ x ⊗ y¯⟩  ¯ j∈x u∈S x ¯,¯ y ∈E0 (E,u)  ∑ 1 ¯ u)| |E0 (E,  =  ∑  ∏   ¯ j∈x u∈S x ¯,¯ y ∈E0 (E,u)    =  2|∂ E|−1 ¯  ∏  j∈x  ∏   ∏  (γ⊕δ)k  Ze   |¯ x ⊗ y¯⟩  ¯1 ∩δv e∈E j  (γ⊕δ)k   Ze   ¯ ∪ G2 (E))⟩ ¯ |K(G1 (E)  (4.6)  ¯1 ∩δv e∈E j  where we have used Equation 3.8 in the final step, and in the last two lines ¯ ∪ G2 (E) ¯ 3. δvj denotes the set of edges incident on vj in the graph G1 (E) ¯ ¯ As before, we can add edges to the graph G1 (E) ∪ G2 (E) to create an ¯ that is a punctured cylinder graph. This is depicted effective graph G′ (E) in Figure 4.10 below in the step just after edge a has been measured. The 3 Note that the choice of E1 instead of E2 in the third line of Equation 4.6 is arbitrary, so we make it out of convenience. We might have also chosen (∏ )γk (∏ )δk ∏ ∏ |¯ x ⊗ y¯⟩. ¯1 ∩δv Ze ¯2 ∩δv Ze j∈x e∈E j∈x e∈E j  j  39  4.3. Crossing Holes procedure for adding edges in subsequent steps is basically the same, done in such a way that the subgraph of G around the k th hole is reconstructed. In ¯ during a few other important Appendix G we show the construction of G′ (E) steps while crossing a hole.  Figure 4.10: A stage of computation just after edge a of Figure 4.6 has been measured. Two copies of the relevant part of G are shown on the left, and the ¯ are shown on the right. relevant part of the effective graph G′ (E)  ¯ will have 2k − 1 holes: two copies each of the holes The graph G′ (E) ¯ and one more hole (the one 1...k−1 from G which are contained entirely in E, depicted in Figure 4.10) corresponding to the k th hole of the original graph ¯ will hence encode 4k−2 logical qubits. G. A surface code on the graph G′ (E) We associate the first 2k − 2 of these qubits with the k − 1 holes coming from the subgraph GE¯1 , and associate encoded qubits 2k + 1...4k − 2 with the k − 1 holes from GE¯2 . Encoded qubits 2k − 1 and 2k are associated with the hole which is completed by adding edges via vertex splitting and edge addition. The non-trivial cocycles corresponding to encoded qubits 2k − 1 and 2k are shown in Figure 4.10 as dotted lines. To avoid confusion, we will ¯ as C ′ (to distinguish denote the associated non-trivial cocycles on G′ (E) k ′ them from the Ck , which were defined on G). ¯ ∪ G2 (E) ¯ to construct Let E denote the edges which are added to G1 (E) ′ ¯ G (E), and let |ϕE ⟩ denote a tensor product of the |+⟩ state for each of the horizontal edges (added by vertex splitting), and |0⟩ for each of the vertical edges (added by edge addition). From the properties of vertex splitting and  40  4.3. Crossing Holes edge addition, we know that  ∏  j∈x  ∏  (γ⊕δ)k  Ze   ∗  ¯ ∪ G2 (E))⟩ ¯ |K(G1 (E)  ¯1 ∩δv e∈E j  √ =   ∏ 2|E| ⟨ϕE |  j∈x  ∏  (γ⊕δ)k  Ze   ¯ |K(G′ (E))⟩  ¯1 ∩δv e∈E j  (4.7) We are able to commute the Pauli Z operators to the right of the bra ¯1 and E are disjoint sets of qubits. ⟨ϕE | because E ′ Now observe from Figure 4.10 that for each of the edges in C2k−1 that is ′ ˆ ¯ in the set E, there is a new edge in the effective lattice G (E) that belongs to ′ the dual graph cycle C2k−1 . As a result, there is a one-to-one correspondence ′ between the indices in the set x and edges in C2k−1 ∩ E. Further, for each ′ ej ∈ C2k−1 ∩ E:   ¯ Zej |K(G′ (E))⟩ =   ∏   ¯ Ze  |K(G′ (E))⟩  ¯1 ∩δv e∈E j  because Zej and the operator  (∏  )  differ by the stabilizer ¯ we associate In the graph G′ (E), operator Avj of the surface code for the vj with the left hand side of hole k, as shown by the squares (light blue) in Figures 4.10 and 4.11. Using the above fact along with Equations 4.6 and ¯1 ∩δv e∈E j  Ze  ¯ G′ (E).  41  4.3. Crossing Holes 4.7, we may rewrite Equation 4.5 (  p|ψ⟩ {|ϕe ⟩}e∈E¯  )  =  ∗  √  ∑  1 ¯ E| 2|∂ E|−1−|      ∗  γ,ρ,δ,ϵ  ⟨ϕE¯1 ⊗ ϕE ⊗ ϕ∗E¯2 |     g ∏  δγj ,δj  j=k+1  k−1 ∏  g ∏  δρj ,ϵj  j=k  ¯  ¯  ¯  ¯  E1 E1 ρj E2 E2 ϵj (Z2j−1 )γj (Z2j ) (Z2j−1 )δj (Z2j )  j=1   ∗  cγ,ρ c∗δ,ϵ    ¯ ¯  (ZeE1 )γk (ZeE2 )δk  ∗   ∏ ′ ¯ e∈C2k−1 ∩E  ∏  ρk ZeE1 ZeE2  ¯  ¯  ′ ∩E ¯ e∈C2k  (γ⊕δ)k  Ze   ∏  ¯ |K(G′ (E))⟩  (4.8)  ′ e∈C2k−1 ∩E  Our final tool will be two more transformations that we can make to Equation 4.8. The first transformation is:    (γ⊕δ)k ¯ ¯ ¯   ∏   ∏ (ZeE1 )γk (ZeE2 )δk  →  ZeE1   ′ ¯ e∈C2k−1 ∩E  ′ ¯ e∈C2k−1 ∩E  To see why this is possible, consider the example depicted in Figure ′ ¯ On the 4.11. At the stage depicted, there is one edge in the set C2k−1 ∩ E. ′ ¯ effective lattice G (E) we have two copies of this edge, marked e1 and e2 in Figure 4.11. By applying the stabilizer operators As and At , we can see that ¯ ¯ ⟨ϕE |Ze1 |K(G′ (E))⟩ = ⟨ϕE |Ze2 |K(G′ (E))⟩. For any of the vertical couplings e in E, the operator Ze has no effect because the associated qubit is in the |0⟩ state in |ϕE ⟩. In a subsequent stage of computation when more than one ′ ¯ the same argument can be made for each. edge is in the set C2k−1 ∩ E, The second( transformation )ρkthat we can make in Equation 4.8 is to insert ∏ the operator Z inside the inner product. We can do this e∈C ′ ∩E e 2k  ′ ∩ E are vertical edges, so the associated qubits because all of the edges C2k are all in the |0⟩ state in |ϕE ⟩. Thus the inserted operator has no effect. Our two transformations are useful, because  (γ⊕δ)k  (γ⊕δ)k  (γ⊕δ)k ¯   ∏  ∏   ∏  (ZeE1 ) Ze  = Ze    ′ ¯ e∈C2k−1 ∩E  ′ e∈C2k−1 ∩E  ′ e∈C2k−1  42  4.3. Crossing Holes  ¯ during the step depicted in Figure 4.9. Figure 4.11: The relevant part of G′ (E)  and     ρk  ¯ ¯  ZeE1 ZeE2    ∏    ′ ∩E ¯ e∈C2k  ρk  ∏   Ze   ρk      ∏ Ze  =  ′ ∩E e∈C2k  ′ e∈C2k  Therefore ) ( p|ψ⟩ {|ϕe ⟩}e∈E¯ = ∗  √  ¯ E| 2|∂ E|−1−|  ρk    ∏  Ze    cγ,ρ c∗δ,ϵ  γ,ρ,δ,ϵ  ⟨ϕE¯1 ⊗ ϕE ⊗ ϕ∗E¯2 |   ∗  ∑  1  δγj ,δj  j=k+1  k−1 ∏  ¯  g ∏  δρj ,ϵj  j=k ¯  ¯  ¯  E1 E1 ρj E2 E2 ϵj (Z2j−1 )γj (Z2j ) (Z2j−1 )δj (Z2j )  j=1  (γ⊕δ)k   ∏   ′ e∈C2k  g ∏   Ze   ¯ |K(G′ (E))⟩  (4.9)  ′ e∈C2k−1  Note that the summand in Equation 4.9 depends on (γ⊕δ)k but not upon γk and δk individually. With this in mind we define, similar to Equation 4.3: ∑ c¯γ1 ...γk δ1 ...δk−1 ,ρ1 ...ρk ϵ1 ...ϵk−1 := cγ1 ...γk−1 (γ⊕δ)k ,γk+1 ...γg ,ρ1 ...ρg c∗δ1 ...δg ,ϵ1 ...ϵg  ∗  γk+1 ...γg ρk+1 ...γg δk ...δg ϵk ...ϵg g ∏  g ∏  j=k+1  j=k  δγj ,δj  δρj ,ϵj  (4.10)  43  4.3. Crossing Holes Relabeling indices, Equation 4.9 now takes the same form as Equation 4.4: ∑ ( ) 1 p|ψ⟩ {|ϕe ⟩}e∈E¯ = √ c¯γ1 ...γ2k−1 ,ρ1 ...ρ2k−1 ¯ E| γ1 ...γ 2|∂ E|−1−| 2k−1 ρ1 ...γ2k−1  ∗  ¯ γ,ρ ⟩ ⟨ϕE¯1 ⊗ ϕE ⊗ ϕ∗E¯2 |K(G′ (E)) (4.11)  ¯ γ,ρ ⟩ is an encoded X eigenstate in the code space of a where |K(G′ (E)) ¯ surface code defined on G′ (E). In this Chapter, we have reduced the problem of simulating MBQC on punctured cylinder code states to the evaluation of surface code product state overlaps ⟨ψ|ϕ⟩, assuming a natural measurement pattern moving across the lattice. In the next chapters, we will use the mapping of this problem onto the evaluation of a 2D classical Ising model partition function to construct schemes of evaluating it on certain graphs, including punctured cylinder graphs.  44  Part II Evaluation of the Ising Model Partition Function  45  Chapter 5  The 2D Classical Ising Model Partition Function 5.1  Introduction and History  The classical Ising model is a widely studied model of spins on a lattice in which each spin σ has one of two discrete states: σ ∈ {+1, −1}. Pairs of neighbouring spins interact with some coupling −J, such that the energy contribution from the interaction is −J times the product of the two spin variables involved. When the couplings J are positive it is energetically preferable for neighbouring spins to align, and the Ising model provides a simple model of ferromagnetism. For a 2D classical Ising model, the spins live on the sites of a two dimensional lattice, typically taken to be a regular quadratic lattice. In this case, each spin has four neighbours that it is coupled to energetically, and the model exhibits a phase transition [42], [49]. The 2D classical Ising model is sometimes considered in the presence of an external magnetic field, but we will focus exclusively on the case of no applied field, and will use the term “Ising model” to refer to the model without such a field unless otherwise stated. As the statistical properties of any system are derived from its partition function, the problem of evaluating the partition function of the 2D classical Ising model has a long and varied history. On first glance, explicit computation of it appears intractable, since the direct evaluation of a partition function involves summing over all of the possible states of the system in question. For a 2D MxN lattice of binary spin variables, this sum contains 2M N terms. Nevertheless, in one of the most well-known accomplishments of statistical physics in the 20th century it was discovered that the partition function of the 2D classical Ising model can in fact be evaluated efficiently, i.e. in a number of steps that grows polynomially with M and N. In fact, there have been several different formulations of this result, which can be roughly categorized into two basic approaches. The first approach historically is entirely algebraic in nature: the partition function is solved by  46  5.1. Introduction and History mapping it to some problem in matrix algebra. The second approach uses a combinatoric method and makes use of graph theory. The first and most celebrated computation of the partition function Z of a 2D classical Ising model was an algebraic one and is that of Onsager in 1944 [49]. The result assumes homogeneous couplings and a large lattice thermodynamic limit, and uses the so-called transfer matrix formalism, which we will study in detail in Section 5.3. The transfer matrix formalism maps the summation over states in the partition function to the multiplication of a set of 2N x2N matrices called transfer matrices. The idea was introduced by Kramers and Wannier in 1941 [42], who in the same paper identify the phase transition, but they were unable to solve the matrix problem to obtain the explicit result of Onsager. The Onsager solution was simplified in 1949 by Kaufman [38], and again in 1964 by Shultz, Mattis and Lieb [55], who showed that the transfer matrices could be identified in a natural way with non-interacting fermion operators. The combinatoric approach to finding the partition function was initiated by Kac and Ward in 1952, by reducing it to the problem of counting closed polygons on the lattice, and then performing the counting by evaluating a certain matrix determinant [34]. Hurst and Green re-derived the result of Kac and Ward algebraically in 1960, while showing that the partition function can also be expressed as the Pfaffian of a matrix [32]. An expression in terms of Pfaffians was subsequently found by Kastleyn in 1961 through a mapping to the so-called dimer problem [36], [37], [35]. The connection between the Ising problem and the dimer problem was anticipated by Fisher in 1961 [16]. In 1985, Barahona derived a similar solution for a quadratic lattice on the plane and the torus that explicitly accommodates inhomogeneous real couplings [1]. An extension of this result to the case of complex couplings for a planar lattice is implicit in Raussendorf and Brayvi’s result on MBQC with planar code states [52]. Since complex couplings are unphysical for the classical Ising model, they have typically not been studied in the statistical mechanics literature. However, they arise in our mapping from surface code state overlaps, so we will need to accommodate them. Our approach will focus on the algebraic transfer matrix formalism, and make a novel application of the method to graphs of higher genus. Combinatoric approaches to the Ising model have recently been applied to graphs of genus g>1 in [57], [50], [20], [9], and [7]. The connection between combinatoric and algebraic approaches to the Ising model has been explored in [30] and [31]. The emergence of topological considerations will compel us to seek parallels with the combinatorial approaches, in particular those of [19] and [37]. However, for us the algebraic approach will be useful because it allows 47  5.2. Formulation of the Problem for an Arbitrary Graph us to make a mapping to fermion operators. The fermions will provide a physical picture of the evaluation of the 2D classical partition function in terms of a 1D quantum system.  5.2  Formulation of the Problem for an Arbitrary Graph  Consider a graph G = (V, E), not necessarily embedded on any surface. Associate with each vertex vj a classical spin σj ∈ {−1, +1}, and with each edge e ∈ E an arbitrary coupling strength Je . We can assume there are no self-loops on G, because they contribute only an overall factor to the partition function, which can be easily evaluated separately. For a given edge e, let e1 & e2 denote the two vertices that it connects, in no particular order. Then we can define an Ising Hamiltonian for this graph: H(σ) = −kB T  ∑  Je σe1 σe2  e∈E  where σ represents a spin configuration for the whole graph. We factor out the temperature in the Hamiltonian above in order to simplify the expression for the classical partition function, which is defined as Z(T ) ≡  ∑ σ  −1  e kB T  H(σ)  =  ∑  ∑  e  e∈E  Je σe1 σe2  (5.1)  σ  where the above summation over σ is a summation over all of the 2|V | distinct spin configurations {σ1 , σ2 ...σ|V | }. We can without loss of generality assume that G is a connected graph. If it is not, then the partition function will factorize into a product of partition functions for each connected component of G, and we can consider each separately. In our analysis we will take the couplings Je to be arbitrary complex numbers. Thus the Hamiltonian H(σ) will be in general complex, which as mentioned before is not a physical regime for the classical Ising Model. But, as complex Ising couplings map to a physical regime of simulating MBQC on surface code states, we allow it. We note then that since the partition function depends only on the exponential of the Hamiltonian H(σ) an addition of any integer times 2π to the imaginary part of any coupling Je leaves the partition function unchanged. The addition of any odd integer times π to the imaginary part of Je for a single edge e introduces an overall minus sign to the partition function, because eiπσe1 σe2 = e±iπ = −1, regardless of whether the plus or the minus sign holds in the exponential. Another 48  5.2. Formulation of the Problem for an Arbitrary Graph interesting property of the Ising Model partition function is that it is invariant under flipping the sign of the coupling strengths along any trivial (i.e. contractible) cocycle on G. This can be verified by noticing that such a transformation is equivalent to flipping the spin state of one or more of the classical spins, which merely permutes the terms in the partition function summation. Several algebraic manipulations allow the partition function to be mapped into a combinatorial problem: ∑∏ ∑∏ eJe σe1 σe2 = Z = (cosh(Je ) + σe1 σe2 sinh(Je )) σ e∈E  ( =  ∏  )  cosh(Je )  σ e∈E  ∑∏  (1 + σe1 σe2 tanh(Je ))  σ e∈E  e∈E  ∏ The term e∈E (1 + σe1 σe2 tanh(Je )) can be expanded into a sum over subsets of the set of all edges e ⊆ E: ( ) ∏ ∑∑∏ Z = cosh(Je ) σe1 σe2 tanh(Je ) (  e∈E  (  e∈E  =  =  ∏ ∏  ) cosh(Je ) ) cosh(Je )  e∈E  σ e⊆E e∈e  ∑∑  (  σ e⊆E  ∑ e⊆E  (  ∏  )( tanh(Je )  e∈e  ∏  )  tanh(Je )  e∈e  ∏  ) σe1 σe2  e∈e  ∑∏  σe1 σe2  σ e∈e  ∏ ∏ Ne The quantity e∈e σe1 σe2 can be rewritten as v∈V σv v , where Nve denotes the ∑ ∑ number of edges in the set e incident on vertex v. Then, since σ = σ1 ...σ|V | ∑ ∏  σ1 ..σ|V | v∈V  e  σvNv =  ∏  ∑  e  αNv  v∈V α∈{−1,+1}  is odd for any v ∈ V , because then ∑ The aboveNquantity ∑ vanishes if e v = α∈{−1,+1} α α∈{−1,+1} α = −1 + 1 = 0. If, on the other hand e is even for all v ∈ V , we get a factor of 2 for each vertex since then N ∑ ∑v Nve = α∈{−1,+1} 1 = 2. So, we may rewrite the partition α∈{−1,+1} α function in the simple form ( ) ( ) ∏ ∏ ∑ |V | tanh(Je ) (5.2) Z=2 cosh(Je ) Nve  e∈E  e∈E0 (G)  e∈e  49  5.3. Square Lattices: Transfer Matrix Formalism where E0 (G) is the set of all subsets of the edges such that for any e ∈ E0 (G), Nve is even for all v. E0 (G) is precisely the set of all closed subgraphs (also known as Eulerian subgraphs, as in [20]) of G that we met in Section 2.4 regarding surface code states.  5.3  Square Lattices: Transfer Matrix Formalism  Consider a N*M plane square 2D lattice with sites labeled by (j, k), where j is a row number between 1 and N, and k is a column number between 1 and M. On each site we consider a classical spin σi,j ∈ {−1, +1}. As above, denote an entire configuration of the N × M spin variables by σ. On this graph, the Ising Hamiltonian can be written H(σ) = −kB T  N ∑ M ∑  Kjk σj,k σj,k+1 + Jjk σj,k σj+1,k  j=1 k=1  Here the coefficients Kjk and Jjk are dimensionless and are the horizontal and vertical coupling strengths (as a ratio of temperature), respectively. The above expression is written in such a way as to allow periodic boundary conditions in the horizontal direction, vertical direction, or both. In the case of ordinary (non-periodic boundary conditions), we simply set KjM = 0 for all j and JN k = 0 for all k. To accommodate periodic boundary conditions, define σN +1,k := σ1,k and σj,M +1 := j, 1. The partition function of the system is, with some algebra: Z=  ∑  −1  e kB T  H(σ)  =  σ  M ( ∑∏  e  ∑N j=1  Kjk σj,k σj,k+1  )( e  ∑N j=1  Jjk σj,k σj+1,k  )  σ k=1  ∑ If we ∑ denote a whole column of spins as ui := {σ1,i , σ2,ik...σN,i }, then = σ u1 ,u2 ,...uM . Furthermore, if we define the functions h (uk , uk+1 ) := e  ∑N  j=1  Kjk σj,k σj,k+1  and v k (uk ) := e Z=  ∑  ∑N  j=1  M ∏  Jjk σj,k σj+1,k  , then  v k (uk )hk (uk , uk+1 )  (5.3)  u1 ,u2 ,...uM k=1  where with our definitions uM +1 := u1 . Note that in the case of ordinary boundary conditions in the horizontal direction Kj,M = 0, so hM (uM , u1 ) = 1 for all u1 and uM . The so-called transfer matrix method ([42], [60]) of computing the partition function is to consider the functions hk and v k as 50  5.3. Square Lattices: Transfer Matrix Formalism giving the entries of matrices in a 2N dimensional space of column configurations. That is, letting |uk ⟩ = |σ1,k , σ2,k , ...σN,k ⟩ denote an orthonormal basis, we seek matrices H k and V k such that ⟨uk |H k |ul ⟩ := hk (uk , ul ) and ⟨uk |V k |ul ⟩ := δuk ,ul v k (uk ) Then we can rewrite the partition function as ∑ Z = ⟨u1 |V 1 |u1 ⟩⟨u1 |H 1 |u2 ⟩⟨u2 |V 2 |u2 ⟩...⟨uM |V M |uM ⟩⟨uM |H M |u1 ⟩ u1 ,u2 ,...uM  =  ∑  ⟨u1 |V 1 |w1 ⟩⟨w1 |H 1 |u2 ⟩⟨u2 |V 2 |w2 ⟩...⟨uM |V M |wM ⟩⟨wM |H M |u1 ⟩  u1 ,u2 ,...uM w1 ,w2 ,...wM  where we have introduced further spin-column variables w1 ...wM and used that the vertical transfer matrices V k are diagonal in the column configuration basis. With cyclic boundary conditions in the horizontal direction ∑ Z= ⟨u1 |V 1 H 1 V 2 H 2 ...V M H M |u1 ⟩ = tr(V 1 H 1 V 2 H 2 ...V M H M ) u1  With ordinary boundary conditions in the horizontal direction ∑ ⟨u1 |V 1 H 1 V 2 H 2 ...V M −1 H M −1 V M |wM ⟩ Z = u1 ,u2 ,...uM w1 ,w2 ,...wM  (  =  ∑  )  (  ⟨u| V H V H ...V 1  1  2  2  M −1  H  M −1  u  V  M  ∑  ) |u⟩  u  = 2N ⟨+|V 1 H 1 V 2 H 2 ...V M |+⟩  where |+⟩ :=  √  1 2  N  ∑  u |u⟩  and u ranges over all N-component bitstrings.  Now we explicitly construct the matrices H k and V k . First, notice that the 2N dimensional space is the tensor product of N two dimensional spaces, one corresponding to each row. Column configuration states are product states with respect to this tensor product decomposition: |uk ⟩ = |σ1,k ⟩|σ2,k ⟩...|σN,k ⟩ 51  5.3. Square Lattices: Transfer Matrix Formalism We can define Pauli matrices Xj , Yj , Zj for each of these factor spaces. Let the spin states |σj,k ⟩ be eigenstates of the Zj operator. The state |+⟩ defined above is then the mutual +1 eigenstate of all of the Pauli X operators: √  1 2  |+⟩ :=  N  ∑ u  N ⊗ ) 1 ( √ |0⟩j + |1⟩j |u⟩ = 2 j=1  The vertical transfer matrices can now be written [60] as Vk = e  ∑N j=1  Jjk Zj Zj+1  (5.4)  where following our periodic boundary condition convention ZN +1 := Z1 . To see why Equation 5.4 holds, note that Zj Zj+1 |uk ⟩ = σj,k σj+1,k |uk ⟩. So, e  ∑N  j=1  Jjk Zj Zj+1  |uk ⟩ = e  ∑N  j=1  Jjk σj,k σj+1,k  ⟨uk |V k |ul ⟩ = δuk ,ul e  ∑N j=1  |uk ⟩, and thus  Jjk σj,k σj+1,k  = δuk ,ul v k (uk )  For the horizontal transfer matrices, the result is [60]:   N N ∑N ∏ ∏ Ajk  e j=1 γjk Xj = Hk =  Ajk eγjk Xj j=1  (5.5)  j=1  √ where Ajk := 2 sinh(2Kjk ), γjk := tanh−1 (e−2Kjk ). Equation 5.5 can also be written as 4 : Hk =  N ∏  eγjk Xj +ln(Ajk )I  (5.6)  j=1  Equation 5.6 allows us to see that H k is a bounded, non-zero operator, even when sinh(2Kjk ) = 0, or tanh−1 (e−2Kjk ) is infinite. The eigenvalues of the operator in the exponent in Equation 5.6 are ln(Ajk )±γjk . Using the relations: tanh−1 (z) = 12 (ln(1+z)− √ ln(1 − z)) and ln( z) = 21 ln(z), we may obtain: ln(Ajk ) ± γjk = ln(eKjk ± e−Kjk ) + iπnjk . Here njk is an integer which may arise because since the natural logarithm is multivalued in the complex plane, and as a result: ln(z) + ln(z) = 2 ln(z) + 2iπn for some integer n which may not be equal to zero. This proves that Ajk eγjk Xj is always a bounded operator for any finite Kjk . However, since Ajk evaluates to zero and γjk is infinite for certain finite values of Kjk (e.g. Kjk = iπ), in practical calculations involving such singular Kjk one would need to take Ajk and γjk as limits as Kjk approaches this offending value. 4  52  5.3. Square Lattices: Transfer Matrix Formalism Now we prove Equation 5.5. To begin, note that since the above operator is a tensor product of local operators for each row ⟨uk |H k |ul ⟩ = ⟨uk |  N ⊗  Ajk eγjk Xj |ul ⟩  j=1 N ∏  =  Ajk ⟨σj,k |eγjk Xj |σj,l ⟩  j=1 N ∏  =  ( Ajk ⟨σj,k |  j=1  )  cosh(γjk ) sinh(γjk ) sinh(γjk ) cosh(γjk )  |σj,l ⟩  where the last equality can be verified by inspection of the Taylor series sinh(γ ) of eγjk Xj . Now, since cosh(γjk ) = tanh(γjk = sinh(γjk ) ∗ e2Kjk , we have that jk ) (  )  cosh(γjk ) sinh(γjk ) sinh(γjk ) cosh(γjk )  ( = sinh(γjk )e  Kjk  eKjk e−Kjk  e−Kjk eKjk  )  From the definition of Ajk √ √ Ajk sinh(γjk )eKjk = eKjk e2Kjk − e−2Kjk ∗sinh(γjk ) = sinh(γjk ) e4Kjk − 1 (5.7) 2 1 Using the identity coth (z) = 1 + sinh2 (z) : Ajk sinh(γjk )eKjk = since then e4Kjk =  √  sinh2 (γjk ) + 1 − sinh2 (γjk ) = ±1  sinh2 (γjk )+1 . sinh2 (γjk )  ⟨uk |H |ul ⟩ = k  N ∏  Thus (  Ajk sinh(γjk )e  j=1  =  N ∏  ( ±⟨σj,k |  j=1  = ±  N ∏  (5.8)  Kjk  eKjk e−Kjk  eKjk e−Kjk  e−Kjk eKjk  e−Kjk eKjk  )  ) |σj,l ⟩  eKjk σj,k σj,l  j=1  = ±hk (uk , ul ) 53  5.4. Mapping to Fermions The√sign ambiguity in the above expression comes from the definition of Aij := e2Kjk√− e−2Kjk , and the subsequent replacement of the expression eKjk Aij with e4Kjk − 1 in Equation 5.7. Since Kjk is an arbitrary complex number, this replacement is only true if we do not specify which square root we are talking about. Otherwise, there may be an overall minus sign due to √ the branch cut in the function z for complex z. In practical calculations, we need to know whether to take the principal square root or its additive inverse when calculating Aij , such that the minus sign does not occur. In Appendix B, we show that one should choose either the principal square root or its inverse depending on the sign of the real part of eKij sinh(γjk ). This can be done in a way that ensures that Ajk sinh(γjk )eKjk = 1 with no sign ambiguity.  5.4  Mapping to Fermions  At this point we will make a brief introduction to the mathematical treatment of fermions, before showing that the transfer matrices can be rewritten as fermionic operators. Fermions are indistinguishable particles that are antisymmetric under interchange of any two particles, and are almost always dealt with using creation and annihilation operators in the language of second quantization. There are two main reasons for this, the first being that writing ordinary wave functions that possess the antisymmetric property is cumbersome (though possible) for any number of particles greater than two. The second and more important reason is that physical processes may change the number of fermions in a state, so a Hilbert space with a fixed number of particles will not do. Fermionic quantum states live in a mathematical state space known as Fock space. Suppose that a single fermion were to exist in an N dimensional Hilbert space H with some meaningful basis {|1⟩...|N ⟩}. Fock space can be constructed by taking the direct sum of the antisymmetric subspaces of H⊗k for k=0...N. By the Pauli exclusion principle, there can only be N fermions in any state, and the dimension of Fock space turns out to be 2N . Fermion states and operators are built out of N operators: a1 ...aN and their Hermitian conjugates: a †1 ...a†N . The a†j operators are known as creation operators and have the interpretation of creating a fermion in the state labeled by j. The aj operators are known as annihilation operators and have the interpretation of annihilating a fermion in the state labeled by j. The fermionic nature of the particles is encoded in the canonical  54  5.4. Mapping to Fermions anticommutation relations: {aj , a†k } = δjk I  {aj , ak } = 0  where {A, B} = AB + BA is the anticommutator of A and B. From the second relation it follows that: a2j = (a†j )2 = 0 for all j, which is a mathematical expression of the Pauli exclusion principle. We define a socalled vacuum state by the eigenvalue equation aj |vac⟩ = 0 for each j, and build up an orthonormal basis of Fock space from it. For any bitstring α = α1 , ...αN with αj ∈ {0, 1} ( )α1 ( )α2 ( )αN |α⟩ := a†1 a†2 ... a†N |vac⟩ The states |α⟩ are called occupation number eigenstates and have an interpretation as states with a definite physical occupation of each fermionic mode. They are eigenstates of the so-called number operator a†j aj for each j, which counts the number of fermions (0 or 1) in the mode labeled by j. Notice that the Fock space corresponding to N fermionic modes has the same dimensionality as the Hilbert space of N qubits. In fact, we can establish an isomorphism between products of Pauli operators and fermion operators, using a mapping known as the Jordan-Wigner transformation. There are differing versions of this mapping (e.g. compare those used in [47], [39] and [58]), but we will use the following: c2j−1 :=  ( j−1 ⊗  ) ⊗ Zj  Xk  k=1  c2j := −  ( j−1 ⊗  ) Xk  ⊗ Yj  k=1  where the ck are a set of 2N hermitian operators known as Majorana fermion operators, defined as c2j−1 := aj + a†j and c2j := −i(aj − a†j ). The canonical anticommutation relations are equivalent to {cj , ck } = δjk I for the Majorana operators. One can verify that the canonical anticommutation relations are obeyed by the Jordan-Wigner transformation using the anticommutation relations of the Pauli operators. We are interested in rewriting the Pauli operators appearing the transfer matrices as fermion operators. Firstly, note that −ic2j−1 c2j = iZj Yj = Xj 55  5.5. Partition Function on a Cylinder or Torus and for j < N −ic2j c2j+1 = i  ( j−1 ⊗  ) ⊗ Yj ⊗  Xk  k=1  ( j ⊗  ) Xk  ⊗ Zj+1  k=1  = iYj Xj Zj+1 = Zj Zj+1 If we have ordinary boundary conditions in the vertical direction, we can immediately rewrite the transfer matrices as k  V =  N −1 ∏  e−iJjk (c2j c2j+1 ) = e−i  ∑N −1 j=1  Jjk (c2j c2j+1 )  j=1  Hk =  N ∏ j=1   Ajk e−iγjk (c2j−1 c2j ) =   N ∏   Ajk  e−i  ∑N j=1  γjk (c2j−1 c2j )  (5.9)  j=1  For horizontally cyclic boundary conditions the Ising partition function is then given as a trace of a product fermionic of operators: Z = tr(V 1 V 2 ...V M H M )  (5.10)  and for ordinary boundary conditions in the horizontal direction Z = 2N ⟨vac|V 1 H 1 ...V M −1 H M −1 V M |vac⟩  (5.11)  where |vac⟩ is the fermionic vacuum state with the property that aj |vac⟩ = 0 for all j. Under our Jordan-Wigner transformation, |vac⟩ = |+⟩. To verify this, note that the mutual +1 eigenvector of all of the Xj is unique, and that Xj |vac⟩ = −ic2j−1 c2j |vac⟩ = −(aj + a†j )(aj − a†j )|vac⟩ = |vac⟩  5.5  Partition Function on a Cylinder or Torus  If we have cyclic boundary conditions in the vertical direction, then we still need to deal with the operator ZN Z1 for the final vertical coupling in each  56  5.5. Partition Function on a Cylinder or Torus column. This operator is not quadratic in the fermion operators. Rather ZN Z1 =  c2N −1 (X1 ...XN −1 )Z1  =  −c2N −1 Z1 (X1 ...XN −1 )  =  −c2N −1 c1 (X1 ...XN −1 )  =  −c2N −1 c1 XN (X1 ...XN )  =  ic2N −1 c1 c2N −1 c2N (X1 ...XN )  =  ic2N c1 (X1 ...XN )  (5.12)  Fermion occupation number states are eigenstates of the Pauli X operator (because Xj = −ic2j−1 c2j = aj a†j − a†j aj ), so for any occupation number eigenstate |α1 ...αN ⟩, αj ∈ {0, 1}: ZN Z1 |α1 ...αN ⟩ = (−1)n(α) ic2N c1 |α1 ...αN ⟩ ∑ 5 where n(α) := N i=1 αi is the fermionic parity of the state |α1 ...αN ⟩ . So, acting upon any state of definite parity the operator ZN Z1 acts exactly as a quadratic fermion operator, with a sign that depends on the parity of that state. We can decompose the identity operator on Fock space as: I = Podd + Peven , where Podd and Peven are the projectors onto the odd and even parity fermion number eigenspaces, respectively. Then we can write eJN k ZN Z1 = eJN k ZN Z1 (Podd + Peven ) = e−iJN k c2N c1 Podd + eiJN k c2N c1 Peven Introducing some notation, and combining eJN k ZN Z1 with the rest of the vertical transfer matrix V k = V o,k Podd + V e,k Peven where  (5.13)  ∑  N −1 V o,k := e−i( j=1 Jjk c2j c2j+1 )−iJN j c2N c1 ∑N −1 V e,k := e−i( j=1 Jjk c2j c2j+1 )+iJN j c2N c1  All of the transfer matrices are parity preserving: that is as matrices they are block diagonal with respect to the even and odd parity subspaces. From this it follows that all transfer matrices commute with the even or odd parity 5 Note that this is not the same as the parity of the Ising spins in a column. Those were associated with the Pauli Z operators while the fermionic parity is associated with the Pauli X operators  57  5.6. Partition Function On a Double Torus projectors. Since Peven Podd = Podd Peven = 0, if the vertical transfer matrices in a product of transfer matrices such as V k H k V k+1 are each expanded in terms of the even/odd parity subspaces just 2 terms survive: ( ) ( ) V k H k V k+1 = V o,k Podd + V e,k Peven H k V o,k+1 Podd + V e,k+1 Peven = V o,k H k V o,k+1 Podd + V e,k H k V e,k+1 Peven For vertically cylindrical boundary conditions (i.e. ordinary boundary conditions in the horizontal direction, but cyclic in the vertical direction), we are interested in the following product of transfer matrices: ( ) ( ) V 1 H 1 ...H M −1 V M = V o,1 H 1 ...H M −1 V o,M Podd + V e,1 H 1 ...H M −1 V e,M Peven The state |vac⟩ has definite even parity. So for vertically cylindrical boundary conditions, only the Peven term contributes, and the partition function is Z = 2N ⟨vac|V e,1 H 1 ...V e,M −1 H M −1 V e,M |vac⟩  (5.14)  For boundary conditions that are cyclic in both directions (toroidal), the partition function gets a contribution from both terms ) ) ( ( Z = tr V o,1 H 1 ...H M −1 V o,M H M Podd + tr V e,1 H 1 ...H M −1 V e,M H M Peven (5.15) We will later obtain an explicit means of evaluating Equation 5.15. For now though, we use it as a stepping stone to understand the structure of the partition function on a more general family of lattices.  5.6  Partition Function On a Double Torus  The approach taken above for the torus can be extended to certain families of square lattices of a higher genus, including the punctured cylinder graphs from Chapter 4. For clarity, we will next describe the procedure for the simplest case of genus greater than one: a square double toroidal graph. The square double toroidal graph can then be easily generalized to a class of lattices that we will call “square g-graphs” (of arbitrary genus g), as we will see in Section 5.7. Finally, we will consider the Ising model on punctured cylinder graphs in Section 5.8. The double torus graph that we will study here has been previously considered for a homogeneous Ising Model at criticality [9]. The approach in Ref [9] took a combinatoric approach, 58  5.6. Partition Function On a Double Torus  Figure 5.1: A genus two square lattice  whereas we will show that the problem can also be solved algebraically via the mapping to fermions. We consider the lattice depicted in Figure 5.1. The twenty-nine pairs of numbers represent edges which are identified with one another. All of the faces of the lattice are squares except two octagonal faces that arise due to the boundary conditions: the face visiting edges 1-8-7-29-5-4-20, and the face visiting edges 11-12-21-11. All vertices have four incident edges. We parameterize by (N1 , N2 , K, M ) a family of double-torus graphs like the above, but with differing sizes. For a lattice in this family, the number of vertices |V | is N M , the number of edges |E| is 2N M , and the number of faces |F | is N M − 2. It can be cellularly embedded onto a closed orientable surface has a genus of 1 − 12 χ = 2 (see Appendix A). For such an interaction lattice, the Ising Hamiltonian is: H(σ) = HH (σ) + HV0 (σ) + HVB (σ) + HV′ (σ)  59  5.6. Partition Function On a Double Torus where we let N := N1 + N2 and (M −1 ) N ∑ ∑ HH (σ) := −kB T Kjk σj,k σj,k+1 + KjM σj,M σj,1 j=1  k=1  is the energy contribution due to all of the horizontal edges, N −1 ∑  HV0 (σ) := −kB T  M ∑  Jjk σj,k σj+1,k  j=1,j̸=N1 k=1  is the energy of the vertical edges that belong to one sub-torus aside from the cyclic ones, HVB (σ)  := −kB T  K ∑  JN1 k σN1 ,k σN1 +1,k  k=1  is the energy due to the “bridge” connecting the two tori, and finally ) (K M ∑ ∑ JN k σN,k σ1,k + (JN1 k σN1 ,k σ1,k + JN k σN,k σN1 +1,k ) HV′ (σ) := −kB T k=1  k=K+1  is the energy due to the vertical cyclic edges. Denoting as before σ = {u1 ...uM }, where each uk denotes a column of spins e  −1 kB T  HH (σ)  =  M ∏  hk (uk , uk+1 )  k=1 ∑N  where as before hk (uk , uk+1 ) := e j=1 Kjk σj,k σj,k+1 and we let uM +1 := u1 . Furthermore, we can write )( M ) (K ∏ ∏ −1 0 (σ)+H B (σ)+H ′ (σ) H ( ) k k V V V v (uk ) e kB T = v (uk ) k=1  k=K+1  } J σ σ + J σ σ where as before v k (uk ) := exp jk j,k j+1,k N k N,k 1,k , and j=1 we define   −1  N∑  v k (uk ) := exp Jjk σj,k σj+1,k + JN1 k σN1 ,k σ1,k + JN k σN,k σN1 +1,k   {∑ N −1  j=1,j̸=N1  60  5.6. Partition Function On a Double Torus The partition function of the system can now be written in a manner similar to Equation 5.3: ) )( M (M )( K ∏ ∏ ∏ ∑ hk (uk , uk+1 ) v k (uk ) v k (uk ) (5.16) Z= u1 ,u2 ,...uM  k=1  k=1  k=K+1  As before, hk (uk , uk+1 ) = ⟨uk |H k |uk+1 ⟩ with H k given by Equation 5.5, and v k (uk ) = ⟨uk |V k |uk ⟩ with V k given by   −1  N∑ k V = exp Jjk Zj Zj+1 + JN k ZN Z1   j=1  This is the same as Equation 5.4 but now we dispense with the notational convention ZN +1 := Z1 to avoid confusion in what follows. By the same arguments that justified Equation 5.4: v k (uk ) = ⟨uk |V k |uk ⟩, where   −1  N∑  V k = exp Jjk Zj Zj+1 + JN1 k ZN1 Z1 + JN k ZN ZN1 +1 (5.17)   j=1,n̸=N1  With these definitions, Equation 5.16 becomes ( ) Z = tr V 1 H 1 ...V K H K V K+1 ...V M H M  (5.18)  Now we make the mapping to fermions. For the horizontal transfer matrices H k , Equation 5.9 is still valid, and for the first K vertical transfer matrices V k we can still use Equation 5.13. To express Equation 5.17 as a fermion operator, we need to deal with the operators ZN1 Z1 and ZN ZN1 +1 . By repeating the algebraic steps used to obtain Equation 5.12 for ZN Z1 , we find that in general, for any integers m, n such that 1 ≤ m < n ≤ N Zn Zm+1 = ic2n c2m+1 (Xm+1 ..Xn )  (5.19)  Thus, ZN1 Z1 = ic2N1 c1 (X1 ...XN1 ) and ZN ZN1 +1 = ic2N c2N1 +1 (XN1 +1 ..XN ). Equation 5.17 becomes V k = exp{−i  N −1 ∑  Jjk c2j c2j+1  j=1,n̸=N1  + iJN1 k c2N1 c1 (X1 ...XN1 ) + iJN k c2N c2N1 +1 (XN1 +1 ..XN )}(5.20) Recall that the Pauli matrix Xj acts on fermion occupation number eigenstates as (−1)nj , where nj ∈ {0, 1} is the occupation of fermion mode j. With this in mind, we will find it convenient to make the following definition:  61  5.6. Partition Function On a Double Torus Definition 5.6.1 Consider a fermionic system of N modes. For any subset of the modes x ⊂ {1...N }, define the even(odd) parity subspace with respect to x be the +1(-1) eigenvalue eigenspace of the operator ∏  †  (−1)aj aj =  j∈x  ∏  (−ic2j−1 c2j )  j∈x  †  where the operator (−1)aj aj is defined by its action on occupation number †  eigenstates: (−1)aj aj |α1 ...αN ⟩ = (−1)αj |α1 ...αN ⟩ Note that with our Jordan-Wigner transformation, Xj = −ic2j−1 c2j , so ∏ the operator in Definition 5.6.1 is in fact j∈x Xj . Denote the projector into the even parity subspace with respect to x as Px0 , and the projector into the odd parity subspace with respect to x as Px1 . Note that the operators Peven 0 1 and Podd introduced in section 5.5 are P{1...N } and P{1...N } respectively, in this more general notation. In the case of the double torus, we want to consider a bipartition of modes into the sets {1...N1 } and {N1 ...N }. Now denote the projector into the subspace where modes {1...m} have parity α1 and modes {m + 1...N } α2 α1 α2 := P α1 have parity α2 (α1 , α2 ∈ {0, 1}) by Pm {1...m} ∗ P{m+1...N } . One can decompose the identity as I = P 0 + P 1 = (PN001 + PN111 ) + (PN011 + PN101 ) = PN001 + PN111 + PN011 + PN101 So from 5.20 we can write V k = V k (PN001 + PN111 + PN011 + PN101 ) k 00 k 11 k 10 k 10 = V00 PN1 + V11 PN1 + V01 PN1 + V10 PN1  where Vαk1 α2    := exp −i   N −1 ∑  Jjk c2j c2j+1 + (−1)α1 iJN1 k c2N1 c1 + (−1)α2 iJN k c2N c2N1 +1  j=1,n̸=N1  which is an exponential of quadratic fermion operators. Now comes a crucial step: None of the H k or V k transfer matrices change the parity of modes {1...N1 } or the parity of modes {N1 + 1...N }. Thus, [H k , PNα11 α2 ] = [V k , PNα11 α2 ] = 0. This can be verified by observing the form of Equation 5.17, where in all products of two Pauli Z operators, the subscripts either both belong to {1...N1 } or to {N1 +1...N }, and never one to each. By commuting 62      5.7. Partition Function on a Square g-Graph projectors to the left through the transfer matrices, we can rewrite Equation 5.18 as ( ) Z = tr V 1 H 1 ...V K H K V K+1 ...V M H M (PN001 + PN111 + PN011 + PN101 ) ) ∑ ( M M = tr V 1 H 1 ...V K PNα11 α2 H K VαK+1 ... V H α α α 1 2 1 2 α1 ,α2  The transfer matrices V k will change the fermionic parity of modes {1...N1 } and {N1 + 1...N }, so we cannot continue commuting PNα11 α2 to the left. However, any state with definite parity of modes {1...N1 } as well as definite parity of modes {N1 +1...N } has definite overall parity, which is just sum of the parities of the two sets of modes. So Peven PNα11 α2 = δα1 α2 PNα11 α2 and Podd PNα11 α2 = δα1 ,α2 ⊕1 PNα11 α2 , where α2 ⊕ 1 denotes addition modulo 2. So we can write out the four terms of the partition function as ( ) K+1 M M Z= tr V e,1 H 1 ...V e,K H K V00 ...V00 H ∗ PN001 ( ) K+1 M M + tr V e,1 H 1 ...V e,K H K V11 ...V11 H ∗ PN111 ( ) o,1 1 o,K K K+1 M M 01 + tr V H ...V H V01 ...V01 H ∗ PN1 ( ) K+1 M M (5.21) + tr V o,1 H 1 ...V o,K H K V10 ...V10 H ∗ PN101 where in each term we have commuted the PNα11 α2 operator back to the right. Again we postpone any further manipulation of Equation 5.21 until we derive its analogue for a more general family of lattices.  5.7  Partition Function on a Square g-Graph  It is straightforward to define a square lattice corresponding to a g-fold torus. One imagines many square toroidal graphs stitched together via a sequence of bridges as in Figure 5.2. Call the graph G corresponding to a lattice such as the one depicted in Figure 5.2 a square g-graph. Here g is the number of tori glued together, and as we will see below it is also the genus of the graph. Also, each torus is taken to have the same “circumference of M cells (in Figure 5.2, M = 21). This ensures that∑ G can be represented by a rectangular N × M grid, where we define N := gj=1 Nj and Nj is the number of rows in the j th torus. Any square g-graph lattice can be analyzed via the transfer matrix formalism, but we will consider a slightly less general family of lattices that admits of a simpler notation. In particular, we will 63  5.7. Partition Function on a Square g-Graph  Figure 5.2: A lattice composed of four toroidal graphs stitched together. The boundary conditions are given by the following rule: the two endpoints of each line defining the grid are identified with each other. The leftmost vertical lines are slightly elongated to distinguish them from the vertical lines on the fourth torus. This square g-graph lattice has a “staircase” structure.  consider square g-graphs that have a “staircase” structure, meaning that the bridges between subsequent tori are nondecreasing in width as one goes from top to bottom. A square g-graph of this type will have bridge lengths satisfying K1 ≤ K2 ≤ ...Kg−1 . An example is depicted in Figure 5.2. A square g-graph graph with a staircase structure is parameterized by the integers (N1 , ...Ng , K1 , ...Kg−1 , M ). To clean up notation we define Nj := ∑j k=1 Nk . For a lattice in this family, the number of vertices |V | is N ∗ M . Each vertex has 4 incident edges and there are no self-loops so the number of edges |E| is 4 ∗ N2M = 2N M . There are 2(g − 1) octagonal faces and the rest of the faces are rectangular, and each vertex lies at the meeting of four faces. 64  5.7. Partition Function on a Square g-Graph This means that |V | = N M = 41 (4(|F |−2(g−1))+8∗2(g−1)) = |F |+2(g−1), where |F | is the number of faces. So |F | = N M − 2(g − 1). The graph is cellularly embedded on a closed orientable surface has a genus of 1 − 21 χ = g (see Appendix A). It is straightforward to construct a set of non-trivial cycles on a square g-graph which form a basis of the first homology group of a surface of genus g. Figure 5.3 shows a set of non-trivial cycles C1 ...C2g and non-trivial co′ on a square g-graph with g = 4. These cycles satisfy the cycles C1′ ...C2g requirement |Cj ∩ Ck′ | = δjk (mod 2) of Section 2.3.  Figure 5.3: Non-trivial cycles of a square 4-graph. The dotted lines indicate cycles on the lattice itself, while the solid lines indicate cocycles (cycles on the dual lattice).  Now we consider the Ising partition function on a square g-graph. A square g-graph is constructed such that it can still be thought of as an N × M square lattice. All that has changed is boundary conditions, and this allows us to still use the transfer matrix formalism and the mapping to fermions. The partition function of an Ising Model on a square g-graph is   Kj g ∏ ∏ Z = tr  V [j],k H k  (5.22) j=1  k=Kj−1 +1  where we let K0 := 0 and Kg := M . The H k are still given by Equation 65  5.7. Partition Function on a Square g-Graph 5.9, and we define V [j],k as V  [j],k  = exp{  +  g−1 ∑  g ∑  N l −1 ∑  l=1  m=Nl−1 +1  Jmk Zm Zm+1 +  j−1 ∑  JNl k ZNl ZNl−1 +1  l=1  JNl k ZNl ZNl +1 + JN k ZN ZNj−1 +1 }  (5.23)  l=j  where we let N0 := 0. The first term in the exponential is the energy contribution from all of the edges internal to any of the sub-tori, the second is the cyclic contribution for each torus still “unbridged” in column j, the third term is the contribution of all bridging edges between tori, and the fourth term is the vertical cyclic contribution from the final row N. Recall from Equation 5.19 that for any integers m, n such that 1 ≤ m < n ≤ N : Zn Zm+1 = ic2n c2m+1 (Xm+1 ..Xn ). Thus Equation 5.23 can be written V [j],k = exp{i  + i − i  j−1 ∑ l=1 g−1 ∑  g ∑  N l −1 ∑  l=1  m=Nl−1 +1  Jmk c2m c2m+1  JNl k c2Nl c2Nl−1 +1 (XNl−1 +1 ..XNl ) JNl k c2Nl c2Nl +1 + iJN k c2N c2Nj−1 +1 (XNj−1 +1 ..XN )}  l=j  We now wish to decompose the identity into projectors corresponding to the parities of the subsets in the partitioning of modes: {1..N1 }, {N1 + 1...N2 }, ...{Ng−1 + 1...N }. For a set of integers m1 < m2 < ... < mj , denote α ,α ,...α Pm11 ,m22 ,...mj j  =  j ∏  αk P{m k−1 +1...mk }  (5.24)  k=1  where m0 := 0 and Pxα is (as before) the projector onto the α ∈ {0, 1} subspace with respect to the modes x ⊆ {1...N }. Since the sets {1..N1 }, {N1 + 1...N2 }, ...{Ng−1 + 1...N } partition the whole set of modes {1...N } we can write ∑ α ,α ,...α I= PN11 ,N22 ,...Ngg α1 ,α2 ,...αg  66  5.7. Partition Function on a Square g-Graph So the partition function can be expressed as a summation over 2g terms:    Kj g ∑ ∏ ∏ α ,α ,...α Z= tr  V [j],k H k  PN11 ,N22 ,...Ngg  (5.25) α1 ,α2 ,...αg  j=1  k=Kj−1 +1  We now make some transformations in direct analogy with the double toroidal graph. Using that for each each value of 1 ≤ j ≤ g and k > Kj−1 : α ,α ,...α  α ,α ,...α  [H k , PN11 ,N22 ,...Njj−1 ,N ] = [V [j],k , PN11 ,N22 ,...Njj−1 ,N ] = 0 and α ,α ,...α  (α  j−1 PN11 ,N22 ,...Nj−2 j−2 ,N  ⊕αj )  α ,α ,...α  α ,α ,...α  PN11 ,N22 ,...Njj−1 ,N = PN11 ,N22 ,...Njj−1 ,N  it follows that    Kj ∏ α ,α ,...α  V [j],k H k  PN11 ,N22 ,...Njj−1 ,N =  k=Kj−1 +1 α ,α ,...α  (α  V [j],k PN11 ,N22 ,...Njj−1 ,N H k  α ,α ,...α  k=Kj−1 +1    j−1 = PN11 ,N22 ,...Nj−2 j−2 ,N    Kj ∏    Kj ∏  ⊕αj )   Vα[j],k H k  PN11 ,N22 ,...Njj−1 ,N 1 ,α2 ,...αj α ,α ,...α  k=Kj−1 +1  where Vα[j],k 1 ,α2 ,...αj  := exp{i −  i  j−1 ∑  (−1)αl JNl k c2Nl c2Nl−1 +1 + i(−1)αj JN k c2N c2Nj−1 +1  g ∑  l=1 N l −1 ∑  l=1  m=Nl−1 +1  Jmk c2m c2m+1 − i  g−1 ∑  JNl k c2Nl c2Nl +1 } (5.26)  l=j  is an exponential of a sum of quadratic fermion operators. Then commuting projectors to the right and recombining we arrive at    Kj g ∑ ∏ ∏ α ,α ,...α [j],k Z= tr  V Hk P 1 2 g  α1 ...αj−1 (αj ⊕αj+1 ⊕...αg )  α1 ,α2 ,...αg  j=1  N1 ,N2 ,...Ng  k=Kj−1 +1  (5.27) α ,α ,...α We can further expand out PN11 ,N22 ,...Ngg as a polynomial in fermion operators. The easiest way to do this is to consider the projector into a single  67  5.7. Partition Function on a Square g-Graph parity subspace x: Px± . Recall that the the fermionic parity is associated with the Pauli X operators, so     ∏ ∏ 1 1 Xj  = I ± −ic2j−1 c2j  Px± = I ± (5.28) 2 2 j∈x  j∈x  α ,α ,...α  Then from the definition of PN11 ,N22 ,...Ngg in Equation 5.24 α ,α ,...α PN11 ,N22 ,...Ngg  ( )g g ∏ ) 1( 1 αk = I + (−1) X[k] = 2 2 k=1  ∑  g ∏  (−1)αk ∗βk (X[k] )βk  β1 ,β2 ,...βg ∈{0,1} k=1  ∏ k g where X[k] := N j=Nk−1 +1 Xj . This expansion contains 2 terms, as does Equation 5.27. Combining with Equation 5.27, we get a summation over 22g terms ∑ (−1)α·β Z= ∗ tr (Γα,β ) (5.29) 2g α1 ,α2 ,...αg β1 ,β2 ,...βg  where α · β is the mod 2 dot product of the bitstrings α and β and we define   g g ∏ ∏ [j] Γαβ :=  Γα1 ...αj−1 (αj ⊕αj+1 ⊕...αg )  (X[j] )βj (5.30) j=1  j=1  and Γα[j]1 ...αj  :=  Kj ∏  Hk Vα[j],k 1 ...αj  k=Kj−1 +1  In Equation 5.30, the bitstrings α and β enter in very different ways. The αj show up as a coefficient (−1)αj in the vertical transfer matrices, whereas the βj show up as the exponent of a Pauli X operator in 5.30. However, we can rewrite Γα,β in a way that is more symmetric between α and β. Consider the definition of the horizontal transfer matrices H k given by Equation 5.5:   N N ∑N ∏ ∏ Hk =  Ajk  e j=1 γjk Xj = Ajk eγjk Xj (5.31) j=1  j=1  √ where Ajk := 2 sinh(2Kjk ), γjk := tanh−1 (e−2Kjk ). We assume that Ajk and γjk are always chosen such that Ajk eKjk sinh(γjk ) = 1, as explained in Section 5.3 and Appendix B. We may now ask the question: what would 68  5.7. Partition Function on a Square g-Graph be the impact of flipping the sign of the horizontal couplings Kjk for all j in some set of modes x? That is, how do Ajk and γjk transform if we let ′ = −K ′ ′ Kjk → Kjk jk for all j ∈ x? Let Ajk → Ajk and γjk → γjk . It is easy √ √ to see that A′jk = 2 sinh(−2Kjk ) = −2 sinh(2Kjk ) = (−1)mjk ∗ iAjk , for ′ , we will use the relation for the some integer mjk ∈ {0, 1}. To calculate γjk inverse hyperbolic tangent: tanh−1 (z) = 12 (ln(1 + z) − ln(1 − z)). Then ′ γjk =  = = = =  ) 1( ln(1 + e2Kjk ) − ln(1 − e2Kjk ) 2 ) ( )) 1 ( ( 2Kjk ln e (1 + e−2Kjk ) − ln −e2Kjk (1 − e2Kjk ) 2 ( ) ( ) ( )) 1 ( ( 2Kjk ) ln e + ln 1 + e−2Kjk − ln(−1) − ln e2Kjk − 1 − e2Kjk 2 ) 1( ln(−1) + ln(1 + e2Kjk ) − ln(1 − e2Kjk ) 2 iπ + γjk + iπnjk 2  where njk is some integer, which is necessary in this expression because tanh−1 (z) = 21 (ln(1 + z) − ln(1 − z)) is multivalued for complex z, since since ln(z) is a multivalued function. So while we cannot conclude that iπ iπ ′ = γ ′ γjk jk + 2 , we can conclude that γjk = γjk + 2 + iπnjk for some ∗  iπ  iπ  integer njk . Further: eγjk = eγjk Xj +( 2 +iπnjk )Xj = eγjk Xj e( 2 +iπnjk )Xj = i(−1)njk Xj eγjk Xj , since eiϕXj = cos(ϕ)I + i sin(ϕ)Xj . So upon flipping the signs of the couplings Kjk , ∀j ∈ x, the horizontal transfer matrix transforms as    ∏ ∏ ′ H k → H k =  (−1)mjk iAjk (−1)njk iXk eγjk Xj   Ajk eγjk Xj   =    j∈x N ∏  j=1    Ajk eγjk Xj    ∏  j∈x    j ∈x /  (−1)njk +mjk +1 Xk  = H k  ∏  Xk  j∈x  In the last equality we use that njk + mjk + 1 works out to be even for all  69  5.7. Partition Function on a Square g-Graph ′  ′ ) = 1, because: j, k. This follows from: Ajk eKjk sinh(γjk ) = A′jk eKjk sinh(γjk ′  ′ 1 = A′jk eKjk sinh(γjk )  ) ( π = (−1)mjk iAjk e−Kjk sinh γjk + i (2njk + 1) 2 mjk −Kjk = (−1) iAjk e cosh (γjk ) i(−1)njk = (−1)mjk +njk +1 Ajk eKjk sinh (γjk ) = (−1)mjk +njk +1  where we have used the fact that tanh(γjk ) = e−2Kij and the addition formula for the hyperbolic cosine. Let us now define HβN as the horizontal transfer matrix for the M th column where we flip the signs of KjM for all j in the set: j ∈ {Nk−1 + 1...Nk } iff βk = 1. Then it follows from the above analysis that HβM  =H  M  g ∏     βk  Nk ∏  Xj   =H  j=Nk−1 +1  k=1  M  g ∏  (X[k] )βk  k=1  Now we can rewrite Equation 5.30 as   g−1 ∏ [j] [g] Γ αβ =  Γα1 ...αj−1 (αj ⊕αj+1 ⊕...αg )  Γα,β  (5.32)  j=1    where  Γα,β :=  [g]  M −1 ∏   Vα[j],k H k  Vα[g],M HβM 1 ...αg 1 ...αg  k=Kg−1 +1  Now Γαβ can be interpreted as a simple product of transfer matrices, where the signs certain Ising coupling constants have been flipped according to the bitstrings α and β. The couplings that that get multiplied by minus one when the binary variables α and β are changed are edges along certain cocycles on G, which are shown in Figure 5.4. In Figure 5.5, we show the transformation between the non-trivial cycles in Figure 5.4 and the natural ones shown in Figure 5.3. The transformation amounts to a choice of basis of the first homology group on the surface S.  70  5.8. Partition Function on a Punctured Cylinder Graph  Figure 5.4: A set of non-trivial cocycles on a square 4-graph, different from those in Figure 5.3. The binary variables α and β from Section 5.7 determine whether or not the Ising couplings along these non-trivial cocycles get multiplied by minus ′ ′ correspond to the binary variable αj and the cocycles C2j−1 one. The cocycles C2j correspond to the binary variables βj  5.8  Partition Function on a Punctured Cylinder Graph  We now consider the partition function of an Ising Model defined on the punctured cylinder graph, introduced in Chapter 4. Recall that such a graph is obtained by cutting n horizontal slots in an N × M lattice with vertically cylindrical boundary conditions, and extending vertical edges through the slots (see Figures 4.1 and 4.2). A punctured cylinder graph is specified by the parameters:{N, M, {x1 , y1 , K1 }, ...{xn , yn , Kn }}, where N and M are the height and width of the initial grid, xj and yj are the coordinates of the j th slot, and Kj is its width. By the transfer matrix method, the partition function on a punctured cylinder graph is Z = ⟨+|Γ|+⟩  (5.33)  71  5.8. Partition Function on a Punctured Cylinder Graph  Figure 5.5: The left-most part of Figure a) shows an alternative basis for the first homology group of the dual graph of the square 4-torus. Figure a) shows the linear transformation from this basis to the one shown in Figure 5.3. Each equation, e.g. A′ = A + B, is true up to the addition of a trivial cocycle of edges. Figure b) shows the trivial cocycles on the dual lattice that must be added to demonstrate the third equation down (blue) for each step.  where    Γ :=   n ∏  x +Kj −1  V xj−1 +Kj−1 +1 H xj−1 +Kj−1 +1 ...V xj −1 H xj −1 V[j]j H xj ...V[j]j x  j=1  ) H xj +Kj −1 V xj +Kj H xj +Kj V xn +Kn +1 H xn +Kn +1 ...V M  (5.34) 72  5.8. Partition Function on a Punctured Cylinder Graph where x0 = K0 := 0, and the H k and V k are given as before by Equations k are vertical transfer matrices for columns 5.9 and 5.4, respectively. The V[j] within the j th slot, and are defined as       −1 N∑  k V[j] = exp Jlk Zl Zl+1 + Jyj ,k Zyj Z1 + JN k ZN Zyj +1 (5.35)      l=1  l̸=yj  We note that a graph obtained by endowing a punctured cylinder graph with horizontal cyclic boundary conditions (obtaining a punctured torus graph) can also be accommodated by taking Γ → Γ∗H M . Then the partition function is given by Z = tr(Γ). We now make the mapping to fermions. Recall that we can write V k = V1k Podd + V0k Peven where Podd and Peven are the projectors into the odd and even parity fermion occupation number eigenspaces, respectively, and ∑N −1  Vαk := e−i(  l=1  Jlk c2l c2l+1 )+i(−1)α JN k c2N c1  where α is a binary variable: α ∈ {0, 1}. Since Podd |+⟩ = 0, only the k operators, Equation 5.19 lets us α = 0 term contributes. As for the V[j] write Zyj Z1 = ic2yj c1 (X1 ...Xyj ) and ZN Zyj +1 = ic2N c2yj +1 (Xyj +1 ..XN ) Again, we express each vertical transfer matrix as a sum over exponentials of quadratic fermion operators, by decomposing the identity into k with j = 1...n, we fermion number parity subspaces. For the operator V[j] want to bipartition the modes into the sets {1...yj } and {yj + 1...N }. As in Section 5.6, denote the projector into the subspace where the modes in some set x have parity α ∈ {0, 1} as Pxα . Also following Section 5.6, we define α1 α2 α1 α2 Pm := P{1...m} P{m+1...N }  Notice that all of the transfer matrices preserve the overall parity of the fermion occupation number, so the projector Peven can be commuted arbitrarily throughout a product of transfer matrices. This means that we 73  5.8. Partition Function on a Punctured Cylinder Graph will only ever need to consider α1 and α2 such that α1 = α2 . We can decompose the identity within the total even parity subspace as ∑ αj αj ∑ αj α P{1...yj } P{yjj +1...N } (5.36) Peven = Pyj = αj  αj  α  j In each term of Equation 5.36, the first factor P{1...y will suffice instead j}  α αj  of the full projector Pyjj α α Pyjj j Peven  because α  α  Now:    ∑  k k V[j] Peven = V[j]  α  j j = P{1...y P j Peven = P{1...y Peven j } {yj +1...N } j}  j P{1...y Peven =  j}  α  k V[j],α := exp j          −i  x +Kj  and V[j]j we write αj P{1...y j}  1 = 2  k  Peven V[j],α P j j {1...yj } α  Jlk c2l c2l+1 + i(−1)αj Jyj ,k c2yj c1 + i(−1)αj JN k c2N c2yj +1  l=1 l̸=yj  The operator x V[j]j  N −1 ∑    αj  αj ,βj  where  ∑  αj P{1...y j}          (5.37) can be commuted arbitrarily among the operators  in Equation 5.34, but does not commute with the V k . Now  ( αj  I + (−1)  yj ∏  ) Xl  l=1  1∑ = (−1)αj ∗βj 2 βj  ( yj ∏  )βj Xl  (5.38)  l=1  where βj is another binary variable, and we have used Equation 5.28 to write the projector in terms of Pauli operators. We then obtain an expansion of the operator Γ ∗ Peven into a sum over 2n binary variables:   ∑ (−1)α·β Γ ∗ Peven =  Γα,β  ∗ Peven 2g α,β  where α and β are bit strings of length g = n, and  ( yj )βj n ∏ ∏ x +K +1 x −1 xj j−1 j−1 j x +K +1 x −1 Γα,β :=  V0 H j−1 j−1 ...V0 H j Xl V[j],α j j=1 xj +Kj −1 xj +Kj −1 xj +Kj xj +Kj H xj ...V[j],α H V0 H j  )  l=1  V0xn +Kn +1 H xn +Kn +1 ...V0M 74  5.9. Summary By the same considerations as in Section 5.7, we can replace the Pauli X operators appearing in Equation 5.39 by flipped coupling constants in the adjacent horizontal transfer matrices. We then write Equation 5.39 as  n ∏ x +K +1 x −1 x −1 xj xj +Kj −1 Γα,β :=  V0 j−1 j−1 H xj−1 +Kj−1 +1 ...V0 j Hyjj,βj V[j],α H xj ...V[j],α j j j=1  H xj +Kj −1 V0 j  x +Kj  ) H xj +Kj V0xn +Kn +1 H xn +Kn +1 ...V0M  (5.39)  where Hykj ,βj is the H k horizontal transfer matrix with the couplings Klk for l ≤ yj multiplied by minus one iff βj = 1. Expressed in this way, each Γα,β is a simple product of exponentials of quadratic fermion operators. The bits α and β determine whether or not couplings are flipped along the nontrivial cocycles shown in Figure 4.3, back in Chapter 4. In Figure 4.3, the ′ correspond to the binary variable α and the cocycles C ′ cocycles C2j j 2j−1 correspond to the binary variables βj . The partition function on a punctured cylinder graph is Z=  ∑ α1 ,α2 ,...αg β1 ,β2 ,...βg  (−1)α·β ∗ ⟨vac|Γα,β |vac⟩ 2g  (5.40)  which is a sum over 22g terms, each of which is proportional to a trace of a product of exponentials of quadratic fermion operators.  5.9  Summary  In this Chapter, we have considered a large class of square lattices, which are either planar or are cellulary embedded on some closed-orientable surface of genus g. Using the transfer matrix formalism, we have reduced the evaluation of the Ising partition function on these graphs to a problem involving fermion operators. In every case, the partition function is given by a summation over 22g terms, each of which involves a computation involving the exponential of a quadratic fermion operator. In the next Chapter, we will see that such operators admit of an efficient description and introduce a method of evaluating each term in poly(N) time. Using the mapping to fermions allows an interesting interpretation of the transfer matrices as time evolution operators, and the physical picture that this provides will be further investigated in Chapter 7. However, we had to restrict ourselves to square lattices to obtain this mapping. The 75  5.9. Summary combinatoric approach to the Ising partition function on higher genus graphs has greater explicit generality, and we will briefly discuss its connection to our approach. In particular, we will see that the combinatoric approach also involves a summation over two g-component bitstrings α, and β, and that the factor (−1)α·β also appears as a coefficient in the combinatoric approach. It is well known that the partition function of a classical Ising model on a quadratic lattice can be identified with the generating function of perfect matchings (also known as dimer coverings) on a certain modified graph [36], [37], [17], [1]. A perfect matching is a subset of the edges of a graph that visits each vertex exactly once. For a planar graph, the generating function of perfect matchings can be evaluated as the Pfaffian of the adjacency matrix of the graph, weighted and antisymmetrized in an appropriate way (see Section 6.1 for an introduction to the Pfaffian function). In [19], Galluccio and Loebl develop a formalism for evaluating the generating function of perfect matchings on a nonplanar graph. The result is that for a graph G of genus g, the generating function can be expressed as a linear combination of 22g Pfaffians. Each of the 22g terms corresponds to a particular way of specifying an orientation for the edges of a modified graph G′ . The result is (Theorem 3.9, [19]) 4 ∑ g  ′  P(G , x) = ±  c(r(Dj ))Pf (A(Dj ))  (5.41)  j=1  where the overall sign depends on an arbitrary choice (one has to pick some fixed perfect matching), so we ignore it. Here each Dj is a particular orientation of the edges, and x is a vector of the weights associated with the edges of G’, which are derived from the Ising couplings on G. P(G′ , x) is the generating function of perfect matchings on the modified graph G′ with weights x, which is proportional to the Ising partition function on G. A(Dj ) is a weighted, antisymmetrized adjacency matrix for the graph G’, defined by  ∃ an edge e from v to w in orientation Dj  xe : −xe : ∃ an edge e from w to v in orientation Dj A(Dj )v,w =  0: no edge between v and w for all vertices v, w in G′ . The set of 4g orientations that are being summed over are called relevant orientations. ( )gr(Dj ) is a bitstring over 2g variables, and c(r(Dj )) is a scalar factor of ± 12 , where the sign depends on r(Dj ). To compare with our result, we replace the summation over j = 1...4g 76  5.9. Summary with a summation over two sets of g-component bitstrings: α1 ...αg , β1 ...βg , which are in one-to-one correspondence with the possible values of r(Dj ). Then we rewrite Equation 5.41 as: ∑ P(G, x) = cα,β Pf (A(Dα,β )) (5.42) α1 ,α2 ,...αg β1 ,β2 ,...βg  Note that we have ( dropped the possible overall minus sign for brevity. It ∏g (−1)αj ∗βj ) turns out that cα,β = (Definition 3.4 in [19]). Comparing j=1 2 with Equations 5.29 and 5.40 suggests the following one-one correspondence: f (Γα,β ) ↔ Pf (A(Dα,β ))  (5.43)  where the function f (Γα,β ) is either tr(Γα,β ) or ⟨vac|Γα,β |vac⟩ depending on the family of graphs. To make Correspondence 5.43, we need to understand how the bitstrings α and β enter into the orientation Dα,β . The variables α, β are associated with the cuts that one can make to unfold a surface of genus g into a flat 4g-gon [19]. The binary variables α and β are associated with the orientation of edges in G′ that cross these cuts. From the definition of A(Dj ), changing the orientation of an edge has the same effect as flipping the sign of the associated weight. The construction of G′ can be performed in such a way that this is in turn equivalent to flipping the sign of Ising couplings for edges in G that cross the cuts (see for example [1] or [9]). Thus, Correspondence 5.43 is established if the cuts can be chosen along cocycles that are homologous to the ones we encountered when evaluating the Ising partition function on higher genus graphs, shown in Figure 5.4 for square g-graphs and in Figure 4.3 for punctured cylinder graphs. We leave Correspondence 5.43 as a conjecture. We note that the presence of the factor (−1)α·β in both approaches to the Ising model suggests an interesting interplay between the two types of non-trivial cycle associated the handles of a surface. In particular, both our approach and that of [19] distinguish between two types of non-trivial cycle. In our case, the αj bits were associated with cocycles composed of vertical edges, traversing around holes in the surface. The βj bits were associated with the cocycles that pass through holes in the surface, composed of horizontal edges. See [15] for technical definitions of these two types of cycle on a surface. We will see in Chapter 7 that this division of the 2g homology classes into two groups of g will have consequences for the efficient simulatibility of surface code states.  77  Chapter 6  Efficient Simulation of Fermion Gaussian Operators In this Chapter we will consider the task of evaluating the terms in Equations 5.29 and 5.40, which are quantities in the Fock space of N fermionic modes. We will find that the operators Γα,β in these equations belong to a restricted class of fermion operators known as Gaussian operators. Certain computations with Gaussian operators can be performed efficiently in the number of fermionic modes, and we will see that the terms in the Ising partition function are of this type. This provides an exponential speedup over doing explicit matrix algebra with operators in Fock space, which has a dimensionality of 2N . We will see that the Gaussian operators have a close connection with so-called non-interacting fermion Hamiltonians, which are Hermitian operators that are quadratic in Majorana operators: iH ∈ spanR {cj ck ∀j, k ∈ {1...2N }}. We discuss non-interacting fermion operators and their Lie algebraic properties in detail in Appendix F. A fermionic system restricted to evolution under the corresponding unitaries is also referred to as fermionic linear optics, because it is the fermionic analogue of linear optical operations with photons. The efficient classical simulation of fermionic linear optics has also been studied in references [58],[41] and [56], but we will find that the approach due to Brayvi in [3] based on Gaussian operators is best suited to the task at hand.  6.1  Fermion Gaussian Operators  Consider the linear space C2N of operators on a fermion Fock space of N modes. A basis for this vector space is the identity operator along with all products of distinct sets Majorana fermion operators, such that any operator O can be expressed as O = O0 I +  2N ∑  2N ∑  Ox1 ,x2 ,...xM ∗ cx1 cx2 ...cxM  (6.1)  M =1 x1 <x2 <...xM =1  78  6.1. Fermion Gaussian Operators Physical operators preserve the parity of fermion occupation numbers, which is equivalent to M being even for each term in Equation 6.1. Consider now the algebra G2N of 2N distinct Grassman numbers θ1 , θ2 , ...θ2N , where Grassman numbers obey θj θk = −θk θj for all j, k ∈ {1...2N }. Define a mapping ω : C2N → G2N by ω(cx1 ...cxM , θ) = θx1 ...θxM  (6.2)  ω(I, θ) = 1  (6.3)  for any product cx1 ...cxM with no repeating indices. We extend the mapping ω by linearity to accommodate any operator in C2N . Call ω(X, θ) a Grassman representation of an operator X ∈ C2N . For any two operators X, Y ∈ C2N , there exists a result relating the trace of the fermionic operator XY and a Gaussian integral over the Grassman representations of X and Y [3]: ∫ T tr(XY ) = (−2)N DθDµeθ µ ω(X, θ)ω(Y, µ) (6.4) The definition of exponentiation of and integration over Grassman numbers is given in Ref [3], but we omit it here because we will only be interested in a certain simplification of Equation 6.4. Define a Gaussian operator as any operator who’s Grassman representation has a Gaussian form. That is, for an operator X such that tr(X) ̸= 0, we call X a Gaussian operator if i T Mθ  ω(X, θ) = Ce 2 θ  for some complex scalar C and some complex antisymmetric matrix C. For an operator X with vanishing trace, we call X Gaussian if X = limm→∞ Xm for some converging sequence of operators Xm that are Gaussian and have non-vanishing trace. The matrix M is called the covariance matrix of X, and it satisfies tr(X) ∗ Mab = itr(Xca cb ) for a ̸= b. In Appendix D we prove that Gaussian operators provide an operator basis for all physical fermionic operators. If X is a density operator and is Gaussian, we call X a Gaussian state. This is true iff the following three conditions are satisfied: C = 2−N , M is a real matrix, and M T M is a positive matrix with eigenvalues less than or equal to one [3].  79  6.2. Mapping from Coefficient Matrices to Covariance Matrices i T Mθ  Consider two Gaussian operators X and Y: ω(X, θ) = Ce 2 θ i T ω(Y, θ) = De 2 θ Qθ . By Equation 6.4 ∫ i T i T T N tr(XY ) = (−2) C ∗ D ∗ DθDµeθ µ e 2 θ M θ e 2 µ Qµ = (2)N C ∗ D ∗ Pf(Q) ∗ Pf(M − Q−1 )  , and  (6.5)  where the two integrals are evaluated using identities given in [3], and the exponentials are rearranged and combined based on the commutation of the exponents (they are all quadratic in anticommuting variables). Furthermore, it has been assumed that Q−1 exists. The function Pf(M ) for any 2N × 2N antisymmetric matrix M is called the Pfaffian of M, and is defined as Pf(M ) :=  N ∑ ∏ 1 σ (−1) Mσ2j−1 ,σ2j 2N N ! σ∈S2N  j=1  where S2N is the symmetric group on 2N elements and (−1)σ is the signature of the permutation σ. Up to a sign, the Pfaffian √ of a matrix is given by a relationship with the determinant: Pf(M ) = det(M ), which can be computed efficiently in N. Polynomial time algorithms to evaluate Pfaffians directly with the correct sign also exist and are mentioned in [20]. An immediate consequence of Equation 6.5 is that the expectation value of any Gaussian operator X with known covariance matrix M and constant C can be evaluated in any Gaussian state ρ with known covariance matrix Q, in poly(N) time, as tr(ρX) = C ∗ Pf(Q) ∗ Pf(M − Q−1 )  6.2  (6.6)  Mapping from Coefficient Matrices to Covariance Matrices i  ∑  In Appendix D, we prove that any operator O of the form O = e 2 jk Gjk cj ck is a Gaussian operator. In this section, we develop a fairly general method to determine the covariance matrix M corresponding to such an operator O from the coefficient matrix G appearing in the exponential. This generalizes a comment made in [43] regarding Gaussian states (which have real covariance matrices) to Gaussian operators. We will proceed in several steps. ∑ Lemma 6.2.1 Consider an operator V such that V ca V −1 = b Rab cb for −1 some 2N × 2N complex ∑ matrix R. Then for any operator O ω(V OV , θ) = ω(O, η) where ηa = b Rab θb . 80  6.2. Mapping from Coefficient Matrices to Covariance Matrices Proof Express O as O = O0 I +  2N ∑  2N ∑  Ox1 ,...xM cx1 ...cxM  M =1 x1 <x2 <...xM =1  Note that due to the anticommutation of the cxi , we can assume that the tensor Ox1 ,...xM is antisymmetric with respect to any swap of two neighboring indices xi and xi+1 . V OV  −1  = O0 I +  2N ∑  2N ∑  Ox1 ,...xM V cx1 V −1 ...V cxM V −1  M =1 x1 <x2 <...xM =1  = O0 I +  2N ∑  2N ∑  Ox1 ,...xM  = O0 I +  2N ∑  Rx1 ,y1 ...RxM ,yM cy1 ...cyM  y1 ,...yM =1  M =1 x1 <x2 <...xM =1 2N ∑  2N ∑  2N ∑  Ox1 ,...xM (Rx1 ,y1 ...RxM ,yM ) cy1 ...cyM  M =1 y1 ,...yM =1 x1 <x2 <...xM =1  At this point it is to note that we can replace the summation ∑necessary ∑2N 2N y1 ̸=y2 ,...̸=yM =1 . That is, the total contribution from y1 ,y2 ,...yM =1 with terms where any of the y’s are equal to each other is zero. To see this, consider a set of index values y1 ...yM that contains one or more repeats. In this case, we can always find an i < j such that yi = yj for and for no i < k < j does yk = yj . The contribution to the sum for this fixed y1 ...yM is 2N ∑  ) ( Ox1 ,...xM Rx1 ,y1 ...Rxi ,yi ...Rxj ,yj ...RxM ,yM cy1 ...cyi ...cyj ...cyM  x1 <x2 <...xM =1  ∝  2N ∑  ) ( Ox1 ,...xM Rx1 ,y1 ...Rxi ,yi ...Rxj ,yi ...RxM ,yM  x1 <x2 <...xM =1  ) ( Notice that the product Rx1 ,y1 ...Rxi ,yi ...Rxj ,yi ...RxM ,yM is symmetric with respect to interchange of xi and xj . The quantity Ox1 ,x2 ,...xM , however, is antisymmetric with respect to swapping xi and xj because it involves an odd number (2(j-i)-1) of swaps between adjacent indices. Thus, the sum over xi and xj vanishes, and hence the entire sum for this fixed y1 ...yM . So, ∑ V OV −1 = O0 I + Ox1 ,x2 ,...xM (Rx1 ,y1 Rx2 ,y2 ...RxM ,yM ) cy1 cy2 ...cyM  81  6.2. Mapping from Coefficient Matrices to Covariance Matrices where  ∑  :=  ∑2N M =1  ∑2N  y1 ̸=y2 ,...̸=yM =1  ω(V OV −1 , θ) = O0 + = O0 + = O0 +  ∑  ∑  ∑2N  x1 <x2 <...xM =1 .  Thus  (Rx1 ,y1 Rx2 ,y2 ...RxM ,yM ) ω(cy1 cy2 ...cyM , θ) (Rx1 ,y1 Rx2 ,y2 ...RxM ,yM ) θy1 θy2 ...θyM  2N ∑  2N ∑  Ox1 ,x2 ,...xM ηx1 ηx2 ...ηxM  M =1 x1 <x2 <...xM =1  = ω(O, η) Note that the reason it was necessary to establish that the indices y can be assumed to be unique is for the mapping ω(cy1 ...cyM , θ) = θy1 ...θyM to be valid. Corollary 6.2.2 Consider an invertible operator V such that V ca V −1 = ∑ b Rab cb for some 2N*2N ∑matrix R. Then for any operator O ω(O, θ) = ω(V −1 OV, η) where ηa = b Rab θb . ∑ Corollary 6.2.3 Suppose furthermore that V −1 ca V = b Rab cb for some i  ∑  i  ∑  T  R. Consider an operator O = e 2 jk Gjk cj ck . Then ω(O, θ) = ω(e 2 jk [R GR]jk cj ck , η) ∑ ∑ Proof Since V −1 V ca V −1 V = bd Rab Rbd cd = d [RR]ad cd = ca , we coni ∑ clude that R = R−1 . Now using Corollary 6.2.2: ω(O, θ) = ω(V −1 e 2 jk Gjk cj ck V, η) = i ∑ i ∑ −1 −1 T ω(e 2 jk Gjk V cj V V ck V , η) = ω(e 2 jk [R GR]jk cj ck , η). ∑  In Appendix F we show that for any V = e ∑ Rab cb V ca V −1 =  jk  Ajk cj ck  b  where R is given by a matrix exponential: R = e−4A . The matrix A can be taken without loss of generality to be antisymmetric 6 . If it is real, then V is unitary and R is real orthogonal. If A is complex, then V is not generally unitary, but R is still complex orthogonal (RRT = RT R = I) with determinant 1. So, for any A, R ∈ SO(2N, C). Furthermore any SO(2N, C) rotation can be implemented by a product of operators Vj of the above form, because the group SO(2N, C) is connected. Take any matrix R ∈ SO(2N, C). Since SO(2N, C) is connected, 6  Due to the antisymmetry of the cj operators, the only non-vanishing contribution from the symmetric part of A would be from terms like cj cj = I. A term in the exponent of V proportional to the identity will commute through ca and be canceled by the corresponding term in V −1  82  6.2. Mapping from Coefficient Matrices to Covariance Matrices R = eA1 eA2 ...eAn for some antisymmetric matrices A1 ...An (Corollary 3.26 in [26]). So, we can implement R with the fermionic operator V1 ...Vn , where the coefficients in the exponent of Vj are given by the matrix −Aj /4. i  ∑  Lemma 6.2.4 Consider an operator O = e 2 jk Gjk cj ck . If the matrix can be block diagonalized by an SO(2N, C) transformation R as: RGRT i ∑ ∑ T ⊕ l ∏ l jk βjk cj ck , η) where η 2 β , then ω(O, θ) = ω(e a = b Rab θb , and l l is the matrix β l extended to the full 2N∑dimensional space (nonzero only its associated block) so that: RGRT = l β l .  G = βl in  Proof Use Corollary 6.2.3 with V = V1 ...Vn corresponding to the desired R (if R has an antisymmetric matrix logarithm, then simply set A = − 14 ln(R)). We see that ∑ l ∏ i∑ l i ∑ ω(O, θ) = ω(e 2 jk l βjk cj ck , η) = ω( e 2 jk βjk cj ck , η) l  The last equality follows from the fact that β l and β k are never nonzero along the same row or column for l ̸= k. Then use that ω(AB, η) = ω(A, η)ω(B, η) if A and B share no Grassman numbers in their expansions. Now we state the main result of this section: i  ∑  Theorem 6.2.5 Consider an operator O = e 2 jk Gjk cj ck where G is complex antisymmetric and furthermore is diagonalizable. Then (N ) ∏ i T ω(O, θ) = cosh(zl ) ∗ e 2 θ [tan(G)]θ l=1  where ±izl are the eigenvalues of G and tan(G) is defined by diagonalizing and taking functions of eigenvalues. Proof A diagonalizable complex antisymmetric matrix G can be block diagonalized by a SO(2N,C) transformation (see Appendix E) as ) N ( ⊕ 0 zl RGR = −zl 0 T  l=1  where each zl is a∏complex number which could be zero. So, by Lemma 6.2.4: ω(O, θ) = l ω(eizl c2l−1 c2l , η). The mapping ω(eizl c2l−1 c2l , η) can be evaluated explicitly: ω(eizl c2l−1 c2l , η) = ω (cosh(zl ) + i sinh(zl )c2l−1 c2l , η) = cosh(zl )ei tanh(zl )η2l−1 η2l 83  6.2. Mapping from Coefficient Matrices to Covariance Matrices Combining commuting exponents (N ) (N ) ∑N ∏ ∏ i T ω(O, θ) = cosh(zl ) e l=1 i tanh(zl )η2l−1 η2l = cosh(zl ) e 2 θ M θ l=1  where M =R  l=1  [ N ( ⊕ l=1  0 tanh(zl ) − tanh(zl ) 0  )] RT  (6.7)  To prove that M = tan(G), note that the block diagonal form of RGRT can easily be fully diagonalized by the matrix ( ) N ⊕ 1 1 −i √ H := 2 1 i l=1  (6.8)  The matrix H is unitary: H −1 = H † . In fact, H is simply the operator † N which changes basis from Pauli Y to Z operators: H(⊕N l=1 Yl )H = ⊕l=1 Zl . This means that ) N ( ⊕ izl 0 −1 (HR)G(HR) = 0 −izl l=1  So, using Equation 6.7 tan(G) = −i tanh(iG) −1  = −i(HR) = −iRT  ) N ( ⊕ − tanh(zl ) 0 (HR) 0 tanh(zl )  l=1 ( N ⊕ l=1  0 i tanh(zl ) −i tanh(zl ) 0  ) R  = M ∏ A similar sequence of steps shows that the pre-exponential factor N l=1 cosh(zl ) ∏N can be written in terms √ of a determinant: det(cos(G)) = l=1 cosh(zl ) cosh(−zl ), ∏N so l=1 cosh(zl ) = det(cos(G)). The square root introduces a sign ambiguity, ∏N so if the sign is important one must use the less elegant expression l=1 cosh(zl ). Corollary 6.2.6 For any complex antisymmetric matrix G √ i ∑ Gjk cj ck jk 2 tr(e ) = det (e2iG + I) 84  6.2. Mapping from Coefficient Matrices to Covariance Matrices Proof First we note that for any Gaussian operator X with covariance matrix M and coefficient C: tr(X) = 2N ∗ C. This can be proven by inspecting the Grassman representation of X: ω(X, θ) = Ce  iC 2  ∑ jk  Mjk θj θk  =C+  i∑ Mjk θj θk + O(θ4 ) 2 jk  Note that none of the terms contains a product of two Grassman numbers with the same index. Indeed, any such term would be zero. Since the mapping ω is an isomorphism between non-repeating products of Majorana operators and non-repeating products of Grassman numbers, it can be inverted, so we may write X =C ∗I +  iC ∑ Mjk cj ck + O(θ4 ) 2 jk  If one takes the trace of the above expression only the first term will contribute, because the trace of any non-repeating product of Majorana operators is zero. This can be proven by converting all of the Majorana operators back into creation and annihilation operators and taking the trace in the occupation number eigenbasis. So, tr(X) = C ∗ tr(I) = C ∗ 2N . Assume for the moment that G is a diagonalizable matrix. We can then i ∑ use Theorem 6.2.5 with X = 2 jk Gjk cj ck to write tr(X)2 = 22N det(cos(G)) = det(eiG + e−iG ) = det(e−iG ) ∗ det(e2iG + I) = det(e2iG + I)  (6.9)  where we have used that det(e−iG ) = e−itr(G) = 1 since G is antisymmetric. The set of diagonalizable complex antisymmetric 2N × 2N matrices is a dense open subset of the set of all complex antisymmetric 2N ×2N matrices, which is isomorphic to the space CN (2N −1) . One can extend Equation 6.9 to non-diagonalizable matrices using the denseness of diagonalizable matrices in the set of matrices along with the continuity of both sides of Equation 6.9 in the matrix elements of G. Indeed, express an arbitrary antisymmetric matrix as the limit of a converging sequence of antisymmetric diagonalizable matrices, and Equation 6.9 can then be proven in the limit. Thus √ i ∑ G c c j jk k ) = det (e2iG + I) for all complex antisymmetric matrices tr(e 2 jk G. 85  6.3. Lie Algebraic Multiplication Corollary 6.2.7 For any complex antisymmetric matrix G and Gaussian state ρ with covariance matrix Q √ i ∑ Gjk cj ck −N jk 2 tr(ρe )=2 det (−ie2iG Q + iQ − e2iG − I) Proof Again, we will first prove the result in the case that G is diagonalizable. The result can then be extended to all complex antisymmetric G 6.2.6. If G is diagonalizable, then by Theby continuity as in Corollary i ∑ orem√ 6.2.5 the operator e 2 jk Gjk cj ck is a Gaussian state with coefficient C = det(cos(G)) and covariance matrix M = tan(G). From Equation 6.6, we have √ i ∑ tr(ρe 2 jk Gjk cj ck ) = det(cos(G))Pf(Q)Pf(tan(G) − Q−1 ) Since for an antisymmetric matrix the Pfaffian is a square root of the determinant, and by the multiplicity property of determinants, we have i  tr(ρe 2  ∑ jk  ) = det (cos(G) tan(G)Q − cos(G))  Gjk cj ck 2  Now using cos(G) tan(G) = sin(G), 2 cos(G) = eiG + e−G and 2 sin(G) = −i(eiG − e−iG ), we rewrite i  tr(ρe 2  ∑ jk  Gjk cj ck 2  )  ( ) = 2−2N det −ieiG Q + ie−iG Q − eiG − e−iG ( ) = 2−2N det(e−iG )det −ie2iG Q + iQ − e2iG − I  Again, det(e−iG ) = etr(iG) = 1 because G is antisymmetric. This proves the result in diagonalizable case, which can then be extended to the general case by taking the limit of a converging sequence of diagonalizable matrices. i  ∑  Corollaries 6.2.6 and 6.2.7 allow simple expressions for tr(ρe 2 jk Gjk cj ck ) i ∑ G and tr(e 2 jk jk cj ck ) in terms of the matrix G, at the expense of an overall sign ambiguity from the square root.  6.3  Lie Algebraic Multiplication  The trace result of Equation 6.5 allows one to perform computations involving Gaussian operators and Gaussian states in poly(N) time, if one knows the relevant covariance matrices and pre-exponential coefficients. Consider a product of two Gaussian operators X and Y. In [3], it is proved that the operator XY is also a Gaussian operator. However, finding the covariance 86  6.3. Lie Algebraic Multiplication matrix for a product of Gaussian states is not a simple matter of summing together the covariance matrices for each. This is again due to the fact that the mapping ω is not generally a homomorphism, i.e. ω(X)ω(Y ) ̸= ω(XY ) in general for fermionic operators X and Y. Thus to find the covariance matrix of the product of two Gaussian operators, it appears that we must do the operator multiplication explicitly before using Theorem 6.2.5. The cost of doing this will in general scale polynomially with the size of the Hilbert space: 2N , which undermines the efficiency of using the Gaussian operator formalism for practical calculations. In Corollaries 6.2.6 and 6.2.7, we found a way to use Equation∑6.4 to do i calculations with fermion Gaussian operators of the form X = e 2 jk Gjk cj ck using their coefficient matrices G instead of their covariance matrices. It turns out that this representation allows us to circumvent the problem of updating the covariance matrix. The following results come from the fact that the operators cj ck form a representation of a Lie algebra: namely a 2N dimensional representation of SO(2N, C), the complexification of the Lie algebra of 2N × 2N orthogonal matrices. A brief introduction to Lie algebras, and in particular the Lie algebra of quadratic fermion operators is provided in Appendix F. Consider an arbitrary product of exponentials of quadratic fermion operators k k ∏ ∏ j i ∑2N Γ= Γj := Dj e 2 ab=1 Gab ca cb j=1  j=1  It can be seen with some algebra that the commutator of two quadratic fermion operators is a linear combination of quadratic fermion operators. Given this fact, the Baker-Campbell-Haussdorf theorem implies that the set of operators that are exponentials of quadratic fermion operators is closed under multiplication, and thus Γ can be written in the same form as each i ∑ Gjk cj ck jk 2 , where of the Γj . In other words, for some matrix G: Γ = De ∏ D = kj=1 Dj . Our task then is to find a suitable matrix G. Operators of the form Γj form with real coefficients belong to a representation of the Lie group SO(2N,R), and operators of this form with complex coefficients belong to a representation of SO(2N,C). The defining representation of the Lie group SO(2N,C) is the set of all orthogonal 2N ×2N matrices with complex entries, and the associated Lie algebra SO(2N, C) is the set of all 2N × 2N antisymmetric matrices. This vector space is spanned by 87  6.3. Lie Algebraic Multiplication the matrices T¯ab := 2(|a⟩⟨b| − |b⟩⟨a|), and we take these as generators of the defining representation of SO(2N, C). These matrices have the commutator ∑ ef [T¯ab , T¯cd ] = −2(δac T¯bd − δad T¯bc + δbd T¯ac − δbc T¯ad ) := ifab,cd T¯ef ef  Now consider the commutator of the quadratic Majorana fermion operators, where we take our generators to be Tab := ca cb . With some algebra ∑ ef ifab,cd Tef [Tab , Tcd ] = −2(δac Tbd − δad Tbc + δbd Tac − δbc Tad ) = ef ef The so-called structure constant tensor fab,cd is identical in the two cases, N despite the fact that the Tab are matrices in a 2 dimensional vector space while the T¯ab are matrices in a 2N dimensional vector space. In other words, the generators Tab and T¯ab are two representations of the same Lie algebra. A multiplication rule for matrix exponentials is given by the Baker Campbell Hausdorff formula, which gives a solution to ln(eA eB ) as an infinite series over nested commutators of A and B. With A and B set to SO(2N, C) generators  ∑ 1 1 ¯ ¯ BCHab,cd,ef T¯ef ln(eTab eTcd ) = T¯ab +T¯cd + [T¯ab , T¯cd ]+ [T¯ab , [T¯ab , T¯cd ]]+... := 2 12 ef  where the tensor of coefficients BCHab,cd,ef is just shorthand for the above infite series. From the linearity of commutators then, for arbitrary matrices α and β ∑  e  ab  ∑ ¯ αab T¯ab cd βcd Tcd  e  ∑  =e  ∑  ab  ∑ cd  ef  αab βcd BCHab,cd,ef T¯ef  The coefficients BCHab,cd,ef depend only on the structure constants of the SO(2N, C) matrices, so the fermionic representation of SO(2N,C) has the analogous multiplication rule ef fab,cd  ∑  e  ab  αab Tab  ∑  e  cd  βcd Tcd  ∑  =e  ∑  ab  ∑ cd  ef  αab βcd BCHab,cd,ef Tef  If one could sum the infinite series given by the BCH formula, then repeated application of the above would yield a matrix G(BCH) such that i  Γ = De 2  ∑ jk  G(BCH)jk cj ck  (6.10)  88  6.4. Synthesis: Characterizing a Product of Gaussian Operators 7 . The matrix G(BCH) is We may assume G(BCH) to be antisymmetric i ∑ not the only matrix G satisfying Γ = De 2 jk Gjk cj ck , because the matrix logarithm is a multivalued function. Furthermore, we have no gaurantee that the BCH formula actually converges to give a definite matrix G(BCH). Nevertheless, Equation 6.10 still holds if we let G(BCH) stand in for the infinite series over nested commutators of the cj ck , because the BCH formula holds as a formal power series. ¯ := ∏k Γ ¯ Now consider the 2N*2N matrix Γ j=1 j where the barred matri¯ ces are exponentials of sums of the Tab with the same coefficients on the generators as their fermionic counterparts Γj :  ¯ j := Dj e 2i Γ  ∑2N ab=1  Gjab T¯ab  Since the T¯ab generators have the same structure constants as the fermion generators, it must be the case that ¯ = De 2i Γ  ∑ jk  G(BCH)jk T¯jk  Note that for any antisymmetric matrix G ∑ ∑ i∑ Gjk |j⟩⟨k| = 2iG Gjk (|j⟩⟨k| − |k⟩⟨j|) = 2i Gjk T¯jk = i 2 jk  jk  jk  ¯ = e2iG(BCH) . So, Γ  6.4  Synthesis: Characterizing a Product of Gaussian Operators  Suppose we have an operator Γ which is product of exponentials of quadratic fermion operators with known coefficient matrices Gj as in Section 6.3. The results of Sections 6.2 and 6.3 can be combined in order to completely determine the Grassman representation of Γ. From the considerations of Section 6.3, we know that for ∑ a product of quadratic fermion exponentials j ∏k 2N i Γ = j=1 Γj , with Γj := Dj e 2 ab=1 Gab ca cb   k ∏ i ∑2N Γ= Dj  e 2 ab=1 Gab ca cb j=1 mn 2 This can be proven as follows: fij,kl nl −δik δmj δnl +δjl δmk δni −δil δmk δnj ). ∑= i (δjk δmi δmn 4 Using the antisymmetry of α and β, ijkl αij βkl fij,kl = i ([αβ]mn − [αβ]nm ), which is antisymmetric with respect to m and n. The matrix of coefficients given by the BCH formula is a sum of terms of this form, so it is a sum of antisymmetric matrices. 7  89  6.4. Synthesis: Characterizing a Product of Gaussian Operators ¯ = e2iG , where where the matrix G is antisymmetric and satisfies Γ ∏ k ¯ ¯ = Γ j=1 Γ (here we are simplifying notation by replacing G(BCH) from Section 6.3 with G). Due to the multivalued nature of the matrix logarithm, ¯ In particular, we know each of we cannot fully infer the matrix G from Γ. its eigenvalues only up to a non-trivial addition of some integer times π to its imaginary part. The challenge then is to determine the Grassman representation of Γ without complete information about G. If G is diagonalizable, then Theorem 6.2.5 allows us to write the Grassman representation of Γ as   k N ∏ ∏ i ∑2N ω(Γ, θ) =  Dj ∗ cosh(zl ) e 2 ab=1 tan(G)ab θa θb j=1  l=1  where ±izl are the eigenvalues of the matrix G. Let P be the diagonalizing matrix of G such that G = P diag(λl )P −1 where λ2l−1 = ±izl , ¯ which has eigenvalues of e2iλl . λ2l = ∓izl . Then P also diagonalizes Γ, 2iλ From P and e l we can compute ) ( 2iλl e −1 −1 ¯ + I)(Γ ¯ − I)−1 P = i(Γ tan(G) = iP diag e2iλl + 1 ¯ So we may compute the covariance matrix tan(G) from the matrix Γ, without explicit knowledge of G. Unfortunately, the situation is more com∏N plicated for the pre-exponential factor l=1 cosh(zl ). Since we know only ∏ e2iλl and not eiλl , computing N l=1 cosh(zl ) using the matrix Γ could lead to an overall minus sign error, if we took the wrong square root in inferring some of the eiλl . Note that if we are not concerned with this possible overall minus sign, then √ Corollary 6.2.6 allows straightforward computation of ∏ N −N ¯ + I) in terms of Γ. ¯ det(Γ l=1 cosh(zl ) = 2 Fortunately we can determine tr(Γ) ∏Nwith the correct minus sign, despite the uncertainty regarding the sign of l=1 cosh(zl ). The price to pay is that the process is iterative and less straightforward. The trick is to combine the Lie algebraic multiplication method with the trace result for Gaussian operators from Section 6.1. If we allow M (X) to denote the covariance matrix of a Gaussian operator X, and C(X) to denote its coefficient, we can write the Gaussian integral trace result (Equation 6.5) as tr(XY ) = tr(X) ∗ C(Y ) ∗ Pf(M (Y )) ∗ Pf(M (X) − M (Y )−1 )  (6.11)  where X and Y are Gaussian operators. We have used that the coefficient of a Gaussian operator is related to its trace by tr(X) = 2N ∗ C (see 90  6.5. Application to Ising Model Corollary 6.2.6). Equation 6.11 provides a rule for updating the trace of a Gaussian operator upon multiplication by another Gaussian operator. If each of the coefficient matrices Gj is known, this equation allows us to iteratively calculate the pre-exponential coefficient of a product Γ of Gaussian operators. The result is that C(Γ) = Ek , and M (Γ) = Mk , where { El−1 ∗ Cl ∗ Pf(tan(Gl )) ∗ Pf(Ml−1 − tan(Gl )−1 ) : l > 1 El = C1 :l=1 with the definitions ¯ l + I)(Γ ¯ l − I)−1 Ml := i(Γ ¯ l := Γ  l ∏  ¯j Γ  j=1  and Cl :=  N ∏  cosh(z(Gl )j )  j=1  where ±iz(Gl )j are the eigenvalues of the coefficient matrix Gl . We note that in practice it would make the most sense numerically to use Corollaries 6.2.6 and 6.2.7 to find tr(Γ) or ⟨vac|Γ|vac⟩ up to a sign, and then use the above method to find the correct sign.  6.5  Application to Ising Model  In Chapter 5, we reduced the problem of computing the 2D Ising Model partition function on a square, cylindrical, toroidal, square g-graph, or punctured cylinder graph to computing the trace or a matrix element of of a fermionic operator. In all cases, this operator is an exponential of a quadratic fermion operator and is thus a Gaussian operator. In the square lattice case, each transfer matrix has a form (see Section 5.3) k  V =  N −1 ∏  e−iJjk (c2j c2j+1 )  j=1  k  H =  N ∏  Ajk e−iγjk (c2j−1 c2j )  i=1  91  6.5. Application to Ising Model Considering each factor separately, one can show that the Grassman representations of the horizontal transfer matrices are N ∏  ω(H k , θ) =  j=1   =   N ∏  Ajk ω(e−iγjk c2j−1 c2j , θ) =  j=1    N ∏  Ajk cosh(γjk )e−i tanh(γjk )θ2j−1 θ2j  T Ajk cosh(γjk ) e 2 θ Bθ i  j=1  ∑ where B = N j=1 tanh(γjk ) (|2j⟩⟨2j − 1| − |2j − 1⟩⟨2j|). The general vertical transfer matrices defined by Equation 5.26 for the Ising model on a square g-graph and Equation 5.37 for a punctured cylinder or torus graph have a slightly more complicated structure, but they still have a 2x2 block diagonal structure. In particular, they can all be written in the form N ∏ V = e−iJj c2j c2P(j)+1 j=1  where P is bijective map: P : {1, 2, ...N } → {0, 1, ...N − 1} and Jj is a complex coefficient. P and the coefficients Jj can be simply read off from Equation 5.26. Similar to the horizontal transfer matrices ω(V, θ) =  N ∏  ω(e  −iJj (c2j c2P(j)+1 )  j=1  =  (N −1 ∏  , θ) =  N ∏  cosh(Jj )e−i tanh(Jj )θ2j θ2P(j)+1  j=1  ) i T Aθ  cosh(Jj ) e 2 θ  i=1  ∑ where A = N j=1 tanh(Jj ) (|2P(j) + 1⟩⟨2j| − |2j⟩⟨2P(j) + 1|). Since the property of being a Gaussian operator is closed under operator multiplication, an arbitrary product Γ of transfer matrices is also a Gaussian operator. ¯ The “barred” transfer matrices To characterize it, we will need the matrix Γ. are, explicitly ¯k = H  N ∏  Ajk e  j=1    =   N ∏ j=1  −iγjk T¯2j−1,2j   Ajk   =  N ∏  e−2jγjk (|2j−1⟩⟨2j|−|2j⟩⟨2j−1|)  j=1 N ( ⊕ j=1  cosh(2γjk ) i sinh(2γjk ) −i sinh(2γjk ) cosh(2γjk )  )  92  6.5. Application to Ising Model  V¯  =  N ∏ j=1  =  N ∑  e−iJj T2j,2P(j)+1 = ¯  N ∏  e−2iJj (|2j⟩⟨2P(j)+1|−|2P(j)+1⟩⟨2j|)  j=1  cosh(2Jj ) (|2j⟩⟨2j| + |2P(j) + 1⟩⟨2P(j) + 1|)  j=1  + i sinh(2Jj ) (|2P(j) + 1⟩⟨2j| − |2j⟩⟨2P(j) + 1|) ¯ α,β corresponding to a term Γα,β as in Equation 5.32 is The matrix Γ then given by matrix multiplication of such 2N*2N matrices, which can be ¯ α,β up done in poly(N) time. The quantity tr(Γα,β ) can be computed from Γ to a sign using Corollary 6.2.6 √ ( ) ¯ α,β + I tr(Γα,β ) = det Γ If the explicit sign is required, one can use the method of Sections 6.4 to compute tr(Γα,β ). For the Ising partition function on a torus or square g-graph each term of Equation 5.29 to be computed efficiently in N and M. There are 22g such terms to compute, so to compute the partition function on such a graph, the overall scaling is 22g ∗ poly(N, M ). The 2D classical Ising model partition function on an N*M square lattice with ordinary boundary conditions has a slightly different structure from the Ising model on a lattice with periodic boundary conditions. Instead of being proportional to the trace of a Gaussian operator or a sum of such terms, it is proportional to the single Gaussian operator matrix element, as given by Equation 5.11 Z = 2N ⟨vac|Γ|vac⟩ where Γ is the product of transfer matrices: Γ := V 1 H 1 ...V M −1 H M −1 V M . We can rewrite Z as Z = 2N ⟨vac|Γ|vac⟩ = 2N ∗ tr (|vac⟩⟨vac|Γ) The operator |vac⟩⟨vac| is a Gaussian state. Its covariance matrix can be found by writing it as |vac⟩⟨vac| = a1 a†1 ...aN a†N (I + ic1 c2 ) (I + ic3 c4 ) (I + ic2N −1 c2N ) = ... 2 2 2 93  6.5. Application to Ising Model Since no two factors in the above product contain any of the same indices ω(|vac⟩⟨vac|, θ) =  N ∏  ( ω  j=1  = 2  −N  1 (I + ic2j−1 c2j ), θ 2  N ∏  eiθ2j−1 θ2j  j=1 ∑N  where Q =  ∑N  )  = 2−N e  j=1  = 2−N e  i T θ Qθ 2  iθ2j−1 θ2j  j=1 |2j −1⟩⟨2j|−|2j⟩⟨2j −1|.  In fact, this result generalizes i T  −N 2 θ Q(α)θ , where to any occupation number pure ∑Nstate as:αjω(|α⟩⟨α|, θ) = 2 e α is a bitstring and Q(α) = j=1 (−1) (|2j − 1⟩⟨2j| − |2j⟩⟨2j − 1|). It can be verified that Q−1 = −Q. Also, by the Pfaffian identity Pf(A ⊕ B) = Pf(A) ∗ Pf(B)  ( ( ))N 0 1 Pf(Q) = P f = 1N = 1 −1 0 To evaluate the partition function, all we need is coefficient C and the correlation matrix M for the operator Γ. Then by Equation 6.6 Z = 2N ∗ C ∗ Pf(M + Q) If we are interested in Z only up to a minus sign, then we may use Corollary 6.2.7 to write this as √ ( ) ¯ + iQ − Γ ¯−I Z = D det −iΓQ ∏ ∏N −1 where D := N j=1 k=1 Ajk . If the explicit sign is required, it can be determined in poly(N,M) time using the method in Section 6.4. For punctured cylinder graphs, the partition function is given in Equation 5.40 as a summation over 22g terms, each of which takes the form ⟨vac|Γα,β |vac⟩. To compute the partition function on such a graph, the overall scaling is 22g ∗ poly(N, M ).  94  Part III Surface Code MBQC as a Fermion-Qubit Problem  95  Chapter 7  Qubits in the Ising Model Partition Function 7.1  Introduction  In Section 5.8 we found that the 2D classical Ising model partition function on a punctured cylinder graph can be written as 1 ∑ (−1)α·β ⟨vac|Γα,β |vac⟩ 2g  Z=  (7.1)  α,β  ∑  where α · β := j αj ∗ βj and α and β are each bitstrings of length g. The integer g is the genus of the graph G and Γα,β is a product of exponentials of quadratic fermion operators. In Chapter 6, we found that such an operator Γα,β is a fermion Gaussian operator, and we provided a means of computing ⟨vac|Γα,β |vac⟩ (or tr (Γα,β ), in the case of periodic horizontal boundary conditions) in a time that is polynomial in the size of G. In this section, we show how one can interpret the binary degrees of freedom in αj and βj as the encoded qubits of a surface code, and how Equation 7.1 can be interpreted as a computation involving a system of N fermionic modes entangled with these 2g qubits. Consider a Hilbert space H which is the tensor product of a N-mode fermion Fock Space and a 2g qubit Hilbert space. We associate the binary variable αj with the qubit labeled 2j −1, and the binary variable βj with the qubit labeled 2j. Then Equation 7.1 can be rewritten as a matrix element of an operator in H   g ∑∏ Z = 2g ⟨vac ⊗ +|  (−1)αj ∗βj |α, β⟩⟨α, β| ⊗ Γα,β  |vac ⊗ +⟩  = 2g ⟨vac ⊗ +|   α,β j=1 g ∏    CZ (2j − 1, 2j) CΓ|vac ⊗ +⟩  (7.2)  j=1  96  7.1. Introduction where |α, β⟩ denotes states in a Pauli Z-eigenbasis: Z2j−1 |α, β⟩ = (−1)αj |α, β⟩ and Z2j |α, β⟩ = (−1)βj |α, β⟩, CZ(j, k) is the controlled phase operator between the qubits j and k, for example CZ(2j − 1, 2k − 1)|α, β⟩ = (−1)αj ∗αk |α, β⟩ and CΓ is a “controlled Gaussian” operation on the fermions, implementing Γαβ on the fermions iff the qubits are in state |α, β⟩: ∑ |α, β⟩⟨α, β| ⊗ Γα,β CΓ := α,β  =  ∏  CΓα,β  α,β  ( ) ∑ where CΓα,β := |α, β⟩⟨α, β| ⊗ Γα,β + (γ,ρ)̸=(α,β) |γ, ρ⟩⟨γ, ρ| ⊗ I is an operator that implements Γαβ on the fermions if the qubits are in state |α, β⟩, and does nothing otherwise. For graphs that we considered in Chapter 5 with horizontal period boundary conditions, such as a torus or a square g-graph, the analogue of Equation 7.1 was 1 ∑ Z= g (−1)α·β tr (Γα,β ) (7.3) 2 α,β  and the analogue of Equation 7.2 is    g ∏ 1  Z = tr CZ (2j − 1, 2j) CΓ 2g  (7.4)  j=1  To accommodate both forms simultaneously, define the scalar valued matrix function: { ⟨vac|Γα,β |vac⟩ : if G is a punctured cylinder graph f (Γα,β ) = 1 tr (Γα,β ) : if G is a square g-graph or punctured torus graph 22g Then for all of the graphs that we considered 1 ∑ (−1)α·β f (Γα,β ) Z= g 2  (7.5)  α,β  We will focus exclusively on these families of graphs in the remainder of this chapter. However, since the Ising model on each family takes the common form of Equation 7.5, it seems a reasonable conjecture that Equation 7.5 has considerably more generality than this. See Section 5.9 for further discussion of this point. 97  7.2. The Ising Bits as Surface Code Qubits  7.2  The Ising Bits as Surface Code Qubits  Now we will argue that the qubits αj , βj in Equation 7.2 can be thought of as the encoded logical qubits on a surface code which is defined on the same graph G as the Ising model. In Chapter 2, we showed that for the surface code on a graph G cellularly embedded on a surface S, we have Equation 2.12 ( 2 ) ∏ ( ( γ,ρ ) ) 2 ∑ e∈E Sqrt ae − be ∗ −1 b √ ⟨ψ|ϕ⟩ = cγ,ρ Z {T anh } (7.6) a 2|V | |E0 (G)| γ,ρ ∏ where |ϕ⟩ is an arbitrary product state |ϕ⟩ = e∈E ae |0⟩+be |1⟩ (arbitrary up to a phase choice, see Section 2.5) ∑ and |ψ⟩ is an arbitrary state in the code space, with expansion |ψ⟩ = γ,ρ cγ,ρ |Xγ,ρ ⟩. γ and ρ are bitstrings of length g, and the 2g logical qubits are associated with the 2g non-trivial homology classes of cycles on S. The modified coefficients bγ,ρ e are obtained ′ by multiplying be by −1 for any edge e ∈ C2j−1 such that γj = 1 and for ′ such that ρ = 1. If an edge e occurs on more than one any edge e ∈ C2j j such cocycle Ck′ , be needs to be multiplied by −1 once for each. Let us adopt the following notation for Equation 7.5: ( ( ) ) ( ) 1 ∑ b b = g (−1)α·β f Γ Z (7.7) a 2 a α,β α,β  where Γ  (b) a α,β  := Γα,β . Then we can combine Equations 7.6 and 7.7 as:  ⟨ψ|ϕ⟩ = N  ∑ α,β,γ,ρ  c∗γ,ρ (−1)α·β f  ( ( ) ) bγ,ρ Γ a α,β  (7.8)  which is a summation over 24g terms, each of which is efficiently computable. We have defined the coefficient ( 2 ) ∏ 2 e∈E Sqrt ae − be √ N := 2g 2|V | |E0 (G)| The bitstrings γ, ρ in Equation 7.8 have the effect of multiplying by −1 the Ising couplings along the non-trivial cocycles Ck′ of G. Indeed, if for any edge, ( ) be → −be with ae constant, the corresponding coupling Je = T anh−1 abee  will also flip sign because T anh−1 (z) is an odd function of z (based on the branch choice defined in Section 2.5). Similarly, the bitstrings α and β  98  7.2. The Ising Bits as Surface Code Qubits enter into the definition of the operator Γα,β as specifying whether or not to multiply by −1 the Ising couplings along a certain set of non-trivial cocycles of G. These cocycles are shown in Figure 5.4 for square g-graphs and in Figure 4.3 for punctured cylinder graphs. We will call the particular cocycles occurring in these two cases the Ising cocycles. We note that different Ising cocycles may be obtainable using the combinatoric approach to the partition function in [19]. Suppose the cocycles chosen to define the encoded operators of the surface code were pairwise homologically equivalent (on the dual lattice) to the Ising cocycles. Then flipping the bit γj in Equation 7.8 could be exactly undone by flipping the value of the bit αj in Equation 7.8. Indeed, for any edge e, multiplying be by −1 twice leaves it unchanged. The cycles need only be homologically equivalent for this to be true because ∏ the surface code space is invariant under action by the vertex operators e∈δs Ze for any s ∈ V , as we saw in Section 2.3. Thus in the notation of Equation 7.8: ( γ,ρ ) ( ) b b Γ = Γα⊕γ,β⊕ρ (7.9) =Γ γ,ρ a a α⊕γ,β⊕ρ α,β We can then combine Equations 7.6 and 7.7 as ∑ c∗γ,ρ (−1)α·β f (Γα⊕γ,β⊕ρ ) ⟨ψ|ϕ⟩ = N α,β,γ,ρ  = N  ∑  c∗γ,ρ (−1)(α⊕γ)·(β⊕ρ) f (Γα,β )  (7.10)  α,β,γ,ρ  where the second line follows from re-labeling the summation over the bitstrings α and β: (α, β) → (α ⊕ γ, β ⊕ ρ). To make Equation 7.9 simple to appreciate, we made the assumption that the surface code cocycles are pairwise homologically equivalent to the Ising cocycles. But it turns out this assumption is not necessary. If the surface code encoding cocycles Ck′ are any set of 2g homologically independent cocycles, then an equation analogous to Equation 7.9 will hold. Let us denote the Ising cocycles by Dk′ . Then there exists a binary linear transformation relating Ck′ and Dk′ ; for some 2gx2g matrix B with entries in the binary field Z2 : ⊕ Ck′ Blk Dl′ l  where the bitwise summation operator indicates the symmetric difference of sets, and denotes homological equivalence. The matrix B gives a basis 99  7.2. The Ising Bits as Surface Code Qubits  Figure 7.1: Two bases Ck′ and Dk′ for the first homology group on a double torus.  transformation for the first homology group on S, and is invertible over Z2 . Then analogous to Equation 7.9 we have: ( γ,ρ ) b Γ = Γα⊕Eγ⊕Hρ,β⊕Gγ⊕F ρ (7.11) aγ,ρ α,β if we denote B with the block matrix form ( ) E H B= G F where we have arranged the cocycle indices as (odd, even) and E, F, G and H are gxg matrices with entries from Z2 . Figure 7.1 shows an example of two bases Ck′ and Dk′ for the first homology group on a double torus. No graph is drawn, but one can imagine the curves as cocycles on some graph embedded on this surface. In this case, the basis transformation B does not mix the even and odd numbered cocycles Ck′ , so H = G = 0 and B decomposes as a direct sum of the matrix E acting on the odd numbered cocycles and F acting on the even numbered cocycles. Both E and F are then invertible over Z2 : ( ) ( ) 1 0 1 1 E= , F = 0 1 0 1 . 100  7.2. The Ising Bits as Surface Code Qubits Another example is shown in Figure 5.5. For any Z2 -invertible matrix B, Equation 7.10 generalizes as ∑ ⟨ψ|ϕ⟩ = N c∗γ,ρ (−1)(α⊕Eγ⊕Hρ)·(β⊕Gγ⊕F ρ) f (Γα,β ) (7.12) α,β,γ,ρ  While Equation 7.12 is generally a summation over 24g terms, we can isolate a single term by considering the following state: |C α,β ⟩ :=  1 ∑ (−1)α·β+(α⊕Eγ⊕Hρ)·(β⊕Gγ⊕F ρ) |Xγ,ρ ⟩ 2g γ,ρ  (7.13)  The state |C α,β ⟩ is the encoding into the surface code of the 2g qubit state |C α,β (E, F, G, H)⟩, where |C α,β (E, F, G, H)⟩ =  1 ∑ (−1)α·β+(α⊕Eγ⊕Hρ)·(β⊕Gγ⊕F ρ) |Xγ,ρ ⟩ 2g γ,ρ  But while the state |C α,β (E, F, G, H)⟩, depends explicitely on the choice of homology basis for the surface code cocycles, the state |C α,β ⟩ is independent on this choice. This is because the encoded X-eigenstates |Xγ,ρ ⟩ also depend on E, F, G, H, as   g ∏ ∏ ∏ ρ γ  |Xγ,ρ ⟩ = Ze j  |K(G)⟩ Ze j j=1  =  g ∏ j=1  ′ e∈C2j−1     ∏  ′ e∈D2j−1  ′ e∈C2j  (Eγ⊕Hρ)j  Ze  ∏   (Gγ⊕F ρ)j   Ze  |K(G)⟩  ′ e∈D2j  Let |XIsing,γ,ρ ⟩ denote the encoded X-eigenstates in the simple case in which the surface code cocycles Ck′ are chosen to be the same as the Ising cocycles, so G = H = 0 and E = F = Ig , where Ig is the gxg identity matrix. Then from the above it follows that |Xγ,ρ ⟩ = |XIsing,(Eγ⊕Hρ),(Gγ⊕F ρ) ⟩, so |C α,β ⟩ = =  1 ∑ (−1)α·β+(α⊕Eγ⊕Hρ)·(β⊕Gγ⊕F ρ) |XIsing,(Eγ⊕Hρ),(Gγ⊕F ρ) ⟩ 2g γ,ρ 1 ∑ (−1)α·β+(α⊕γ)·(β⊕ρ) |XIsing,γ,ρ ⟩ (7.14) 2g γ,ρ 101  7.2. The Ising Bits as Surface Code Qubits regardless of the matrices E, F, G and H. To move from the first to the second line, we have used the invertiblity of the matrix B. We will use this fact again, so we pause here to make it explicit. Let ( ) J M −1 B = L K be the inverse of the matrix B over the field Z2 , in the same notation that we used for the block structure of B. For any function f (γ, ρ) ∑ of two g-component bitstrings γ and ρ, it is clear that f (γ, ρ) = γ,ρ ∑ −1 T −1 merely permutes γ,ρ f (B (γ, ρ) ), because the nonsingular matrix B the order of the symmetric summation over all bitstrings (γ, ρ) ∈ {0, 1}⊗2g . Then Equation 7.14 is proven by making this substitution and using that EJ + HL = GM + F K = Ig and EM + HK = GJ + F L = 0. The overlap between the |C α,β ⟩ states and a product state can be evaluated using Equation 7.12, and is proportional to the α, β term in Equation 7.5 for the Ising Model partition function: ⟨C α,β |ϕ⟩ =  N ∑ (−1)α·β+(α⊕Eγ⊕Hρ)·(β⊕Gγ⊕F ρ)(δ⊕Eγ⊕Hρ)·(ϵ⊕Gγ⊕F ρ) f (Γδ,ϵ ) 2g γ,ρ,δ,ϵ  =  ∑ N ∑ δ·ϵ (−1) f (Γ ) (−1)(Gγ⊕F ρ)·(α⊕δ)+(Eγ⊕Hρ)·(β⊕ϵ) δ,ϵ 2g γ,ρ δ,ϵ  = 2 N (−1)α·β f (Γα,β ) g  (7.15)  We move from the second to third equality in two steps. First, we again use the invertibility of the matrix B over Z2 to relabel the summand (Gγ ⊕ F ρ) → γ and (Eγ ⊕ Hρ) → ρ. Then we use the following identity, which can be verified directly8 . For any g-component bit strings α and β ∑ (−1)α·γ+β·ρ = 22g δ(α,β),(0,0) (7.16) γ,ρ  where the Kronecker delta δ(α,β),(0,0) is equal to one if α and β are both strings of all zeroes, and is zero otherwise. If the surface code cocycles are chosen as the same as the Ising cocycles, then the states |C α,β ⟩ are the encoding of the states |C α,β ⟩ := 8  To prove identity 7.16 and the later identity 7.19, it is convenient to break exponentials of dot products into products of exponentials, one for each component. Then one can replace the summation over∑the resulting products ∏ by( a product of sums. In the case of) identity 7.16, for example, γ,ρ (−1)α·γ+β·ρ = gj=1 1 + (−1)αj + (−1)βj + (−1)αj +βj and the identity can now be easily verified by considering each possible pair (αj , βj ).  102  7.2. The Ising Bits as Surface Code Qubits |C(Ig , Ig , 0, 0)α,β ⟩. With this choice of encoding, the |C α,β ⟩ can be generated by a simple encoded quantum circuit from the logical |α, β⟩ Z-eigenstates. In particular:   g ( ) ∏ |C α,β ⟩ =  CZ(2j − 1, 2j)H2j−1 H2j CZ(2j − 1, 2j)  |α, β⟩ j=1  (7.17) where Hk is the Hadamard gate on the k th qubit: ( ) 1 1 1 H := √ 2 1 −1 As a quantum circuit, Equation 7.17 looks like:  Figure 7.2: A quantum circuit preparing the state |C α,β ⟩ in the case of g=2.  Note that this circuit only generates entanglement between blocks of two qubits (2j − 1, 2j). The state |C α,β ⟩ is a tensor product of g Bell pairs, up to a local unitary transformation. The 22g encoded states |C α,β ⟩ form an orthonormal basis of the surface code space. To prove their orthonormality, one can use identity 7.16 and the orthonormality of the |Xγ,ρ ⟩ states (see Section 2.4). Since evaluating the local overlaps of the states |C α,β ⟩ can be done in a number of steps that scales as a polynomial in g, and the 22g such states form a basis of the code space of the surface code, we can evaluate any local overlap in the code space in a number of steps that scales as 22g instead of 24g , if the∑ expansion of an C |C α,β ⟩. α,β arbitrary state |ψ⟩ in the |C ⟩ basis is known. Let |ψ⟩ = α,β ψα,β Then by Equation 7.15 ∑ C ⟨ψ|ϕ⟩ = 2g N ψα,β (−1)α·β f (Γα,β ) (7.18) α,β  103  7.3. Quantum Circuit Interpretation Equation 7.18 can be evaluated in a number of steps that is upper bounded by poly(N, M, g) ∗ 22g , since for each (α, β), f (Γα,β ) can be computed in poly(N, M ) time (see Chapter 6).  7.3  Quantum Circuit Interpretation  In the case of a punctured cylinder graph, we can rewrite Equation 7.12 in the notation of Equation 7.2 as a matrix element of an operator in the combined Hilbert space of N fermionic modes and 2g qubits. First, consider an operator χ in the Hilbert space of 2g qubits with matrix elements in the Pauli-Z eigenbasis: ⟨γ, ρ|χ|α, β⟩ = 21g (−1)(α⊕Eγ⊕Hρ)·(β⊕Gγ⊕F ρ) . Then by Equation 7.12: ∑ ⟨ψ|Xγ,ρ ⟩⟨γ, ρ|χ|α, β⟩⟨vac|Γα,β |vac⟩ ⟨ψ|ϕ⟩ = 2g N α,β,γ,ρ  = 2g N ⟨ψ|Hχ  ∑  |α, β⟩⟨vac|Γα,β |vac⟩  α,β  = 22g N ⟨vac ⊗ ψ| (IF ⊗ Hχ) CΓ|vac ⊗ +⟩ where IF is the identity operator on the fermionic space, |+⟩ = |X0,0 ⟩ is the mutual +1 eigenvector of the 2g qubit ∑Pauli X operators, H is a global Hadamard gate on all qubits, and |ψ⟩ := γ,ρ ψγ,ρ |γ, ρ⟩ is the state encoded into the surface code, using the Pauli Z basis as the computational basis. The operator χ is always unitary, as can be verified using identity 7.16. In the simple case of G = H = 0 and E = F = Ig , it turns out that the operator χ has a particularly simple quantum circuit representation: χ=  g ∏  H2j−1 H2j CZ(2j − 1, j)H2j−1 H2j  j=1  which can be verified by considering an identity similar to 7.16: ∑ (−1)α·γ+β·ρ+γ·ρ = 2g (−1)α·β  (7.19)  γ,ρ  Taking the complex conjugate of our expression for ⟨ψ|ϕ⟩, we have a quantum circuit interpretation of it: ( ) ⟨ϕ|ψ⟩ = 22g N ∗ ⟨vac ⊗ +|CΓ† IF ⊗ χ† H |vac ⊗ ψ⟩ (7.20) 104  7.4. Simulation Cost and Entanglement Equation 7.20 provides insight into why the evaluation of product state overlaps with |C α,β ⟩ is simple. From Equation 7.17   g ∏ |C α,β ⟩ = Hχ  CZ(2j − 1, 2j) |α, β⟩ j=1  So:    g ) ∏ ( CZ(2j − 1, 2j) |vac ⊗ α, β⟩ ⟨ϕ|C α,β ⟩ = 22g N ∗ ⟨vac ⊗ +|CΓ† IF ⊗ χ† χ  j=1 ∗  = 2 N (−1) 2g  α·β  ⟨vac ⊗  +|Γ†α,β |vac  ⊗ α, β⟩  (7.21)  since χ is unitary and H is Hermitian and unitary. Even though the gate CΓ† is an entangling gate between the qubits and the fermions, it is diagonal in the |α, β⟩ basis of the qubits and cannot generate qubit-fermion entanglement without a superposition of such terms. The circuit preparing the state |C α,β ⟩ from |α, β⟩ is canceled by the operation of χ. Figure 7.3 provides a circuit representation of Equation 7.20, with the gate CΓ shown explicitly as an entangling gate between the qubits and the fermions. We see that the transfer matrix formalism provides a natural space-time interpretation of the Ising partition function on a grid-like graph. The spatial direction is associated with a 1D array of N fermionic modes and 2g qubits, and the temporal direction with the matrix multiplication of transfer matrix operators defining the fermion Gaussian operator Γα,β . The transfer matrices take the role of time evolution operators (although they aren’t generally unitary), and entanglement between the fermions and the qubits is generated dynamically in time by the joint fermion qubit gates CΓ.  7.4  Simulation Cost and Entanglement  In Chapter 4, we saw that in the case of punctured cylinder graphs, simulating MBQC on surface code states reduces to the problem of evaluating inner products between surface code states and product states. In this Section, we build upon the findings of Section 7.2 to relate the difficulty of evaluating such inner products to entanglement, in two separate ways. The first approach will consider entanglement in the virtual qubit/fermion system considered in the last section, and the second will consider entanglement in the physical Hilbert space of the surface code. 105  7.4. Simulation Cost and Entanglement  ˜ for the genus Figure 7.3: A more detailed quantum circuit representation of ⟨ϕ|ψ⟩ four punctured cylinder graph from Figure 4.2, pictured above the dotted line. The quantum circuit depicted below the dotted line is composed of a set of N = 8 fermionic modes, and a set of 2g = 8 qubits. There is a quantum gate associated with each edge of the punctured cylinder graph. Each circle (blue) is a fermionic ∗ gate of the form e−iJ c2j c2j+1 , and each triangle (red) is a fermionic gate of the ∗ −iγ ∗ c2j−1 c2j form A ∗ e . Some of these gates (hatched, green and orange) operate with couplings that are controlled by qubits from below, as shown with the vertical lines.  Consider the evaluation of the quantity ⟨ψ|ϕ⟩ for any state |ψ⟩ in the C expanding |ψ⟩ in the code space of the surface code. If the coefficients ψα,β |C α,β ⟩ basis are known, then Equation 7.18 tells us that this evaluation can 106  7.4. Simulation Cost and Entanglement be performed as a summation over a maximum of 22g terms. For certain states in the codespace, or certain outcome states |ϕ⟩, it may be possible to do significantly better than a 22g scaling. An extreme example of this we have already met: ⟨C α,β |ϕ⟩ can be computed in a time that is polynomial in g for any local state |ϕ⟩. In this chapter we will consider intermediate cases, where the complexity of evaluating ⟨ψ|ϕ⟩ might increase with g but scale better than as 22g .  7.4.1  Entanglement Across the Fermion/Qubit Split  One way of viewing the computational complexity of evaluating the quantity ˜ ⟨ψ|ϕ⟩ is as arising from entanglement between the qubits α and β, and the fermions. From Equation 7.21 ∑ C ⟨ϕ|ψ⟩ = 22g N ∗ ψα,β (−1)α·β ⟨vac ⊗ +|Γ†α,β |vac ⊗ α, β⟩ α,β ∗  = 2 N ⟨vac ⊗ +|Γψ ⟩ 2g  (7.22)  ∑ C (−1)α·β Γ† |vac ⊗ α, β⟩ is a joint fermion/qubit where |Γψ ⟩ := α,β ψα,β α,β state, which will generally exhibit entanglement between the fermions and the qubits. Strictly speaking, we can only call |Γψ ⟩ a “state” if it has unit norm, which may not be the case if each of the operators Γα,β is not normpreserving. Nevertheless, this makes no difference from the standpoint of computational complexity, so we will think of |Γψ ⟩ as a quantum state in a slight abuse of language. Lemma 7.4.1 The Schmidt rank of |Γψ ⟩ with respect to the bipartition (f ermions, qubits) is equal to the number of linearly independent Γ†α,β |vac⟩ C ̸= 0, i.e. dim(span({ψ C Γ† |vac⟩})). such that ψα,β α,β α,β  ∑s ψ Proof Let |Γψ ⟩ = j=1 λj |Fj ⟩|Qj ⟩ be a Schmidt decomposition of |Γ ⟩ with respect to the fermion/qubit split, where |Fj ⟩ is an orthonormal basis for the 2N dimensional fermionic Fock space, |Qj ⟩ is an orthonormal basis for the 22g dimensional qubit Hilbert space, and the λj are all nonzero. Then the C Γ† |vac⟩})). Schmidt rank of |Γψ ⟩ is s. We will prove that s = dim(span({ψα,β α,β C ̸= 0, and let Let nonzero denote the set of pairs (α, β) for which ψα,β Indep ⊆ nonzero denote the set of pairs (α, β) for some maximal linearly C Γ independent set of ψα,β α,β |vac⟩. Note that C |Indep| = dim(span({ψα,β Γ†α,β |vac⟩}))  107  7.4. Simulation Cost and Entanglement Let Dep denote the complement of Indep in nonzero. For some tensor of coefficients cα,β,γ,ρ , and any (α, β) ∈ nonzero ∑ Γ†α,β |vac⟩ = cα,β,γ,ρ Γ†γ,ρ |vac⟩ (γ,ρ)∈Indep  For (α, β) ∈ Indep: cα,β,γ,ρ = δ(α,β),(γ,ρ) . Now ∑ ∑ C ψα,β (−1)α·β cα,β,γ,ρ Γ†γ,ρ |vac ⊗ α, β⟩ |Γψ ⟩ = (α,β)∈nonzero (γ,ρ)∈Indep  Then for any (α, β) ∈ Indep C ⟨α, β|Γψ ⟩ = ψα,β (−1)α·β Γ†γ,ρ |vac⟩  But from the Schmidt decomposition we also have that ⟨α, β|Γ ⟩ = ψ  s ∑  λj ⟨α, β|Qj ⟩|Fj ⟩  j=1  Thus, each Γ†γ,ρ |vac⟩ can be written as a linear combination of the |Fj ⟩ for j = 1...s. Since each Γ†γ,ρ |vac⟩ is linearly independent, it must be the case that s ≥ |Indep|. ∑ C α·β c Defining λγ,ρ |B γ,ρ ⟩ := α,β,γ,ρ |α, β⟩ for any (α,β)∈nonzero ψα,β (−1) γ,ρ (γ, ρ) ∈ indep, where λγ,ρ is defined such that |B ⟩ is normalized, we can write |Γψ ⟩ as a superposition over |Indep| fermion/qubit product states: ∑ |Γψ ⟩ = λγ,ρ Γ†γ,ρ |vac⟩ ⊗ |B γ,ρ ⟩ (γ,ρ)∈Indep  The Schmidt decomposition always gives a linear expansion of a bipartite state into the minimal number of product state terms. Thus, it must also be the case that s ≤ |Indep|. Together with s ≤ |Indep|, this implies that s = |Indep|. End of proof. Using the decomposition introduced in Lemma 7.4.1 ∑ ∑ 1 C ⟨vac ⊗ +|Γψ ⟩ = g ψα,β (−1)α·β cα,β,γ,ρ ⟨vac|Γ†γ,ρ |vac⟩ 2 (α,β)∈nonzero (γ,ρ)∈Indep  C = (−1)α·β ⟨α, β|χ† |ψ⟩, we can rewrite Since ψα,β  ⟨vac ⊗ +|Γψ ⟩ =  1 2g  ∑  ⟨γ, ρ|Lχ† |ψ⟩ ∗ ⟨vac|Γ†γ,ρ |vac⟩  (γ,ρ)∈Indep  108  7.4. Simulation Cost and Entanglement where L is a |Indep|x|nonzero| matrix such that ⟨γ, ρ|L|α, β⟩ = cα,β,γ,ρ . Let |Lγ,ρ ⟩ = χL† |γ, ρ⟩ for any (γ, ρ) ∈ Indep. This leads us to the following theorem. Theorem 7.4.2 If a state |ψ⟩ is specified in terms of the |Indep| coefficients ⟨γ, ρ|Lχ† |ψ⟩, then ⟨ϕ|ψ⟩ can be computed in a number of steps that scales linearly in the fermion/qubit entanglement, as specified by the Schmidt rank of |Γψ ⟩: s = |Indep|. Unfortunately, the problem specification that Theorem 7.4.2 asks for is fairly ad-hoc. Since the operator L does not generally have full rank in the 22g qubit Hilbert space, the states χL† |γ, ρ⟩ do not constitute a basis. It is then quite unnatural to require |ψ⟩ to be specified by its inner products with these states.  7.4.2  Effective Entanglement in the Output State  While the last section focused on a bipartite entanglement measure in the virtual fermion/qubit system, our second approach will involve a multipartite entanglement measure in the physical Hilbert space of the surface code. Our first step will be to recall that the states |C α,β ⟩ can be written in terms of the encoded X-eigenstates using the Ising cocycles for the encoded Z operators: 1 ∑ (−1)α·β+(α⊕γ)·(β⊕ρ) |XIsing,γ,ρ ⟩ |C α,β ⟩ := g 2 γ,ρ It is straightforward to prove from this equation that these states are all related to one another by encoded Pauli Z operators, with the encoding cocycles chosen as the Ising cocycles (Ck′ = Dk′ ). In particular   g ( )αj ( )βj ∏  |C 0,0 ⟩ |C α,β ⟩ = (−1)α·β  Z2j−1 Z2j j=1  where |C 0,0 ⟩ indicates the state labeled by the g-component zero bitstring for both α and β. To simplify notation, define: C∗ Ψα,β := (−1)α·β ψα,β  Then we can write any state in the surface code space as ) ( g ∑ ∏ αk β k |C 00 ⟩ |ψ⟩ = Ψ∗α,β Z2k−1 Z2k α,β  k=1  109  7.4. Simulation Cost and Entanglement Now consider the quantity ⟨ψ|ϕ⟩. If we let the Pauli Z operators operate to the right rather than the left we see that ⟨ψ|ϕ⟩ = ⟨C 00 |ϕψ ⟩ where |ϕψ ⟩ :=  ∑ α,β  =  ∑ α,β  ( Ψα,β  g ∏  ) Z2k−1  αk  Z2k  βk  |ϕ⟩  k=1   Ψα,β   g ∏  ∏  k=1  ′ e∈C2k−1  Zeαk  ∏ ′ e∈C2k   Zeβk   ⊗  |ϕe ⟩  (7.23)  e∈E  Thus, evaluating the overlap between an arbitrary surface code state and a product state is equivalent to evaluating the overlap of one of the “easy” states |C 00 ⟩ with an effective state |ϕψ ⟩ which is generally not a product state of the physical qubits. In a sense, the state |ϕψ ⟩ reflects an encoding of the 2g qubit state |ψ⟩ into the |E| physical qubits of the state |ϕ⟩. In terms of simulating MBQC, the state |ϕψ ⟩ combines both the specification of the resource state and the particular sequence of measurement outcomes one is computing the probability of. If |ϕψ ⟩ were to be expanded as a sum over product states, we could evaluate ⟨C 00 |ϕψ ⟩ in a number of steps that grows linearly with the number of terms in the expansion. The base-2 logarithm of the minimal number of product states that are required to expand a multipartite quantum state is an entanglement monotone known as the Schmidt measure [4]. That is, for an N qubit pure state |ψ⟩, the Schmidt measure ESch (|ψ⟩) is the minimum number such that (|ψ⟩) 2ESch ∑ |ψ⟩ = |χj1 ⟩|χj2 ⟩...|χjN ⟩ j=1  for some set of local states |χjk ⟩ for all j = 1...2ESch (|ψ⟩) , k = 1...N . We will call the |χjk ⟩ in such an expansion (with 2ESch (|ψ⟩) terms) an optimal local basis for |ψ⟩. Applying the Schmidt measure to our situation, we immediately have the following result. Theorem 7.4.3 If an optimal local basis for the effective state |ϕψ ⟩ is known, then ⟨ψ|ϕ⟩ can be computed in a number of steps that scales as poly(|E|, g) ∗ 2ESch (|ϕ  ψ ⟩)  . 110  7.4. Simulation Cost and Entanglement Computation of ESch (|ψ⟩) for a generic multiparty state - no less finding an optimal local basis for it - is generally a very hard problem. Yet an efficient means of computing an optimal local basis is necessary to give Theorem 7.4.3 much practical significance. Luckily, the task of evaluating the Schmidt measure is simplified considerably in our case by the definition of |ϕψ ⟩. Since each term Equation 7.23 is a product state, we know that ESch (|ϕψ ⟩) must be less than or equal to 2g, even though |ϕψ ⟩ is a state on possibly many more than 2g qubits. Furthermore, we can show that under very general conditions, Equation in fact 7.23 already provides an optimal local basis for |ϕψ ⟩. First we introduce some notation. In the following, the distinction between the even and odd numbered cocycles will not be important, so we simplify matters by writing the coefficients Ψα,β as Ψα where α is now a 2g component bitstring. Then we can rewrite Equation 7.23 as:   2g ∏ ∏ ⊗ ∑ Ψα  Zeαk  |ϕe ⟩ |ϕψ ⟩ = k=1 e∈Ck′  α  =  ∑ α  Ψα  ⊗  e∈E  (Ze )[M α]e |ϕe ⟩  (7.24)  e∈E  where M is a |E|x2g matrix such that Me,k = 1 if e ∈ Ck′ and Me,k = 0 ∑ if e ∈ / Ck′ . [M α]e := 2g k=1 Me,k ∗ αk . Note that the column rank of the matrix M over the binary field Z2 is 2g, because the edge sets Ck′ are mutually independent. Since the row rank and the column rank of any matrix are the same, it follows that there exists some subset of the edges A = {ek }k=1...2g such that rank (MA ) = 2g, where MA is the 2gx2g matrix obtained from the ek rows of M . For any set X ⊆ E, define the matrix MX analogously. Let G − Z denote the subgraph of G such that for all e ∈ E(G − Z), |ϕe ⟩ is not a Pauli Z eigenstate. Theorem 7.4.4 If the subgraph G−Z contains two disjoint sets of 2g edges A = {ek }k=1...2g and B = {e′k }k=1...2g such that rank (MA ) = rank (MB ) = 2g, then Equation 7.23 yields an optimal local basis for |ϕψ ⟩. It follows then that ESch (|ϕψ ⟩) = log2 (D), where D is the number of nonzero coefficients Ψα .  111  7.4. Simulation Cost and Entanglement Proof We will show that if |ϕ ⟩ = ψ  s ∑  |χj1 ⟩|χj2 ⟩...|χj|E| ⟩  j=1  for any set of single qubit states |χjk ⟩, then s ≥ D. Our first step will be to be to isolate a single term of Equation 7.23 by taking a partial inner product between |ϕψ ⟩ and a particular state on the qubits in A. Write |ϕe ⟩ = ae |0⟩ + be |1⟩ for any edge e. Now we define |ϕ0,⊥ e ⟩ := b∗e |0⟩ + a∗e |1⟩, and |ϕe1,⊥ ⟩ := b∗e |0⟩ − a∗e |1⟩. It is easy to verify that for any edge e and binary variable γk ∈ {0, 1} ⟨ϕγe k ,⊥ | (Ze )αk |ϕe ⟩ = δαk ,γk ∗ 2ae be 0,⊥ In particular, |ϕ1,⊥ e ⟩ is perpendicular to |ϕe ⟩ for any edge e, while |ϕe ⟩ is perpendicular to Ze |ϕe ⟩ for any edge e. First we write Equation 7.23 in the form: ∑ |ϕψ ⟩ = Ψα |ϕαrest ⟩ ⊗ |ϕαA ⟩ ⊗ |ϕαB ⟩ (7.25) α  where |ϕαrest ⟩ is a α-dependent product state on all of the qubits in the complement of A ∪ B in E, and |ϕαA ⟩  :=  2g ⊗  (Zek )  [M α]e  k  |ϕek ⟩ =  2g ⊗  (Zek )[MA α]k |ϕek ⟩  k=1  k=1  The states |ϕγekk ,⊥ ⟩ for any 2g component bitstring γ can now be used to pick out a single term in Equation 7.25, because ( 2g ) ( 2g ) ⊗ ∏ ⟨ϕγekk ,⊥ | |ϕαA ⟩ = 2aek bek ∗ δγ,[MA α] k=1  k=1  and thus ( 2g ) ( 2g ) ⊗ ∏ A γ]k ,⊥ ⟨ϕ[M | |ϕψ ⟩ = Ψγ 2aek bek |ϕγrest ⟩ ⊗ |ϕγB ⟩ (7.26) ek k=1  k=1  The only value of α for which [MA α] = [MA γ] is α = γ, because by assumption the square matrix MA has full rank and hence is invertible. Since |ϕek ⟩ is not a Z-eigenstate, 2aek bek is nonzero for each k. We can 112  7.4. Simulation Cost and Entanglement show that the states {|ϕγrest ⟩ ⊗ |ϕγB ⟩} for various bitsrings γ are a linearly independent family of states. This follows from the assumption of the second set B of non-Z eigenstate edges {e′k } for which MB has full rank. For we can repeat the above trick to show that each |ϕγrest ⟩ ⊗ |ϕγB ⟩ has a component that is perpendicular to subspace spanned by the rest of the |ϕγrest ⟩ ⊗ |ϕγB ⟩: ( ⟨ϕγrest |  ) 2g 2g ∏ ⊗ [M γ] ,⊥ [MB γ]k ,⊥ α α ⟨ϕe′ B k | (Zek )[MB α]k |ϕek ⟩ ⟨ϕe′ | |ϕrest ⟩ ⊗ |ϕB ⟩ ∝ ⊗ k=1  k  k=1  k  = 0 if α ̸= γ ̸= 0 if α = γ The RHS is zero if α ̸= γ, but is a nonzero vector if α = γ. So the state |ϕγrest ⟩ ⊗ |ϕγB)⟩ has a component that lies along the vector |ϕγrest ⟩ ⊗ ( ⊗2g [MB γ]k ,⊥ ⟩ , but all of the other |ϕαrest ⟩ ⊗ |ϕαB ⟩ are orthogonal to it. k=1 |ϕe′ k  Thus |ϕγrest ⟩ ⊗ |ϕγB ⟩ cannot be written as a linear combination of the others, for each γ. ∑s j j j ψ Now let |ϕψ ⟩ = j=1 |χ1 ⟩|χ2 ⟩...|χ|E| ⟩ be any other expansion of |ϕ ⟩ into some number s of product states. Write it as |ϕ ⟩ = ψ  s ∑  |χjE/A ⟩ ⊗ |χjA ⟩  j=1  Then ( 2g ⊗  ) A γ]k ,⊥ | ⟨ϕ[M ek  k=1  |ϕ ⟩ = ψ  s ∑ j=1  ) ) (( 2g ⊗ j [MA γ]k ,⊥ | |χA ⟩ |χjE/A ⟩ ⟨ϕek  (7.27)  k=1  Comparing Equations 7.26 and 7.27, we see that for each γ for which Ψγ is nonzero, |ϕγrest ⟩ ⊗ |ϕγB ⟩ can be written as a linear combination of the s states |χjE/A ⟩. Let D be the number of such nonzero Ψγ . Since each |ϕγrest ⟩ ⊗ |ϕγB ⟩ is linearly independent, there must be enough states |χjE/A ⟩ to span a D dimensional space. So, s ≥ D. Since this applies to any ∑ |E| decomposition of the form |ϕψ ⟩ = sj=1 |χ1j ⟩|χ2j ⟩...|χj ⟩, we conclude that ESch (|ϕψ ⟩) = log2 D. The state |ϕψ ⟩ and the coefficients Ψα,β can be efficiently computed from C and the definition of the surface code cocycles C ′ . The the coefficients ψα,β k C . It number of nonzero Ψα,β is exactly equal to the number of nonzero ψα,β 113  7.4. Simulation Cost and Entanglement C are follows from Theorem 7.4.4 then that if the D nonzero coefficients ψα,β known, and the assumption of the theorem is satisfied, then the quantity ⟨ψ|ϕ⟩ can be evaluated in a number of steps that is polynomial in the size and genus of the lattice, but increases exponentially with the entanglement in |ϕψ ⟩, as measured by the Schmidt number.  114  Chapter 8  Conclusion 8.1  Conclusion  In this thesis we have considered the states in the code space of the surface code, an important scheme of quantum error correction that encodes quantum information topologically. In particular, we studied the task of classically simulating MBQC using a state in the surface code space as a resource state. We found that a simulation of single-qubit measurements on punctured cylinder code states can be performed in a number of steps that grows polynomially in the number of qubits, but exponentially in the genus of the lattice. This result was obtained by expressing partial measurement probabilities as a special case of an inner product between a product state and another punctured cylinder surface code state: ) ( (8.1) p|ψ⟩ {|ϕe ⟩}e∈E¯ ∝ ⟨ϕ|ψ⟩ An inner-product of the form in Equation 8.1 maps onto a summation over partition functions of a 2D classical Ising model, which we considered in Part II. There, we focused on two higher genus generalizations of a square grid, which allowed for a mapping to fermions and a physical interpretation of their partition functions as a 1D quantum system. We found that for both punctured cylinder graphs and square g-graphs, the Ising partition function takes the form: ∑ Z= (−1)α·β f (Γα,β ) (8.2) α,β∈{0,1}⊗g  where f (Γα,β ) is an function that can be computed efficiently in all parameters. In Section 5.9, we conjectured that Equation 8.2 characterizes a general form for the Ising model partition function on a graph embedded on a surface of genus g. This conjecture is based on a similar expression which occurs in the combinatoric approach in [19] to evaluating the partition function for a more general class graphs. In fact, if we drop the requirement that f (Γα,β ) be computable efficiently in the size of the graph, then the form of Equation 8.2 must be general. This 115  8.1. Conclusion is because in principle our methods for solving the Ising partition function can be used for any graph. We obtained an Equation of the form of Equation 8.2 for both square g-graphs and punctured cylinder graphs of genus g, which are both families of graphs of unbounded genus and unbounded face-width (the face-width of a graph G embedded in a surface S is the minimum number of edges from G that any non-contractible cycle on S must cross). A theorem due to Robertson and Seymour [54] states that for any graphs G and H that can be embedded on some surface S, there is an integer k(H) such that if the face-width of G embedded in S is at least k(H), then H is a minor of G (see also [46], Theorem 5.9.2). We can define a square g-graph or punctured cylinder graph G with any genus g ≥ 1 and face width k(H), and the cost of evaluating the Ising partition function on G will scale polynomially in k(H). An Ising partition function on H can be turned into a special case of an Ising partition function on G by taking certain Ising couplings to be zero, and considering a limit as others go to infinity (see footnote in Chapter 4). Of course, whether this is of any practical use would depend on how k(H) scales with the size of H, which is not indicated by the theorem. An analogous theorem in the planar case has a linear scaling: any planar graph with N vertices is the minor of an kxk grid graph, where k = O(N ) [46]. In Part III we combined the results of Parts I and II to define a family of surface code states |C α,β ⟩ for which ⟨ϕ|C α,β ⟩ can be evaluated efficiently in all parameters, including the genus of G. In Appendix G, we show that for the measurement scheme we considered on punctured cylinder code states in Chapter 4, the complexity of simulating MBQC with |C α,β ⟩ as a resource state does not grow with the genus. Finally, we considered whether the task of evaluating inner products between a surface code state and a product state could be related to entanglement. We found two such connections, one in the virtual fermion/qubit system that comes out of the transfer matrix formalism, and one based upon the ease of computing ⟨ϕ|C α,β ⟩ for product states |ϕ⟩ as compared with entangled states. This latter connection is more promising than the first, but it still requires a non-trivial assumption regarding which qubits are measured in the Z basis during a run of MBQC. Under the assumption, the cost of evaluating ⟨ϕ|ψ⟩ scales polynomially in both the size and genus of the lattice, yet exponentially in an entanglement monotone. A study of the quantum computational power of MBQC on surface code states could shed some light on how appropriate the required assumption is. 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An edge e ∈ E corresponds to a set of one or two elements from V. For two distinct elements u, v ∈ V , we say that vertex u is adjacent to vertex v if {u, v} ∈ E, and we say that the edge e = {u, v} is incident on vertices u and v. An edge e = {u} for some u ∈ V is called a selfloop. A self-loop is not counted in the set of edges that are incident on any vertex. Note that some authors forbid self-loops and redundant edges in the definition of a graph, calling an object without this restriction a multigraph. We will use the term graph in the general sense. Graphs are often visualized by drawing them as a set of points connected by arcs in the plane, however they can be drawn on other surfaces as well. In this work we will consider closed, orientable two-dimensional surfaces S. The genus g of S is the number of holes in S, and is a topological invariant [2]. An embedding of a graph G onto a surface is a mapping from G to points on S. The vertices of G are mapped one-to-one with points on S, and edges of G correspond to arcs on S that terminate at the vertices they are incident upon. We say that a graph G can be embedded on surface S if there exists an embedding of G on S such that none of the arcs cross each other, except at vertices. Not every graph can be embedded on any surface. The genus of a graph is the minimum genus surface that the graph can be embedded on. Unless otherwise explicitly noted, an embedding of a graph is taken to be one without edge-crossings. Consider a graph G embedded on a surface S. The points of S which are not in the image of the embedding form a set of connected regions that we will call faces [25]. Denote the set of faces by F. Denote the set of edges which separate a face p from its neighbours the boundary of p and denote it by ∂p. If an edge e is surrounded by the same face f on both sides, then 123  Appendix A. Relevant Topological Graph Theory we do not count e in ∂f . If each face is homeomorphic to an open disk, we call the embedding cellular. For a cellular embedding, Euler’s equation holds [46]: χ = |V | − |E| + |F | where χ is the so-called Euler characteristic surface S, and |X| indicates the cardinality of a set X. For closed, orientable surfaces χ is related to the ˜ of an embedded genus of the surface by: χ = 2 − 2g [2]. The dual graph G ˜ with each graph G is defined in the following way: associate a vertex of G ˜ face of G, and connect vertices v and u of G by an edge iff the boundaries of the faces in G corresponding to u and v share an edge. If there exists a face in G that is on both sides of an edge e, then the corresponding vertex in ˜ has a self-loop for the edge e. There is then a one-to-one correspondence G ˜ but the roles of vertices and faces have between edges in G and edges in G, ˜ can always be cellularly been reversed. For a cellular embedding of G on S, G embedded on the same surface S. This follows from Euler’s formula, which is symmetric between |V | and |F | [46]. A walk on a graph G is a sequence v0 , e1 , v1 , e2 , ...ek , vk where the vi are vertices and each ei is an edge incident on vi−1 and vi , or vi−1 = vi and ei is a self loop on vi . If for any two distinct vertices u, v ∈ V of a graph G there exists a walk where v0 = u and vk = v, then the G is called a connected graph. If for some walk vk = v0 , then the walk is called closed. For a closed walk, we call the unordered set of edges {e1 , ...ek } a cycle. Call the unordered set of edges corresponding to a closed walk on ˜ a cocycle. For two cycles C1 and C2 , call the symmetric the dual graph G difference C1 ⊕ C2 := (C1 ∪ C2 )/(C1 ∩ C2 ) the superposition of C1 and C2 . The superposition of two cycles is called a superposition cycle, and is equal to the union of some set of disjoint cycles. We define the product of two faces p and q to be the superposition of ∂p and ∂q. A cycle is called trivial if it is the null set or is a product of faces, and is non-trivial otherwise. Two cycles are homologically equivalent if their superposition is a trivial cycle. This divides the set of all cycles into equivalence classes called homology classes. A set of cycles is called homologically independent if no superposition over a subset of them results in a trivial cycle. Homology classes for cocycles are ˜ A cocycle is trivial defined by the analogous definitions on the dual graph G. if it is equal to the symmetric difference of δs for some set of vertices s on G, where δs is the set of edges incident on s. Two cocycles are homologous if their symmetric difference is a trivial cocycle. For a graph embedded on a surface of genus g, there can be at most 2g homologically independent cycles, and at most 22g homology classes of cycles. 124  Appendix B  The Signs in Equations 2.6 and 5.8 For any complex number z, let R(z) be the real part of z and I(z) be the imaginary part of z, so that z = R(z) + iI(z). For any real number x, let sgn(x) denote the sign of x: +1 if x > 0, −1 if x < 0, and 0 if x = 0. We will make use of the following three identities sinh(z) = sinh(R(z)) cos(I(z)) + i cosh(R(z)) sin(I(z))  (B.1)  cosh(z) = cosh(R(z)) cos(I(z)) + i sinh(R(z)) sin(I(z)) sinh(2R(z)) cos(2I(z)) + i sin(2I(z)) cosh(2R(z)) tanh(z) = 2(cosh2 (R(z)) cos2 (I(z)) + sinh2 (R(z)) sin2 (I(z)))  (B.2) (B.3)  First we review the definition of the principal square root of a complex number z. If we write z as z = reiϕ with r ≥ 0 and −π < ϕ ≤ π, then the principal square root Sqrt of z is defined as iϕ ϕ ϕ Sqrt(z) = Sqrt(r)e 2 = Sqrt(r) cos( ) + iSqrt(r) sin( ) 2 2  where for x real Sqrt(x) denotes the positive square root of x. Notice that since − π2 < ϕ2 ≤ π2 , R(Sqrt(z)) is always positive or zero, while sgn(I(Sqrt(z)) = sgn(ϕ) = sgn(I(z)) unless ϕ = π. If ϕ = π, then I(Sqrt(z)) > 0. If I(Sqrt(z)) = 0, then ϕ = 0 and I(z) = 0 as well. However, I(z) = 0 does not imply that I(Sqrt(z)) = 0, because if z is real negative then ϕ = π and I(Sqrt(z)) ̸= 0, even though I(z) = 0.  B.1  Sign of the Square Root in Equation 2.7  For the purposes of this section, pick a particular edge e and let a = ae , b = be . Recall that without loss of generality,( a)is real and positive, or zero. We wish to show that if we let J = T anh−1 ab , then ( ) a = Sqrt a2 − b2 cosh(J) 125  B.1. Sign of the Square Root in Equation 2.7 where T anh−1 (z) is defined by the constraint −  π π ≤ I(T anh−1 (z)) ≤ 2 2  where we decide between I(T anh−1 (z)) = ± −  π 2  by insisting that  sgn(I(T anh−1 (z))) = sgn(z) if z is real and has magnitude greater than one. We proceed as follows. From the identity B.2: cosh(J) = cosh(R(J)) cos(I(J)) + i sinh(R(J)) sin(I(J))  (B.4)  Since cosh(x) > 0 for all real x, and cos(x) ≥ 0 for all x ∈ [− π2 , π2 ], we may conclude from Equation B.4 that R(cosh(J)) ≥ 0. We may also then conclude that R(sech(J) ≥ 0, because sech(J) =  R(cosh(J)) − iI(cosh(J)) 1 = (B.5) R(cosh(J)) + iI(cosh(J)) R(cosh(J))2 + I(cosh(J))2  Now we consider the identity: sech2 (z) = 1 − tanh2 (z), which implies 2 that sech2 (J) = 1 − ab 2 . Also, since R(sech(J)) ≥ 0, and it is assumed that a is real positive: R(a ∗ sech(J)) ≥ 0. Thus, we know that if R(cosh(J)) ̸= 0 then: a ∗ sech(J) = Sqrt(a2 sech2 (J)) = Sqrt(a2 − b2 ), since the real part of a principal square root is always nonnegative. If on the other hand R(cosh(J)) = 0, then I(J) = ± π2 , and then by Equation B.4 we have that cosh(J) = sgn(I(J)) ∗ i sinh(R(J)). Thus sech(J) = −sgn(I(J)) ∗ icsch(R(J)), and thus sgn(I(sech(J))) = −sgn(I(J)) ∗ sgn(R(J)) 2  Since 1 − ab 2 = sech2 (J) = −csch2 (R(J)) ≤ 0, it is clear that ab must be real and of magnitude greater or equal ( )to one. And from the identity B.3 with I(J) = ± π2 , sgn(R(J)) = −sgn ab . a is by assumption real nonnegative: sgn(I(a∗sech(J))) = sgn(I(J))∗sgn( ab ). Since I(Sqrt(a2 −b2 ) ≥ 0, it follows that if we choose sgn(I(J)) = sgn( ab ) in accordance with our definition of T anh−1 then sgn(I(a ∗ sech(J))) = 1 and hence a ∗ sech(J) = Sqrt(a2 − b2 ) as desired.  126  B.2. Sign of the Square Root in Equation 5.8  B.2  Sign of the Square Root in Equation 5.8  For the purposes of this and let K = Kjk . √ section, pick a particular−1j, k−2K Then we define A := (2 sinh(2K)) and γ := tanh (e ), where for now we leave the branch choices unspecified. Equation 5.8 says that A sinh(γ)eK = ±1 We wish to determine whether the positive or negative root results from our definition of A and γ. As a first step, let b := eK sinh(γ). Then Ab = R(A)R(b) − I(A)I(b) + i(R(A)I(b) + I(A)R(b)) = ±1 Since we know from Equation 5.8 that I(Ab) = 0: R(A)I(b) = −I(A)R(b). Given that we know both I(Ab) = 0 and Ab ̸= 0: R(A) = 0 iff R(b) = 0 and I(A) = 0 iff I(b) = 0. If R(A) = R(b) = 0, then both I(A) and I(b) are non-vanishing. If I(A) = I(b) = 0, then both R(A) and R(b) are nonvanishing. Assume for the moment that R(A) ̸= 0. If I(A) and I(b) are nonzero, then: sgn(R(A))sgn(R(b)) = −sgn(I(A)) ∗ sgn(I(b)), so so both of the two terms of R(Ab) have the same sign, and sgn(R(Ab)) = sgn(R(A))sgn(R(b)). If I(A) = I(b) = 0, then Ab = R(A)R(b) and again sgn(R(Ab)) = sgn(R(A))sgn(R(b)) Under the assumption that R(A) ̸= 0, R(A) > 0 iff we take the principal square root in defining A. Thus, we can assure that Ab = 1 by taking the principal square root A := Sqrt(2 sinh(2K)) iff R(b) is positive. Suppose on the other hand that R(A) = 0. In that case sgn(Ab) = −sgn(I(A))sgn(I(b)). We assume that A ̸= 0, because otherwise its sign is meaningless. Then, I(A) > 0 iff we take the principal square root in defining A. So, we can assure that Ab = 1 by letting A := Sqrt(2 sinh(2K)) iff I(b) is negative.  127  Appendix C  Bipartite Entanglement of Surface Code States Recall that from the Schmidt decomposition Equation 3.4, we were able to determine the entropy of entanglement of the state |K(G)⟩ for an arbitrary ¯ E) ˆ to be log2 (|S|) = |∂ E| ¯ − (¯ bipartition (E, n+n ˆ − 1). We can extend this result to arbitrary states |ψ⟩ in the code space, for a restricted set of ¯ E). ˆ bipartitions (E, We will see that for these bipartitions the entanglement receives two contributions: one from a log2 (|S|) term reflecting the entanglement area law, and another coming from the entanglement in the state |ψ⟩ being encoded in the surface code. In particular, we will consider partitions of the qubits for which one can choose the non-trivial cocycles ′ to be entirely in the set E, ¯ and non-homologous non-trivial cocyC1′ ...C2k ′ ′ to be entirely in the set E. ˆ We furthermore assume that cles C2k+1 ...C2g ¯ and that there there are cycles homologous to each of C1 ....C2k in the set E, ′ ′ ˆ are cycles homologous to each of C2k+1 ...C2g in E. Then we may write: ∑ α ...α ,β ...β 1 ∑ α ...α ,β ...β cα,β ZE¯1 k 1 k |KE¯ (u)⟩Z ˆk+1 g k+1 g |KEˆ (u)⟩ |ψ⟩ = √ E |S| α,β u∈S The state |ψ⟩ has some Schmidt decomposition with to the bipar∑s respect A tition of qubits (1...2k, 2k + 1...2g); let it be |ψ⟩ = j=1 λj |ψj ⟩|ψjB ⟩, where ∑ A s is the Schmidt rank of |ψ⟩, |ψjA ⟩ = α1 ...αk Uj,α |Xα1 ...αk ,β1 ...βk ⟩ 1 ...αk ,β1 ...βk β1 ...βk ∑ B αk+1 ...αg Uj,α and |ψjB ⟩ = |Xαk+1 ...αg ,βk+1 ...βg ⟩. The |Xα,β ⟩ k+1 ...αg ,βk+1 ...βg βk+1 ...βg  are the Pauli X eigenstates, the matrices U A and U B are unitary, and the λj are real nonzero numbers from j = 1...s. Using the Schmidt decomposition, we have that: 1 ∑∑ B |ψ⟩ = √ λj |ψ(u)A j ⟩|ψ(u)j ⟩ |S| j=1 u∈S s  where |ψ(u)A j ⟩ :=  ∑  α1 ...αk β1 ...βk  (C.1)  A Z α¯1 ...αk ,β1 ...βk |KE¯ (u)⟩ and analUj,α 1 ...αk ,β1 ...βk E  128  Appendix C. Bipartite Entanglement of Surface Code States ogously for |ψ(u)B j ⟩. Equation C.1 is a Schmidt decomposition for |ψ⟩, of rank |S| ∗ s. To see this, we consider the inner product: A ⟨ψ(u′ )A l |ψ(u)j ⟩ =  ∑  †  A Uj,α U A γ1 ...γk ,ρ1 ...ρk ,l 1 ...αk ,β1 ...βk  α1 ...αk β1 ...βk γ1 ...γk ρ1 ...ρk (α⊕γ) ...(α⊕γ) ,(β⊕ρ) ...(β⊕ρ)  ∗  1 1 k k ⟨KE¯ (u)|ZE¯ |KE¯ (u)⟩ ∑ † A = δu,u′ Uj,α U A α1 ...αk ,β1 ...βk ,l 1 ...αk ,β1 ...βk  α1 ...αk β1 ...βk  = = δu,u′ δj,l where the second equality follows from Lemma 3.2.6 and the third from the unitarity of the matrix U A . Orthonormality of the |ψ(u)B j ⟩ can be proven in the same way, and it is clear that all coefficients in Equation C.1 are positive. The entropy of entanglement of the state |ψ⟩ with respect to ¯ E) ˆ is then given by: the bipartition (E, ( ) s ∑ s ( ) ∑ ∑ ( ) ) ( |λj |2 |λj |2 E |ψ⟩ = − log2 = −|λj |2 log2 |λj |2 + log2 (|S|) |S| |S| j=1 u∈S  j=1  ¯ − (¯ = |∂ E| n+n ˆ − 1) + E (|ψ⟩) where E (|ψ⟩) is the entropy of entanglement of the state |ψ⟩ being encoded, with respect to the bipartition of qubits (1...2k, 2k + 1...2g).  129  Appendix D  Fermion Gaussian Operators D.1  Relationship to Exponentials of Quadratic Fermion Operators  Here we state Lemma 0 from [3] Lemma D.1.1 An operator X is a Gaussian operator if and∑only if X is even in the fermion operators and [Λ, X ⊗ X] = 0 where Λ := 2N j=1 cj ⊗ cj , where ⊗ indicates an ordinary tensor product. ∑  Theorem D.1.2 Every operator of the form X = e sian operator.  jk  Mjk cj ck  is a Gaus-  Proof By Lemma D.1.1, X is a Gaussian operator if [Λ, X ⊗ X] = 0 and X is even. As an exponential of an even operator, X is even. Now consider the second condition: [Λ, X ⊗ X] = 0. Multiplying this equation on the right T T by X −1 ⊗ X −1 = e−c M C ⊗ e−c M C , we get [Λ, X ⊗ X]X −1 ⊗ X −1 = Λ − X ⊗ XΛX −1 ⊗ X −1 = 0 Since X −1 ⊗ X −1 is invertible, the condition given by this equation is logically equivalent to the first. Using the definition of Λ X ⊗ XΛX −1 ⊗ X −1 =  2N ∑  Xcj X −1 ⊗ Xcj X −1  j=1  From the results of Appendix F, we know that Xcj X −1 =  2N ∑  [e4M ]kj ∗ cj  k=1  130  D.1. Relationship to Exponentials of Quadratic Fermion Operators Substituting: X ⊗ XΛX  −1  ⊗X  −1  =  2N ∑  [e4M ]kj [e4M ]lj ∗ ck ⊗ cl  j,k,l=1  =  2N ∑  [e4M ]kj [e−4M ]jl ∗ ck ⊗ cl  j,k,l=1  =  2N ∑  δk,l ∗ ck ⊗ cl =  k=1  2N ∑  ck ⊗ ck = Λ  k,l=1  Thus [Λ, X ⊗ X]X −1 ⊗ X −1 = Λ − Λ = 0, and the condition is satisfied. Let M be the set of 2N × 2N complex matrices. Let MA ⊂ M be the set of such matrices which are antisymmetric, and let MN A ⊂ MA be the set of 2N × 2N complex antisymmetric matrices who are both diagonalizable and who have no eigenvalues equal to ±i. We can write this as: MN A = Mno±i ∩ MD , where Mno±i is the set of all 2N × 2N complex matrices for which neither +i nor −i is a root of their characteristic polynomial, and MD is the set of all 2N × 2N complex matrices which are diagonalizable. Lemma D.1.3 Consider a Gaussian state X with covariance matrix M ∈ MN trace: tr(X) = 2N ∗C for some nonzero complex number C. A and nonzero∑ i Then, X = De 2 jk Gjk cj ck for some matrix G ∈ MA and complex number D. Proof By assumption that M ∈ MN A , the matrix M can be diagonalized into a complete set of eigenvalues {mj }. The inverse tangent function tan−1 (mj ) is undefined only at mj = ±i, which we have assumed not to be the case. Thus, we can always define a matrix G such that M = tan(G) by diagonalizing and taking the inverse tangent of each eigenvalue mj . Denote this as G = tan−1 (M ). Similarly, we can define a matrix cos(G) = cos(tan−1 (M ))) so long as all of the mj ̸= ±i. i ∑ jk Gjk cj ck with di2 Theorem 6.2.5 says that for any operator O = e √ i T agonalizable, antisymmetric G: ω(O, θ) = det(cos(G)) ∗ e 2 θ [tan(G)]θ . By assumption that the matrix M is a bounded operator, det(cos(G)) ̸= 0, because otherwise the eigenvalues of M = tan(G) would be infinite. So i  ω(De 2  ∑ jk  tan−1 (M )jk cj ck  i T [tan(tan−1 (M ))]θ  , θ) = Ce 2 θ  i T Mθ  = Ce 2 θ  = ω(X, θ)  131  D.1. Relationship to Exponentials of Quadratic Fermion Operators where D := (det(cos(tan−1 (M ))))−1/2 ∗ C. The mapping ω is inverti ∑ −1 ible, so we can conclude that X = De 2 jk tan (M )jk cj ck . We can assume that G = tan−1 (M ) ∈ MA , because by the anticommutation relations the symmetric part of G will be proportional to the identity, which we could subsume into the pre-exponential factor D. However, this will not be strictly necessary in any of what follows. ∑  jk Gjk cj ck , where Theorem D.1.4 X is a Gaussian operator iff X ∈ De ∑ ∑ De jk Gjk cj ck denotes the topological closure of the set {De jk Gjk cj ck }.  Proof Sufficiency: Any X ∈ De  ∑ jk  Gjk cj ck ∑  X = lim D(s)e  jk  can be written as G(s)jk cj ck  s→∞  Any X as defined above is even, and [Λ, X ⊗ X] = lim D(s)2 ∗ [Λ, e  ∑ jk  G(s)jk cj ck  s→∞  ⊗e  ∑ jk  G(s)jk cj ck  ]=0  using Theorem D.1.2. By Lemma D.1.1, X is a Gaussian operator. Necessity: A general Gaussian operator X (which may have a vanishing trace) can be written as a limit of a sequence of non-trace-vanishing Gaussian operators (see Part V of [3]): ( ) T X = lim X(s) = lim ω −1 C(s)eθ M (s)θ , θ s→∞  s→∞  where C(s) is a convergent sequence of complex scalars (nonzero for all finite values of s), and M (s) is a sequence of antisymmetric covariance matrices. Also, tr(X) = lims→∞ tr(X(s)) = 2N ∗ lims→∞ C(s). We cannot say in whether or not M(s) is a convergent sequence, but the 2N ×2N matrix C(s) ∗ M (s) must converge in the limit (see equation between (31) and (32) in [3]), as the matrix 2i C(s) ∗ M (s) encodes the coefficients of the quadratic terms in the expansion of the exponential. So long as M (s) is diagonalizable and does not have ±i as an eigenvalue, Lemma D.1.3 tells us that ( ) T X(s) = ω −1 C(s)eθ M (s)θ , θ i  = D(s)e 2  ∑ jk  tan−1 (M (s))jk cj ck  where D(s, t) = (det(cos(tan−1 (M (s, t)))))−1/2 ∗ C(s). Any M (s) that contains ±i as an eigenvalue or is not diagonalizable can be expressed as a 132  D.2. Gaussian Basis limit of a sequence of matrices M (s, t) for which none of the eigenvalues are equal to ±i and all eigenvalues are distinct (no degeneracies). Then M (s, t) is diagonalizable and we can define a finite matrix tan−1 (M (s, t)). Thus for any M (s) we can write M (s) = limt→∞ M (s, t) where M (s, t) ∈ MN A. Therefore ) ( T X(s) = ω −1 C(s)eθ limt→∞ M (s,t)θ , θ ( ) T = lim ω −1 C(s)eθ M (s,t)θ , θ t→∞  =  i  lim D(s, t)e 2  ∑ jk  tan−1 (M (s,t))jk cj ck  t→∞  where D(s, t) = (det(cos(tan−1 (M (s, t)))))−1/2 ∗ C(s). ∑  It follows that X(s) ∈ De jk Gjk cj ck , because we could always take make t large enough that X(s, t) be arbitrarily close to an element in the set of operators {De  ∑  jk  Gjk cj ck  }. But since X(s) ∈ De  ∑  jk  Gjk cj ck  ∑  , and De  is a closed set, it must contain lims→∞ X(s). Therefore, X ∈ De for any Gaussian operator X.  D.2  ∑  jk  Gjk cj ck  jk  Gjk cj ck  Gaussian Basis  Here we prove that for any two occupation number eigenstates |α⟩ and |β⟩, the operator |α⟩⟨β| is a Gaussian operator if and only if |α⟩ and |β⟩ have the same number of fermions. This allows one to write an arbitrary physical fermion operator as a sum over Gaussian operators. This result is equivalent to a previous one in [8]. To see this, first rewrite the operator Λ as Λ=  N ∑  c2j−1 ⊗ c2j−1 + c2j ⊗ c2j  j=1  And using c2j−1 = a†j + aj , c2j = −i(a†j − aj ) Λ =  N ∑  (a†j + aj ) ⊗ (a†j + aj ) − (a†j − aj ) ⊗ (a†j − aj )  j=1  = 2  N ∑  a†j ⊗ aj + aj ⊗ a†j  j=1  133  D.2. Gaussian Basis Now consider the operator X = |α⟩⟨β|: [Λ, X ⊗ X] = 2  n ∑  [a†j ⊗ aj , |α⟩⟨β| ⊗ |α⟩⟨β|] + [aj ⊗ a†j , |α⟩⟨β| ⊗ |α⟩⟨β|]  j=1  = 2  = 2  n ∑ j=1 n ∑  [a†j |α⟩⟨β| ⊗ aj |α⟩⟨β|] + [aj |α⟩⟨β| ⊗ a†j |α⟩⟨β|] a†j |α⟩⟨β| ⊗ aj |α⟩⟨β| − |α⟩⟨β|a†j ⊗ |α⟩⟨β|aj  j=1  +aj |α⟩⟨β| ⊗ a†j |α⟩⟨β| − |α⟩⟨β|aj ⊗ |α⟩⟨β|a†j  However, each of the terms in the above summand is equal to zero, because for any j, either a†j |α⟩ = 0 or aj |α⟩ = 0 (depending on whether mode  j is occupied or unoccupied in the state |α⟩), and likewise either ⟨β|a†j = 0 or ⟨β|aj = 0. Thus, [Λ, X ⊗ X] = 0 for any X = |α⟩⟨β|. Furthermore, X is even in fermion operators if and only if |α⟩ and |β⟩ have the same number of fermions. To see this, write the operator X as X = a†x1 ...a†xL |vac⟩⟨vac|ayM ...ay1 = a†x1 ...a†xL a1 a†1 ...aN a†N ayM ...ay1 where {x1 ...xL } and {y1 ...yM } are the sets of occupied modes in the states |α⟩ and |β⟩, respectively. Since a creation or annihilation operator is a sum of two Majorana operators, X is a sum of even products of Majorana operators if L+M is even, and is a sum of odd products of Majorana operators if L+M is odd. Thus, X is a Gaussian operator if and only if |α⟩ and |β⟩ have the same number of fermions, i.e. the bitstrings α and β have the same Hamming weight.  134  Appendix E  Diagonalizable Complex Antisymmetric Matrices Gantmacher [18] states as a Corollary to Thm 6 (p21) that for an arbitrary complex antisymmetric matrix G G = RKR−1 where R is complex orthogonal (R−1 = RT ) and µ ⊕  K=  (p ,p ) Kλkk k  ⊕  ν ⊕  K (qk )  k=1  k=1 (p ,p ) Kλkk k  and K (qk ) are given on pages 19 and where the general forms of 21 of [18], respectively. The matrices are parameterized by their elementary divisors, which are (λ±λk )pk for k = 1, 2...µ, and λqk for k = 1, 2...ν and the qk are odd numbers. Elementary divisors are the same for any two matrices that are similar, so these must also be the elementary divisors of the matrix G. It is known that a matrix is diagonalizable iff all of its elementary divisors are linear in λ (see [44] p158, Corollary 1). So, under the assumption that G is diagonalizable, all of the pk and qk are equal to one. This gives the matrix K a very desirable form, because ) ( 0 λk (1,1) Kλk = −λk 0 and K (1) = 0. This means that G is orthogonally similar to ) ⊕ µ ( ν ⊕ ( ) 0 λk 0 K= ⊕ −λk 0 k=1  k=1  Since in our case the dimension of G is even, the number of 1D zero matrices in the above direct sum must be even, so we may conclude that ) N ( ⊕ 0 zl T RGR = −zl 0 l=1  135  Appendix E. Diagonalizable Complex Antisymmetric Matrices where R is complex orthogonal and the zl are complex numbers which could be equal to zero. Furthermore, we may assume that the matrix R is not only complex orthogonal but complex special orthogonal. The argument is as follows: since RRT = I, det(R) = ±1. If det(R) = 1, then R ∈ SO(2N, C). If on the other hand det(R) = −1, then det(DR) = +1 where D := diag(−1, 1, 1....1). (DR)T (DR) = RT DDR = RT R = I, so DR ∈ SO(2N, C). Now ) ( ) ⊕ ) N ( N ( ⊕ 0 zl 0 −z1 0 zl (DR)G(DR) = D D = ⊕ z1 0 −zl 0 −zl 0 T  l=1  l=2  which fits the exact same form if we let z1 → −z1 . Thus, we can assume this has been done and that the matrix R is complex special orthogonal. A stronger condition for a matrix G than diagonalizability is normality (G is normal iff [G, G† ] = 0). In fact, normal matrices can be diagonalized by a unitary matrix. A normal complex antisymmetric matrix G can be block diagonalized by a real orthogonal transformation (See [29] p217 problem 25) as: ) N ( ⊕ 0 zl T RGR = −zl 0 l=1  where each zl is a complex number which could be zero. So, in the more restricted case that G is normal, we can assume that the orthogonal transformation R is real.  136  Appendix F  Lie Algebraic Time Evolution F.1  General Result  A Lie algebra L is a vector space endowed with a binary operation, called a Lie Bracket, that maps any pair of elements in L to an element in L. A Lie algebra is called complex if the vector space is defined over the field of complex numbers. In our quantum mechanical applications, we consider complex Lie algebras L whom admit of a representation in Hilbert Space where the Lie Bracket is the matrix commutator [A, B] = AB − BA . Thus for our purposes, a L is some linear space of matrices in a Hilbert space that are closed under commutation. Take L to be an n-dimensional space with a basis of matrices {Ta }, a = 1...n. We call the Ta a set of generators of L. Then n ∑ c [Ta , Tb ] = i fab Tc c=1 c which are called structure confor some set of complex coefficients fab stants of L. The structure constant tensor f encodes the structure of the Lie algebra. Lie algebras are important in the analysis of Lie groups, which characterize continuous transformations. In quantum mechanics, one might consider the Heisenberg time evolution of an observable that belongs to a Lie algebra under a Hamiltonian which belongs to the same Lie algebra. Mathematically, this is equivalent ¯ of the form to studying an operator X  ¯ = eY Xe−Y X ¯ ∈L where X, Y ∈ L for some Lie algebra L. We will show that in fact X as well, and we will determine it explicitly.  137  F.1. General Result ¯ = f (1) where f (λ) = eλY Xe−λY . The Taylor series First, we write: X ¯ = f (1) about λ = 0 is expansion of X ¯ = f (1) = X  ∞ ∑ 1 (k) f (0) k! k=0  where f (k) (λ) =  dk f (λ). dλk  Note that f (k) (0) = X, while  f (1) (λ) = Y eλY Xe−λY − eλY XY e−λY = eλY [Y, X]e−λY since [Y, eλY ] = 0. Similarly: f (2) (λ) = eλY [Y, [Y, X]]e−λY , and so on recursively. By repeating this a few times, it is clear that: f (k) (0) = [Y, [Y, ...[Y, X]]] where the number of nested commutators is k. Thus ¯ = X + [Y, X] + 1 [Y, [Y, X]] + 1 [Y, [Y, [Y, X]]] + ... X 2! 3! ∑n ∑n Now let X = a=1 xa Ta and Y = a=1 ya Ta be the expansion of the operators X and Y in terms of the generators of L. Then f (k) (0) = [Y, [Y, ...[Y, X]]] n ∑ 2k = ik ∗ xa0 ∗ ya1 faa12,a0 ya3 faa34,a2 ...ya2k−1 faa2k−1 ,a2k−2 Ta2k a0 ....a2k =1 n ∑  = ik  ( xa0  a0 ....a2k =1  Let Fab  k ∏  ) 2i ya2i−1 faa2i−1 ,a2i−2  Ta2k  i=1  k ∑ b := yc fca be an nxn matrix of complex numbers. We can c=1  use this definition to rewrite: f (k) (0) = ik  n ∑  xa [F k ]ab Tb , where [F k ] is the  a,b=1  matrix F to the k th power. So ¯ = X  ∞ k ∑ N ∑ i k=0  =  N ∑  k!  =  N ∑  a,b=1  xa  a,b=1  xa [F k ]ab Tb  [∞ ] ∑ (iF )k k=0  k!  Tb ab  xa [eiF ]ab Tb  a,b=1  138  F.1. General Result ¯ ∈ L, and provides its expansion in terms of the This establishes that X generators. The result can be extended to a sequence of Lie algebraic gates, in a situation where we are interested in the operator ¯ = OXO−1 = eYM ...eY2 eY1 Xe−Y1 e−Y2 ...e−YM X By repeated application of the above result ¯ = X  N ∑  xa [eiF1 ]a,b1 eYM ...eY2 Tb1 e−Y2 ...e−YM  a,b1 =1 N ∑  =  xa [eiF1 ]a,b1 [eiF2 ]b1 ,b2 ...[eiFM ]bM −1 ,bM TbM  a,b1 ,b2 ...bM =1  =  N ∑  xa [eiF1 eiF2 ...eiFM ]a,b Tb  a,b=1  where [Fk ]ab :=  n ∑  b (yk )c fca and Yk =  ∑n  a=1 (yk )a Ta .  c=1  This implies that for any operator which is an element of a Lie algebra, conjugation of this operator by the exponential of an element in the Lie algebra keeps the operator inside the algebra. Furthermore, it shows that an updated description of the operator in terms of the generators of the Lie algebra can be computed by doing matrix algebra with matrices of the same size as the dimension of the Lie algebra. This fact should have useful applications when the dimension of some physical Lie algebra is much smaller than the dimension of the Hilbert space that the operators are acting on. We will see one such application to fermions in the next Section. First, we provide a trivial example as a consistency check. Example Full Hilbert Space Lie algebra: For any Hilbert space H of dimension D, the space of all linear operators on H forms a Lie algebra. This follows from the closure of all DxD matrices under commutation. This complex Lie algebra is the that of the so-called general linear group GL(D, C), the group of all DxD invertible matrices. The set of generators of this Lie algebra can be taken to be Tab := |a⟩⟨b|, where a and b range from 1 to D, and |a⟩ picks out some orthonormal basis for H. The dimension of the Lie algebra is D2 . In this case, application of the Lie algebraic time evolution ¯ = eY Xe−Y as expected. result simply recovers the equation X  139  F.2. Application to Non-Interacting Fermions  F.2  Application to Non-Interacting Fermions  The term non-interacting fermions describes a system of N fermionic modes described by a Hamiltonian that is quadratic in fermion creation and annihilation operators, that is: H ∈ spanC {a†j ak , aj a†k , a†j a†k , aj ak } ∀j, k = 1...N . For an introduction to fermion operators, see Section 5.4. Such a Hamiltonian is called “non-interacting” because there always exists a linear canonical transformation to a set of quasiparticle operators ηj such that H can be ∑ † written in the free-fermion form: H = N j=1 λj ηj ηj (see [58], [47]). Writing a general sum of the possible quadratic products of creation and annihilation operators is more convenient using the 2N so called Majorana fermionic operators c2j−1 = aj + a†j ,  c2j = −i(aj − a†j )  for j = 1...N. For Majorana fermions, the canonical commutation relations take the form: {ck , cl } = 2δkl I. Since the Majorana operators are linear in the creation and annihilation operators, one can show that spanC {cj ck } = spanC {a†j ak , aj a†k , a†j a†k , aj ak }, where on the left hand side we allow j and k to range from 1 to 2N. Such quadratic products of the Majorana operators can be taken as the generators of a Lie algebra. If we exclude any operators proportional to the identity operator, this turns out to be the Lie algebra SO(2N, C) of the group SO(2N, C), the complex Lie algebra of orthogonal 2N ×2N matrices 9 . Exponentials of general quadratic fermion operators (the so-called Gaussian operators, see Section 6.1) constitute a 2N dimensional representation of a Lie group who’s defining representation is only 2N dimensional. This is a crucial fact underlying the efficient simulatibilty of such operators. We note that this fermionic representation of SO(2N, C) is reducible, because quadratic fermion operators do not change the fermion number parity of a state. We take as our generators of the fermionic representation of SO(2N, C) Tij := ci cj for all i ̸= j ∈ {1...2N }. Instead of indexing the generators by one index as in Ta from the previous section, we index the generators by two indices for notational convenience. Note that the anticommutation relation for the Operators in the subspace spanC {a†j ak , aj a†k } also close under commutation, and constitute the Lie algebra U(N ) of the group U (N ), the Lie group of unitary N × N matrices. 9  140  F.2. Application to Non-Interacting Fermions cj implies that cj cj = I, and for i ̸= j, ci cj = −cj ci . Thus, there are 2N (2N − 1)/2 = 2N 2 − N independent generators, excluding the identity. From the anticommutator {ck , cl } = 2δkl I it follows that [ck , cl ] = 2(δkl I− cl ck ). With a few further steps one can show that: [ci cj , ck cl ] = 2(δki cl cj + δkj ci cl − δli ck cj − δlj ci ck ) So [Tij , Tkl ] = 2(δik Tlj + δjk Til − δil Tkj − δjl Tik ) Note that we can make the following transformation [Tij , Tkl ] = 2(δjk Til − δik Tjl + δjl Tki − δil Tkj )  (F.1)  Where we have used Tlj = −Tjl and Tik = −Tki . This is only actually true when l ̸= j in the first case, and i ̸= k in the second place. Nevertheless, the combined transformation is true in general. To see why, consider the first transformation Tlj → −Tjl . This is incorrect if l=j. But Tlj only affects the expression when i=k, because of the delta function multiplying it. Similarly, Tik → −Tki is incorrect if k=i. But Tik only affects the expression when j=l, because of the delta function multiplying it. So neither transformation makes any difference unless both i=k and l=j. When i=k and l=j, the two terms which are to be changed: δik Tlj and −δjl Tik , are equal to I and -I respectively, and cancel. After the transformation, these terms are: −δik Tjl and δjl Tki , which again cancel if both i=k and l=j. So the overall expression is unchanged in all cases. From this we can extract the structure constants of the Lie algebra: 2 kl fmn,ij = (δni δkm δlj − δmi δkn δlj + δnj δki δlm − δmj δki δln ) i  (F.2)  Let the operator Y to correspond to quantum time evolution under a non-interacting fermion Hamiltonian: ∑ ∑ Y = rH = r Amn cm cn = r Amn Tmn mn  mn  were r = −it , and we leave the summation limits implicit. However, we will not assume that the operator H is Hermitian, even though this would be in the case that it represents a physical Hamiltonian. Hermiticity of H would be equivalent to the coefficient matrix A being pure imaginary. Contracting with the structure constant tensor ∑ 2r kl Fij,kl = r Amn fmn,ij = (Aki δjl − Aik δjl + Alj δik − Ajl δik ) i mn 141  F.2. Application to Non-Interacting Fermions The anticommutation relations for the Majorana fermions allow us to take the matrix A to be antisymmetric. To see this, let α := 12 (A + AT ), β := 21 (A − AT ). Then ∑ ∑ H = βmn cm cn + αmn cm cn mn  =  ∑  mn  βmm cm cm +  m  =  ∑  ∑  βmm I +  m  ∑  βmn cm cn + βnm cn cm + i  m<n  βmn {cm , cn } +  ∑  m<n  = tr(β)I +  ∑  ∑  αmn cm cn  mn  αm cm cn  mn  αmn cm cn  mn  And furthermore ¯ = ertr(β) e X  ∑ mn  αmn cm cn  Xe−rtr(β) e−  ∑ mn  αmn cm cn  ∑  =e  mn  αmn cm cn  Xe−  ∑ mn  αmn cm cn  So without any loss of generality we can take A to be equal to its antisymmetric part. Using the antisymmetry of A, then ∑ −4r kl Fij,kl = r Amn fmn,ij = (Aik δjl + Ajl δik ) i mn F is a (2N )2 x(2N )2 matrix. We can easily exponentiate it by noticing a natural tensor product structure: Fij,kl =  −4r (Aik ⊗ Ijl + Iik ⊗ Ajl ) i  where here I is the 2N x2N identity matrix. The two terms commute, so F exponentiates as ∑ [eiF ]ij,kl = [e−4rA⊗I ]ij,mn [e−4rI⊗A ]mn,kl mn  =  ∑  [e4rA ]mi δjn [e4rA ]ln δmk  mn 4rA  = [e So with U := eY U ci cj U −1 =  ∑  ]ki [e−4rA ]jl  [eiF ]ij,kl ck cl  kl  ( =  ∑ k  )( [e4rA ]ki ck  ∑ [e4rA ]lj cl  )  l  142  F.2. Application to Non-Interacting Fermions This allows us to compute the Heisenberg evolution of a quadratic fermion operator of the form ci cj . However, we may extend this result to calculate the evolution of a linear operator like U cj U −1 . This is possible because although spanC {ci cj , cj } does not close as a Lie algebra under commutation, it is still the case that [ci cj , ck ] ∈ spanC {ci cj }. This allows us to still use the result from Section F.1. To do this in a clean notation, we let the index j for cj range now from 0 to 2N, and define c0 = I. To use the machinery from Section F.1 to calculate the time evolution of a linear operator like T0j = cj , we need to determine the commutators for all cases when either l or k, or both, is equal to zero. With the indices now extended to include zero, the anticommutation relation {ck , cl } = 2δkl I is only true for j, k > 0. The commutators that involve a linear term can be evaluated explicitly. Using the identity [AB, C] = A{B, C} − {A, C}B: [ci cj , ck ] = ci {cj , ck } − {ci , ck }cj = 2δjk ci − 2δik cj So: [Tij , Tk0 ] = 2(δjk Ti0 − δik T0j ), [Tij , T0l ] = 2(δjl Ti0 − δil T0j ), and [Tij Tkl ] = [ci cj , I] = 0 where i, j, k, l ̸= 0. If we now assume only that i, j ̸= 0, then all of these are still special cases of Equation F.1: [Tij , Tkl ] = 2(δjk Til − δik Tjl + δjl Tki − δil Tkj ) So the structure constants given by Equation F.2 for the quadratic fermion Lie algebra still work for determining commutators involving linear terms, provided that i and j are nonzero. This suffices for the case at hand, since the left side of the commutators in Section F.1 only ever include the operator Y, which we still assume to be built only of quadratic fermion operators. So extending our indices to start from 0, we still have that 4rA eiF ]ki [e−4rA ]jl ij,kl = [e  Where, now   e4rA  0 0 ... 0   0 A11 . . . A1N   = exp 4r ∗  .. .. ..   0 . . . 0 AN 1 . . . AN N      1 0 ... 0   0 [e4rA ]11 . . . [e4rA ]1N    =  .. .. ..   0 . . . 0 [e4rA ]N 1 . . . [e4rA ]N N  143        F.2. Application to Non-Interacting Fermions So U cj U −1 =  ∑ [eiF ]0j,kl ck cl kl  =  ( ∑  )( [e4rA ]k0 ck  k  =  ( ∑  )( δk0 ck  k  =  ( ∑  l  ∑  ) [e4rA ]lj cl  ∑ [e−4rA ]jl cl  )  )  [e−4rA ]jl cl  l  (F.3)  l  Equation F.3 is useful in our discussion of Gaussian operators in Chapter 6. We also note that it provides an alternative derivation of one of the key steps in [58], where Terhal and DiVincenzo prove that quantum computation with non-interacting fermion operators can be efficiently classically simulated.  144  Appendix G  Supplement to MBQC on the Punctured Cylinder Code In Chapter 4, reduced the problem of simulating MBQC on punctured cylinder code states to the evaluation of the inner product ( ) ∑ ¯ γ,ρ ⟩ ⟨ϕ ¯ ⊗ ϕ ⊗ ϕ∗¯ | c¯γ,ρ |K(G′ (E)) (G.1) E1  E2  E  γ,ρ  ¯ is an effective lattice of genus 2k or 2k − 1, and k is the where G′ (E) number of holes in the set of qubits that has already been measured. In Section 7.2 we introduced a family of states |C α,β ⟩ for which evaluation of product state overlaps ⟨ϕ|C α,β ⟩ is efficient in both the size of the graph G and independent of its genus g. The states |C α,β ⟩ are labeled by two gcomponent bitstrings α and β, and form an orthonormal basis of the surface code space. If the encoded-Z cocycles are chosen to be homologous to the Ising cocycles (see Section 7.2), then the coefficients of the state |C α,β ⟩ in the encoded X-eigenbasis are 1 ∏ := g ∗ (−1)αj ∗βj +(α⊕γ)j ∗(β⊕ρ)j 2 g  cγ,ρ  j=1  Here we will show that for MBQC with punctured cylinder code states the tensor c¯γ,ρ takes this same form, and thus the state ∑ ¯ γ,ρ ⟩ c¯γ,ρ |K(G′ (E))  |C α,β ⟩,  γ,ρ  in Equation G.1 can be interpreted as a member of the family of states ¯ {|C α,β ⟩} in the code space of the surface code on the effective graph G′ (E). α,β This implies that since ⟨ϕ|C ⟩ can be evaluated in a time independent of the genus of the graph, MBQC starting with the |C α,β ⟩ states in the code space of the punctured cylinder code can be simulated with a number of steps that does not grow with the genus of the embedded graph G. 145  Appendix G. Supplement to MBQC on the Punctured Cylinder Code We begin with the cases where computation is between holes. Then, using the definition of the c¯ coefficients (Equation 4.3): 22(g−k) ∏ ∗ (−1)αj ∗βj +(α⊕γ)j ∗(β⊕ρ)j 22g k  c¯γ1 ...γk δ1 ...δk ,ρ1 ...ρk ϵ1 ...ϵk  =  j=1  ∗  (−1)  αj ∗βj +(α⊕δ)j ∗(β⊕ϵ)j  Renaming in the way we did in Equation 4.4, this is simply ∏ bitstrings (α⊕γ)j ∗(β⊕ρ)j . This is exactly the tensor of coefficients (−1) c¯γ,ρ = 212k ∗ 2k j=1 for a state of the form |C α,β ⟩ in the code space of a punctured cylinder code with 2k slots, where α and β are symmetric between the first and last k entries. When crossing holes |C α,β ⟩, we evaluate c¯γ1 ...γk δ1 ...δk−1 ,ρ1 ...ρk ϵ1 ...ϵk−1 of Equation 4.10 to be: ∏ 1 ∗ (−1)(α⊕γ)j ∗(β⊕ρ)j ∗ (−1)(α⊕δ)j ∗(β⊕ϵ)j 22k j=1 ∑ (−1)(α⊕γ⊕δ)k ∗(β⊕ρ)k (−1)(α⊕δ)k ∗(β⊕ρ)k k−1  ∗  δk  =  1 22k−1  ∗  k−1 ∏  (−1)(α⊕γ)j ∗(β⊕ρ)j ∗ (−1)(α⊕δ)j ∗(β⊕ϵ)j ∗ (−1)γk ∗(β⊕ρ)k  j=1  which is again proportional to tensor describing a |C α,β ⟩ type state in ¯ the code space of the surface code for G′ (E). Below we provide some Figures depicting further steps of MBQC on punctured cylinder code states while crossing holes. These Figures are referenced in Section 4.  146  Appendix G. Supplement to MBQC on the Punctured Cylinder Code  Figure G.1: At certain stages in the computation, more columns must be added ¯ The edge set E ¯ is shown in bold. Compare with Figure 4.10. to the graph G′ (E). Two copies of the relevant part of G are shown on the left, and the relevant part ¯ is shown on the right. of G(E)  Figure G.2: A stage of computation just before edge b of Figure 4.6 is measured. ¯ is shown in bold. Two copies of the relevant part of G are shown The edge set E ¯ is shown on the right. on the left, and the relevant part of G(E)  147  

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