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Measurement-based quantum computation Browne, Dan; Raussendorf, Robert 2010-07-29

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Measurement-based quantum computation  10th Canadian Summer School on QI  Dan Browne  Dept. of Physics and Astronomy University College London Thursday, 29 July 2010  What is a quantum computer?  Thursday, 29 July 2010  The one-way quantum computer A multi-qubit entangled “resource state” E.g. cluster state  +  Single-qubit measurements Choice of bases specify computation Adaptive (some bases depend upon previous outcomes)  = Thursday, 29 July 2010  A universal quantum computer  Overview of Lectures ★  1. Cluster states and graph states I ★  ★  2. The one-way quantum computer ★  ★  Thursday, 29 July 2010  How do we build them?  5. Beyond cluster states (If time permits) ★  ★  Stabilizer formalism and graphical representation  4. Cluster states and graph states III ★  ★  What is the model? How does it work?  3. Cluster states and graph states II ★  ★  What are they? Basic properties.  (other models of MBQC)  6. (Robert Raussendorf) Fault tolerant MBQC  Pauli group and Clifford Group • Pauli group: • Set of all n-fold tensor products of X, Y, Z and I, n  with pre-factors +1, -1. +i, -i for group closure.  • Clifford group: • “Normalizer” of  n  Set of unitaries C such that  ∀σk ∈  n  Equiv,:  Cσk C = σj †  Cσk = σj C = (Cσk C†)C  “Maps Pauli group onto Pauli group” Thursday, 29 July 2010  σj ∈  n  2-qubit gates and universal q.c.  Thursday, 29 July 2010  J  J  J  J  J  J  J  J  J  J  J  J  J  J  J  2-qubit gates and universal q.c.  Thursday, 29 July 2010  J  J  J  J  J  J  J  J  J  J  J  J  J  J  J  2-qubit gates and universal q.c.  Thursday, 29 July 2010  J  J  J  J  J  J  J  J  J  J  J  J  J  J  J  dimensional arrays of ype next-neighbor inltonian Hint ෇ g͑t͒ 3  quirement is not essential as will be explained below [see (d)]. In the first step of the computation, a subset of qubits is measured in the basis of sz which effectively removes them. In Fig. 1 these qubits are denoted by “ Ø.”  Etching with z measurements  ͑a͒ ͑a0 ͒ a0 ͘ sz sz  [7] whose xternally. A possible ussed below. A qubit at “quantum wire” j0͘ z,a or j1͘a ϵ j1͘z,a , forflip logical single sz͑a͒ ase operator qubitthe compustates form ally be in an arbitrary jaj2 1 jbj2 ෇ 1. For z-measurements all qubits in the su, an eigenstate of the 6͘ ෇ 6j6͔͘. Hint is tely chosen finite time , by which a unitary Hint acts uniformly on oring particles become e quantum state jF͘C ,  information flow  quantum gate  FIG. 1. Quantum computation by measuring two-state particles on a lattice. Before the measurements the qubits are in the cluster state jF͘C of (1). Circles Ø symbolize measurements of sz , vertical arrows are measurements of sx , while tilted arrows refer to measurements in the x-y plane.  Figure 1in H. J. Briegel and R. Raussendorf, Phys. Rev. Lett. • 86(22)͞5188(4)$15.00 © 2001 The American Physical Society 86, 5188 (2001)  Thursday, 29 July 2010  qubit measurements on a cluster state. The explanation of why and how these gates work will be given in Section II G.  UΣ,CN OT =  Measurement-patterns for gates in the one-way model (a)  1 2 3 4 5 control X Y Y Y Y Y 8 target X X X Y X 9 10 11 12 13  6 7 Y X 14 15  CNOT-gate  (b)  (c)  1 2 3 4 5 X +_ ! +_# +_ "  1 2 3 4 5 X X +_ # X  general rotation  z-rotation  (d)  (e)  1 2 3 4 5 X Y Y Y  1 2 3 4 5 X X Y X  Hadamard-gate  (c)  γx (t) γx (c) γz (t) γz  = = = =  Therein, the si on the qubits i. dundant cluster state on which t a set {κa } differe cludes the prese its functioning is An arbitrary on a chain of 5 representation  URo  where the rotati  π/2-phase gate  U FIG. 2: Realization of elementary quantum gates on the QCC . square represents a lattice qubit. The A squares in the R. Raussendorf, D.Each E. Browne and H.J. Briegel, Phys. Rev. 68: 022312 extreme left column marked with white circles denote the inThursday, 29 July 2010 put qubits, those in the right-most column denote the output  (2003)  Initially, the firs  3  3  2  3  No. 1 −−−−→ No. 2 −−−−→ No. 3 −−−−→  3  No. 9  3  No. 10 1  2  4  2  3  No. 1  No. 2  2  4  2  No. 5  4  2  3  −→  No. 8 1  For b ∈ Na , we compute 4  2  3  No. 9  4  3  2 3  2  4  = σxa σzNa · σxb σzNb +Na  = Ka Apply LC!Rule  1  4  2 3  No. 11  1  4  4  τ τ † 1 a Kc (Ua ) = Kc = Kc . U  Uaτ Kb (Uaτ )† = (−iσxa ) iσzb σxb σza σzNb \a  3  No. 10 1  3  2  1  No. 7  (44)1  1  2  states  Fig. 4. – An example for a successive application of the LC-rule, which exhibits the whole Proof of Proposition 5: equivalence class associated with graph No. 1. Let G be a graph with correlation operators Kb The rule is successively applied to the vertex, 4 2 4 2 4 and G = τa (G) the corresponding graph under which is colored red in the figure.  No. 6  2  −→  4  · Kb .  4 3  Fig. 4. – An example for a successive applicaFigure inLC-rule, M. Hein, W. exhibits Dür, J. the Eisert, tion of4the which wholeR. Raussendorf, M. Van equivalence associated with graph No. 1. (Varena lectures) Nest, H.-J. class Briegel, quant-ph/0602096 s Kb The rule is successively applied to the vertex, which is colored red in the figure. Thursday, 29 July 2010 nder  •  Apply LC!Rule  local at vertex a3 with correlation operators Kb . For c ∈ V \ Na we find 3 complementation 3  3  −→  3  1  No. 4  No.1 10 −−−−→ No. 1 11  1  ssive first apding wing  2 No. 3  No. 11  1  raph states |G and No. |G4 −are iff6 the 1 LC-equivalent 3 1 − − − → No. 5 − − − − → No. − − − −→ LC-orbit of local Clifford equivalent ce of local complementations, i.e. G =4 τ ◦ . . . ◦1 3 No. 7 −−−−→ No. 8 −−−−→a1No. 9 −−−−→  3  den  ation of Q(x) discussed above does not simplify or change the form of the four stabilize n Fig. 2.  Counter-example to LU-LC conjecture  The two graphs (with / without the red edge) represent locally equivalent FIG. 3: Corresponding Graph States states, but are not related by the LC rule. Zhengfeng Ji et al, Quantum Inf. Comput., Vol. 10, No. 1&2, 97-108, 2010 Thursday, 29 July 2010  of unitary gates can be implemented by local measurements on an elementary unit and these units are combined (bottom-up approach). Here we took, in contrast, a  Universal resources derived via graphical rules (a)  (b)  Hexagonal  Triangular  (c)  (d)  Kagome  Square 3 1 2  Figure can 2. Examples of efficient universal resource forlattices MQC. These are graphmeasurements These regular graphs all be mapped into square via Pauli states corresponding to (a) hexagonal, (b) triangular and (c) Kagome lattices. Deterministic transformation from (a) to (d)for (2Duniversal cluster state)MBQC. via (b) - using the graphical rules.LOCC Hence all are resources and (c) is indicated, where simple graph rules can be used sequentially (σy and σz measurements are displayed by and ♦, respectively). The spatial overhead for the transformation is constant.  M.Van den Nest, et al, New J. Phys. 9 204 (2007). Thursday, 29 July 2010  of unitary gates can be implemented by local measurements on an elementary unit and these units are combined (bottom-up approach). Here we took, in contrast, a  Universal resources derived via graphical rules (a)  (b)  Hexagonal  Triangular  (c)  (d)  Kagome  Square 3 1 2  Figure can 2. Examples of efficient universal resource forlattices MQC. These are graphmeasurements These regular graphs all be mapped into square via Pauli states corresponding to (a) hexagonal, (b) triangular and (c) Kagome lattices. Deterministic transformation from (a) to (d)for (2Duniversal cluster state)MBQC. via (b) - using the graphical rules.LOCC Hence all are resources and (c) is indicated, where simple graph rules can be used sequentially (σy and σz measurements are displayed by and ♦, respectively). The spatial overhead for the transformation is constant.  M.Van den Nest, et al, New J. Phys. 9 204 (2007). Thursday, 29 July 2010  qubit measurements on a cluster state. The explanation of why and how these gates work will be given in Section II G.  UΣ,CN OT =  Measurement-patterns for gates in the one-way model (a)  Generate the full Clifford group  1 2 3 4 5 control X Y Y Y Y Y 8 target X X X Y X 9 10 11 12 13  6 7 Y X 14 15  CNOT-gate  (b)  (c)  1 2 3 4 5 X +_ ! +_# +_ "  1 2 3 4 5 X X +_ # X  general rotation  z-rotation  (d)  (e)  1 2 3 4 5 X Y Y Y  1 2 3 4 5 X X Y X  Hadamard-gate  (c)  γx (t) γx (c) γz (t) γz  = = = =  Therein, the si on the qubits i. dundant cluster state on which t a set {κa } differe cludes the prese its functioning is An arbitrary on a chain of 5 representation  URo  where the rotati  π/2-phase gate  U FIG. 2: Realization of elementary quantum gates on the QCC . square represents a lattice qubit. The A squares in the R. Raussendorf, D.Each E. Browne and H.J. Briegel, Phys. Rev. 68: 022312 extreme left column marked with white circles denote the inThursday, 29 July 2010 put qubits, those in the right-most column denote the output  (2003)  Initially, the firs  6  7  Graphical “Gottesman-Knill Theorem” Quantum Fourier Transform  No. 1 21  22  23  8  24  25  17  18  9  10  26  11  27  28  12  29  19  20  13  14  30  31  32 33  20 1  2  3  16  5  No. 2 21  29  25  17  15  4  33  19  10  8  12  14 16  6  1  6  7  2  3  4  3  5  FIG. 16: The graph associated with the QFT on 3 qubits in the one-way quantum computer is represented in denote the input (left) and output (right) vertices. Graph No. 2 is obtained from the first after performing all P to the protocol in33Ref. [3], except from the σx -measurements at the input vertices. More precisely, it is obt 29 σy -measurements on the vertices 22, 23, 24, 26, 27, 28, 30, 31, 32 and σx -measurements on the vertices 2, 4, 7, performed.  No. 2 21  1  5  Standard-form measurement pattern 25  17  19  Graph state after all Pauli measurements performed  ror correction, quantum communication, and quantum computation in the context of the one-way quantum computer. The 14 Schmidt measure is tailored for a comparably detailed account on the quantum correlations grasping genuine multi-particle 16 entanglement, yet it turns out to be computable for many graph states. We have presented a number of general rules that can be applied when approaching the problem of evalu-  as neutral atoms in optical lattice dates to realize such graph states [  10 8 12 M. Hein, J. Eisert and H.J. Briegel, Phys. Rev. A 69, 062311 (2004) In this paper, the Schmidt mea  Thursday, 29 July 2010  6  quantify the degree of entanglem the Schmidt rank in the bipartite sufficiently coarse to be accessibl  Valence bond PEPS to MPS “Virtual Qudit pairs” d �  n=1  |n�|n�  PEPS “projector” D � d �  j=1 p,q=1  Ajp,q |j��p|�q|  M (j)m,n =  j Ap,q  Constructs a Matrix Product State MPS |ψ� =  Thursday, 29 July 2010  �  s1 ,...sn  Tr[M (s1 )M (s2 ) . . . M (sn )] |s1 �|s2 � · · · |sn �  References • •  •  Thursday, 29 July 2010  Progress Review  •  H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, M.Van den Nest, Nature Physics 5 1, 19-26 (2009)  Tutorials  •  M. Hein, W. Dür, J. Eisert, R. Raussendorf, M.Van den Nest, H.-J. Briegel, quant-ph/0602096 (Varena lectures on graph states)  • •  D. E. Browne and H. J. Briegel, quant-ph/0603226 M.A. Nielsen, quant-ph/0504097  And many more...  •  Search arxiv for One-way, MBQC, Cluster States, Graph States, ....   Fault-tolerant quantum computation with cluster states Robert Raussendorf, UBC  QI10 @ UBC, July 29, 2010  Our setting elementary cell qubit  • 2D/3D: Nearest-neighbor translation-invariant interaction. • High fault-tolerance threshold  Our setting  Fault-tolerant quantum computation Task: • Maintain the quantum speedup in the presence of decoherence. Solution: Fault-tolerance theorem∗: If for a universal quantum computer the noise per elementary operation is below a constant nonzero error threshold then arbitrarily long quantum computations can be performed efficiently with arbitrary accuracy.  *: Aharonov & Ben-Or (1996), Kitaev (1997), Knill & Laflamme & Zurek (1998), Aliferis & Gottesman & Preskill (2005)  Talk outline  1. Universal cluster state computation. • The scheme: computation by local measurements • Cluster states: creation, definition, experiment  2. General introduction to quantum error-correction  3. Making cluster state computation fault-tolerant  Part I: Cluster state quantum computation  1.1 Cluster state quantum computation  measurement of Z ( ), X (↑), cos α X + sin α Y (  )  • Universal computational resource: cluster state. • Information written onto the cluster, processed and read out by one-qubit measurements only. R. Raussendorf and H.J Briegel, PRL 86, 5188 (2001).  1.2 Cluster states - creation |0 a + |1 a √ on d-dimensional qubit 2 a∈C  1. Prepare product state lattice C.  2. Apply the Ising interaction for a fixed time T : −i gT ¯ h  UIsing = e  (i) (j)  i,j  σz σz , with  π gT = . ¯ h 4 • Interaction time T independent of cluster size.  1.2 Cluster states - simple examples  |ψ 2 = |0 1|+ 2 + |1 1|− 2 Bell state |ψ 3 = |+ 1|0 2|+ 3 + |− 1|1 2|− 3 GHZ-state |ψ  4  = |0 1 |+ 2 |0 3 |+ 4 + |0 1 |− 2 |1 3 |− 4 + + |1 1 |− 2 |0 3 |+ 4 + |1 1 |+ 2 |1 3 |− 4  Number of terms exponential in number of qubits!  Z  1.2 Cluster states - definition Z  X  Z  Z  A cluster state |φ C on a cluster C is the single common eigenstate of the stabilizer operators {Ka}, Ka|φ C = |φ C , ∀a ∈ C, with K a = Xa  Zb,  ∀a ∈ C.  b∈N (a)  Therein, b ∈ N (a) if a,b are spatial next neighbors in C.  (1)  1.2 Cluster states - experiment  Cold atoms in optical lattices [1,2]  The QCC with photons [3].  1: Greiner, Mandel, Esslinger, H¨ ansch, and Bloch, Nature 415, 39-44 (2002), 2: Greiner, Mandel, H¨ ansch and Bloch, Nature, 419, 51-54 (2002). 3: P. Walther et al., Nature 434, 169 (2005).  Part II: Introduction to quantum error correction  ... take a break from cluster states  2.1 Quantum vs. classical bits z  y  E  Ψ x  V  1  1 0  0  s = 1,0  Ψ = α 0 +β 1  quantum bit  classical bit  − Measurement affects state − Set of states continuous  − Mmnt does not affect state − Set of states discrete  Despite the differences: • Quantum error-correction (QEC) is possible. • QEC is based on classical error correction.  2.2 Starting point: Classical EC An example: the repitition code. bit 1 encoded state 0 erroneous states encoded state 1  2  3  0 0 0 0 1 0 0 1 1 1 1 1  Individually read out all bits & Perform majority vote  • Procedure on n-bit code corrects n−1 errors. 2 • Error-correction procedure learns encoded state. If state measurement does not help in classical EC why bother that it can’t be done in QEC?  2.2 Starting point: Classical EC Same effect without state measurement: Read out parities only.  bit 1 encoded state 0 erroneous states encoded state 1  2  3  0 0 0 0 1 0 0 1 1 1 1 1  Parity check Syndrome matrix  ( )=( 1 1  1 1 0 0 1 1  )  code word w. errors  () 0  1 0  • Syndrome only reveals error, not encoded state: Sy(c) = 0, ∀ codewords c. Sy(E ⊕ c) ≡ Sy(E).  (2)  Learning the state is not crucial for classical error-correction.  2.3 How Quantum Error Correction works Classical-to-quantum dictionary: c ∈ {000, 111}  −→  |Ψ = α|000 + β|111  Errors: bit flip  −→  spin & phase flips σx, σy , σz  −→  stabilizer operators Z1 ⊗ Z2, Z2 ⊗ Z3  −→  Measured eigenvalues of stabilizer operators.  Parity check matrix 1 1 0 0 1 1 Syndrome  2.3 How Quantum Error Correction works  • Repeated measurement of the stabilizer operators, and conditional correction. • Correctable errors anti-commute with at least one stabilizer operator → error-syndrome. • Syndrome informs about an error, not the encoded state.  Residual error after correction  Emergence of the error threshold  infinite-size code  large code small code  error threshold  error probability p  Fault-tolerance theorem: For a universal quantum computer, an error per gate < is effectively as good as zero error.  So far... • Have explained the basics of quantum error-correction.  • Have ignored: – Errors introduced by error-correction itself. – Computation.  ... but that can be fixed  Part III: Fault-tolerant quantum computation with 3D cluster states  Part III outline  3.1 Topological quantum error-correction with 3D cluster states  3.2 Topologocal quantum gates  3.3 Fault-tolerance threshold, overhead scaling, mapping to 2D  Known threshold values no constraint [1] [2] [3] [4]  0.03, est. 10 -3 , est.  geometric constraint 2D 1D -3  [5]  7 .10 , est.  [6]  2.10 , bd.  -4  10 , est. -5 10 , bd.  -5  [7]  -8  10 , bd.  • Error sources: |+ -Preparation, Λ(Z)-gates, Hadamard gates, measurement. [1] Knill, (2005); [2] Zalka (1999); [3] Dawson & Nielsen (2005); [4] Aliferis & Gottesman & Preskill (2005), [5] Raussendorf & Harrington, quant-ph/0610062; [6] Svore & DiVincenzo & Terhal, quant-ph/0604090, [7] Aharonov & Ben-Or (1999)  Main idea  3D cluster state = faulttolerant substrate  Gates from non-trivial cluster topology  Macroscopic view  Three cluster regions: V (Vacuum), D (Defect) and S (Singular qubits). Qubits q ∈ V : Qubits q ∈ D: Qubits q ∈ S:  local X-measurements, local Z-measurements, local measurements of X ± Y .  R. Raussendorf, J. Harrington and K. Goyal, Ann. Phys. 321, 2242 (2006).  Microscopic view cluster edges  z  2 1 0  0  1  x  2  elementary cell of L qubit location:  (even, odd, odd)  -  face of L,  qubit location:  (odd, odd, even)  -  edge of L,  syndrome location:  (odd, odd, odd)  -  cube of L,  syndrome location:  (even, even, even)  -  site of L.  Lattice duality L ←→ L  Translation by vector (1, 1, 1)T : • Cluster C invariant, • L (primal) −→ L (dual).  face edge site cube  of of of of  L L L L  − − − −  edge of L, face of L, cube of L, site of L,  • Many objects in this scheme exist as ‘primal’ and ‘dual’.  (3)  Part III.1: Quantum Error-correction in 3D cluster states  3.1 Measuring the cluster state stabilizer  2  X-measurement  Z  Z-measurement Z  3  X Z 4  But ...  1  5  Z  K1 = X1Z2Z3Z4Z5  3.1 Measuring the cluster state stabilizer  2  X-measurement  Z  Z-measurement Z  3  X  1  5  Z  K1 = X1Z2Z3Z4Z5  Z 4  λX,1 λZ,2 λZ,3 λZ,4 λZ,5 = +1. ±1 ±1 ±1 ±1 ±1 Measure eigenvalue of K1 by local measurements on qubits 1 - 5. But ...  3.1 Measuring the cluster state stabilizer 2  X-measurement  Z  Z-measurement Z  3  X  1  5  Z  Z 4  But ...  all measurements in cluster region V are in the X-basis.  Are there stabilizer elements that we can measure by local Xmeasurements only? Criterion: Xa.  KJ = a∈J  3.1 Measuring the cluster state stabilizer Criterion:  KJ =  a∈J Xa .  Such stabilizer elements exist! X1  X-measurement 2  Example:  5  X X  X  3  X 6  4  X  X1X2X3X4X5X6 = K1K2K3K4K5K6 Correlation of measured eigenvalues: λX,1 λX,2 λX,3 λX,4 λX,5 λX,6 = +1, if no error. ±1 ±1 ±1 ±1 ±1 ±1  3.1 Measuring the cluster state stabilizer  X1  X-measurement 2  5  X X  X  3  X 6  λX,1λX,2λX,3λX,4λX,5λX,6 = -1  4  X  indicates an error.  Error syndrome • One bit of error syndrome per lattice cell.  3.1 Measuring the cluster state stabilizer Error  Z-error on face qubit yields non-trivial syndrome on adjacent cells. non-trivial non-trivial error syndrome error syndrome -1 -1  • Each error leaves characteristic signature in the syndrome. • Identify error by that syndrome.  3.1 Geometry and topology Error  located on dual edge multiple edges = chain  Error syndrome supported on closed surface  3.1 Geometry and topology error chain  e1  f1 Error syndrome 1  f  2  Error syndrome 2  • An error chain Z(e) is detected by a syndrome Sy(f ) if e and f interesect an even number of times. • Intersection number is a topological invariant.  3.1 Geometry and topology error chain  e1  f f1 Error syndrome 1  e2  equivalent error chain  f  2  Error syndrome 2  • Homologically equivalent error chains have same effect on the computation: e2 = e1 + ∂f −→ Z(e2) ≡ Z(e1). • Only need to identify the homology class of the error.  3.1 Topological error-correction  • Topological error-correction in 3D cluster states described by Random plaquette Z2-gauge model (RPGM) [1]. • FT quantum memory with toric code described by RPGM as well [1]. [1] Dennis et al., quant-ph/0110143 (2001).  3.1 Phase diagram of the RPGM Map error correction to statistical mechanics:  T  Nishimori line  no EC  optimal Error correction [1]  EC  Minimum weight chain matching [2]  3%  p  • Have an error budget of 3%. [1] T. Ohno et al., quant-ph/0401101 (2004). [2] E. Dennis et al., quant-ph/0110143 (2001); J. Edmonds, Canadian J. Math. 17, 449 (1965).  Part III.2: Topological quantum gates  3.2 Encoded quantum gates  Z-measurement removes qubit from the cluster  • Local Z-measurements remove the qubits in region D from the cluster. • Remaining cluster has non-trivial topology.  3.2 Encoded quantum gates  Surface perpendicular to “time” supports a quantum code  3.2 Surface codes  • Storage capacity of the code depends upon the topology of the code surface.  3.2 The surface code One qubit located on every edge  plaquette stabilizer Bp  Z Z Z Z X  X X  X  X X X  syndrome at endpoint  X X X X  site stabilizer As Z = Z Z Z Z Z Z ZX Z Z X  harmless error  harmful error  = X  |ψ = As|ψ = Bp|ψ , ∀|ψ ∈ HC , ∀s, p.  (4)  • Surface codes are stabilizer codes associated with 2D lattices. • Only the homology class of an error chain matters. A. Kitaev,quant-ph/9707021 (1997).  3.2 The surface code syndrome at endpoint  harmful error  Non-correctable error: small weight-distance away from non-trivial cycle.  3.2 Surface code on plane with holes X site stabilizer not enforced X X X primal hole Z plaquette stabilizer not enforced Z dual hole  • There are two types of holes: primal and dual. • A pair of same-type holes constitutes a qubit.  Z Z  3.2 Surface code on plane with holes  Surface code with boundary: primal qubit  dual qubit  Xp  Zd dual hole  primal hole  Zp primal hole rough boundary  Xd dual hole smooth boundary  • X-chain cannot end in primal hole, can end in dual hole. • Z-chain can end in primal hole, cannot end in dual hole.  3.2 Encoded quantum gates  Defect D = worldline of hole.  3.2 Encoded quantum gates  Topological quantum gates are encoded in the way worldlines of primal and dual holes are braided.  primal qubit  dual qubit  Xp  3.2 A CNOT-gate  Zd dual hole  primal hole  Zp primal hole rough boundary  Xc  Xc Xt • Propagation relation: Xc −→ Xc ⊗ Xt. • Remaining prop rel Zc → Zc, Xt → Xt, Zt → Zc ⊗ Zt for CNOT derived analogously.  Xd dual hole smooth boundary  3.2 Topological quantum gates  3.2 Universal gate set • Need one non-Clifford element: √ |A . fault-tolerant preparation of |A := X+Y 2  Encoding of |A .  • FT prep. of |A provided through realization of magic state distillation∗. *: S. Bravyi and A. Kitaev, Phys. Rev. A 71, 022316 (2005).  Part III.3: Threshold and Overhead Scaling  3.3 Sequential cluster state creation  measurement  conveyor belt for sequential cluster state creation  depth d  How large or small can the depth d be?  3.3 Sequential cluster state creation  measurement  conveyor belt for sequential cluster state creation  How large?  depth d  Limitation: • Temporal order of measurements −→ accumulating memory error.  3.3 Sequential cluster state creation  measurement  conveyor belt for sequential cluster state creation  How small?  depth d  •  d ≥ 1.  - d = 1: mapping to circuit model, d ≥ 2: cluster state.  3.3 Error model • Locality: Errors are assocoated with the elementary gates of a quantum computer. Errors act where the gates act.  • Independence: Errors associated with different gates are stochastically independent.  • Probabilistic error-model: The elementary errors are probabilistic Pauli-flips σx, σy , σz on all qubits.  All these assumptions can be relaxed. Example: “Fault-tolerance with long-range correlated noise” (Dorit Aharonov, Alexei Kitaev, John Preskill, Phys. Rev. Lett. 96 (2006) 050504)  3.3 Error Model • Error sources: 2D (d = 1)  3D (d ≥ 2)  1. |+ -preparation  1. |+ -preparation  2. cPhase-gates  2. cPhase-gates  3. Hadamard-gates  3. Memory error  4. Local measurement  4. Local measurement  • Every quantum operation has same error p.  • Instant classical processing.  3.3 Fault-tolerance threshold  Topological threshold in cluster region V : pc = 7.5 × 10−3 (2D), pc = 6.8 × 10−3 (3D).  (5)  Purification threshold for fault-tolerant |A -preparation: pc = 3.7 × 10−2.  (6)  Topological EC sets the overall threshold.  3.3 Fault-tolerance threshold  fidelity of error correction  0.92 0.9 0.88 0.86 0.84 0.82 0.8 0.78 0.76  l l l l l l  = 3 = 5 = 7 = 9 = 11 = 13  0.007  0.0072  0.0074 0.0076 0.0078 error probability p  0.008  Numerical estimate of the fault-tolerance threshold in 2D.  3.3 Overhead and Robustness • Denote by S (S ) the bare (encoded) size of a quantum circuit. Then, for the described method: S ∼ S log3 S.  (7)  • The threshold is robust against variations in the error model such as higher weight elemetary errors, long-distance errors.  3.3 Overhead in absolute terms  Operational cost of a fault-tolerant gate, at 1/3 threshold.  Summary  • The 3D cluster state has error-correction built-in • Encoded gates by topology • High error threshold of 0.7% Reading: Raussendorf and Harrington, Phys. Rev. Lett. 98, 190504 (2007). Raussendorf, Harrington and Goyal, New J. Phys. 9, 199 (2007).  

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