Canadian Summer School on Quantum Information (CSSQI) (10th : 2010)

Topological Quantum Computation Bonesteel, Nick Jul 27, 2010

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 Nick Bonesteel, Florida State University Main original sources: Fault Tolerant Quantum Computation by Anyons, A. Yu. Kitaev, Annals Phys. 303, 2 (2003). (quant-ph/9707021) A Modular Functor Which is Universal for Quantum Computation, M.H. Freedman, M. Larsen and Z. Wang, Comm. Math. Phys. 227, 605 (2002). Some excellent reviews: Non-Abelian Anyons and Topological Quantum Computation, C. Nayak et al., Rev. Mod. Phys. 80, 1083 (2008). (arXiv:0707.1889v2) Lectures on Topological Quantum Computation, J. Preskill, Available online at: www.theory.caltech.edu/~preskill/ph219/topological.pdf Also: NEB, L. Hormozi, G. Zikos, S.H. Simon, Phys. Rev. Lett. 95 140503 (2005). S.H. Simon, NEB, M.Freedman, N, Petrovic, L. Hormozi, Phys. Rev. Lett. 96, 070503 (2006). L. Hormozi, G. Zikos, NEB, and S.H. Simon, Phys. Rev. B 75, 165310 (2007). L. Hormozi, NEB, and S.H. Simon, Phys. Rev. Lett. 103, 160501 (2009).    Stone    =0    =1    =1  The iStone    The iStone: 1 bit    The iStone 4: ~ 20 bits    The iPhone 4: ~ 2.6 x  11 10  bits    The iPod: ~ 1.4 x  12 10  bits    http://en.wikipedia.org/wiki/Hard_disk_drive   A spin-1/2 particle: “spin up”  “spin down”   A spin-1/2 particle: “spin up”  Many spin-1/2 particles:  “spin down”   A spin-1/2 particle: “spin up”  Magnetic Order  “spin down”   A spin-1/2 particle: “spin up”  Magnetic Order  “spin down”   A spin-1/2 particle: “spin up”  Magnetic Order  “spin down”  =0   A spin-1/2 particle: “spin up”  Magnetic Order  “spin down”  =1   A spin-1/2 particle: “spin up”  Magnetic Order  “spin down”  =1 Terrific for storing classical information, but useless for quantum Information.   A valence bond: 1 (↑↓ − ↓↑) = 2   A valence bond: 1 (↑↓ − ↓↑) = 2   A valence bond: 1 (↑↓ − ↓↑) = 2  Many spin-1/2 particles:   A valence bond: 1 (↑↓ − ↓↑) = 2  Use periodic boundary conditions   A valence bond: 1 (↑↓ − ↓↑) = 2  3  1  1  1   A valence bond: 1 (↑↓ − ↓↑) = 2  3  1  1  1  Odd   A valence bond: 1 (↑↓ − ↓↑) = 2  3  1  3  1  Odd   Controlled-Not  Single Qubit Rotation  ψ  Uφ    Uφ ψ   |0  |0  |0  |0  |0  |0  |1  |1  |1  |1  |1  |1  |0  |1  |1  |0  Any N qubit operation can be carried out using these two gates.  Ψf   a11  =   a  M1      a1 M     a MM   Ψi  Universal Quantum Gates Single Qubit Rotation  ψ  U φ  U φ ψ  Controlled Not |0  |0  |0  |0  |0  |0  |1  |1  |1  |1  |1  |1  |0  |1  |1  |0  Any N qubit operation can be carried out using these two gates. Ψf  =   a11    a  M1      a1 M     a MM   Ψi  One way to go…  |0  =  |1  =  Loss and DiVincenzo, ‘98  Manipulate electron spins with electric and magnetic fields to carry out quantum gates. Problem: Errors and Decoherence! May be solvable, but it won’t be easy!  Topological Order  (Wen & Niu, PRB 41, 9377  (1990))  Conventionally Ordered States: Multiple “broken symmetry” ground states characterized by a locally observable order parameter. magnetization  magnetization  m = Sz = +  1 2  m = Sz = −  1 2  Nature’s classical error correcting codes !  Topologically Ordered States: Multiple ground states on topologically nontrivial surfaces with no locally observable order parameter. odd  odd odd  even even  even odd  Nature’s quantum error correcting codes ?  even    U  U  What braid corresponds to this circuit?   A valence bond: 1 (↑↓ − ↓↑) = 2  3  1  3  1  Odd   A valence bond: 1 (↑↓ − ↓↑) = 2  3  1  3  1  Odd   A valence bond: 1 (↑↓ − ↓↑) = 2  3  1  3  1  Odd   A valence bond: 1 (↑↓ − ↓↑) = 2  3  1  1  1  Odd   A valence bond: 1 (↑↓ − ↓↑) = 2  Quantum superposition of many valence-bond states: A “spin liquid.”  Odd   A valence bond: 1 (↑↓ − ↓↑) = 2  2  2  0  2  Even   A valence bond: 1 (↑↓ − ↓↑) = 2  2  0  0  2  Even   A valence bond: 1 (↑↓ − ↓↑) = 2  Even   A valence bond: 1 (↑↓ − ↓↑) = 2  |0 3  1  3  1  Odd   A valence bond: 1 (↑↓ − ↓↑) = 2  |0 Odd   A valence bond: 1 (↑↓ − ↓↑) = 2  |1 2  0  0  2  Even   A valence bond: 1 (↑↓ − ↓↑) = 2  |1 Even            4  4  2  2  2  4  4  2  2  Even      α  +β  Environment can measure the state of the qubit by a local measurement – any quantum superposition will decohere almost instantly.  Bad Qubit!    α  +β Odd  Even  Environment can only measure the state of the qubit by a global measurement – quantum superposition should have long coherence time.  Good Qubit!    α  +β Odd  Even  Topologically Ordered States ) : Multiple ground states on topologically nontrivial surfaces with no locally observable order parameter. odd  odd odd  even even  even odd  Nature’s quantum error correcting codes ?  even        Spin flip: “quasiparticle” with total Sz =+1        Breaking a bond creates an excitation with Sz = 1    Breaking a bond creates an excitation with Sz = 1    Breaking a bond creates an excitation with Sz = 1    Sz = 1 excitation fractionalizes into two Sz = ½ quasiparticles.   B  Electrons  A two dimensional gas of electrons in a strong magnetic field B.   B  Quantum Hall Fluid  An incompressible quantum liquid can form when the Landau level filling fraction ν = nelec(hc/eB) is a rational fraction.   Electron (charge = e)  Quasiparticles (charge = e/3 for ν=1/3)  When an electron is added to a FQH state it can be fractionalized --- i.e., it can break apart into fractionally charged quasiparticles.   41, 9377 (1990)) As in our spin-liquid example, FQH states on topologically nontrivial surfaces have degenerate ground states which can only be distinguished by global measurements. Degeneracy  For the ν = 1/3 state:  1 3 9  … 1  2  3N N   Fractionally charged quasiparticles  Essential features: A degenerate Hilbert space whose dimensionality is exponentially large in the number of quasiparticles. States in this space can only be distinguished by global measurements provided quasiparticles are far apart. A perfect place to hide quantum information!   r1  r2  ψ (r1 , r2 )  One exchange  ψ (r2 , r1 ) = λ ψ (r1 , r2 )  A second exchange  ψ (r1 , r2 ) = λ2 ψ (r2 , r1 )  Two exchanges = Identity  λ = +1 Bosons Photons, He4 atoms, Gluons…  λ2 = 1 λ = −1 Fermions Electrons, Protons, Neutrons…    1 time dimension  2 space dimensions Particle “world-lines” form braids in 2+1 (=3) dimensions    Clockwise exchange  Counterclockwise exchange  Particle “world-lines” form braids in 2+1 (=3) dimensions   iϑ  ψ f = e ψi  Phase  ψi θ=0  Bosons  θ = π/3  ν=1/3 quasiparticles Anyons  θ=π  Fermions  Only possible for particles in 2 space dimensions.   ψf  ~ α ~   a11 ~ a12  α  =α  β~ψ 0= +aβ ψ 1a   β     21 22     α  ψ i =α  βψ 0 + β ψ 1   degenerate states   ψf  ~ α    a11 a12  α  =  ~  =      β   a21 a22   β   α  ψi =   β   Matrix!  Matrices form a non-Abelian representation of the braid group. (Related to the Jones Polynomial, TQFT (Witten), Conformal Field Theory (Moore, Seiberg), etc.)      Ψf  time Ψf     a11  =   a  M1  Ψi      a1 M  a MM     Ψi        Ψf  time Ψf     a11  =   a  M1  Ψi      a1 M  a MM     Ψi        Ψf  time Ψf     a11  =   a  M1      a1 M  a MM     Ψi    Ψi  Matrix depends only on the topology of the braid swept out by anyon world lines! Robust quantum computation?   Controlled-Not  Single Qubit Rotation  ψ  Uφ    Uφ ψ   |0  |0  |0  |0  |0  |0  |1  |1  |1  |1  |1  |1  |0  |1  |1  |0  Any N qubit operation can be carried out using these two gates.  Ψf   a11  =   a  M1      a1 M     a MM   Ψi    U  U  What braid corresponds to this circuit?    J.S. Xia et al., PRL (2004).    J.S. Xia et al., PRL (2004).  ν= 5/2: Probable Moore-Read Pfaffian state. Charge e/4 quasiparticles described by SU(2)2 Chern-Simons Theory. Nayak & Wilczek, ’96    J.S. Xia et al., PRL (2004).  ν= 5/2: Probable Moore-Read Pfaffian state. Charge e/4 quasiparticles described by SU(2)2 Chern-Simons Theory. Nayak & Wilczek, ’96  ν = 12/5: Possible Read-Rezayi “Parafermion” state. Read & Rezayi, ‘99 Charge e/5 quasiparticles described by SU(2)3 Chern-Simons Theory. Slingerland & Bais ’01  Universal for Quantum Computation! Freedman, Larsen & Wang ’02   Braid Group on n-Strands: Bn  Braid Group on n-Strands: Bn Some Elements of B4  Braid Group on n-Strands: Bn Some Elements of B4  Not a Braid:  Braid Group on n-Strands: Bn Some Elements of B4  Group Multiplication  ×  Braid Group on n-Strands: Bn Some Elements of B4  Group Multiplication  ×  =  Braid Group on n-Strands: Bn Some Elements of B4  Group Multiplication  ×  =  Elementary Braid Operations σi : Braid ith strand over i+1st strand 4 3 2  = σ1  = σ3  =σ2  1  σi `s and their inverses generate the braid group  σ1−1  σ2  σ1−1  σ2  σ3  Braid Relations σ i σ i +1 σ i = σ i +1 σ i σ i +1  =  σi σ j = σ j σi ,  |i− j | ≥ 2  Matrix Rep of B3 from Fib Anyons time 3  σ1 = ?  2 1  a 1  Matrix Rep of B3 from Fib Anyons time 3   e − i 4π / 5 σ 1 = R =   0  2 1  a 1  0   i 3π / 5  e   Matrix Rep of B3 from Fib Anyons time 3   e − i 4π / 5 σ 1 = R =   0  2 1  a 1  3  σ2 = ?  2 1  a 1  0   i 3π / 5  e   Matrix Rep of B3 from Fib Anyons time 3   e − i 4π / 5 σ 1 = R =   0  2 1  a  0   i 3π / 5  e   1   − τ e − iπ / 5 σ 2 = F R F =  −i 3π / 5  τe  3 2 1  a 1  τ e −i 3π / 5 −τ       Elementary Braid Matrices  e − i 4π / 5 σ 1 =   0  a  0   i 3π / 5  e   1   − τ e − iπ / 5 σ 2 = F σ 1 F =  −i 3π / 5  τe  a 1  τ e −i 3π / 5 −τ  Ψi  Ψf = 1  σ1  −1  σ2  σ1  −1  σ2 = M  1  MT       Ψi  Qubit Encoding Qubit States  0 =  0  Non-Computational State  1 1  1 =  1  0  1  State of qubit is determined by q-spin of two leftmost particles  Transitions to this state are leakage errors  Initializing a Qubit Pull two quasiparticle-quasihole pairs out of the “vacuum”.  0  0 0  These three particles have total q-spin 1  Initializing a Qubit Pull two quasiparticle-quasihole pairs out of the “vacuum”.  0  0  1  Initializing a Qubit Pull two quasiparticle-quasihole pairs out of the “vacuum”.  0  0  1  Measuring a Qubit Try to fuse the leftmost quasiparticle-quasihole pair.  ?  α 0 +β 1  1  Measuring a Qubit If they fuse back into the “vacuum” the result of the measurement is 0.  0  0  1  Measuring a Qubit If they cannot fuse back into the “vacuum” the result of the measurement is 1  1  1  1  1  Single Qubit: The Bloch Sphere 0  θ  0 +i 1  0 −i 1  2  2  φ 1  ψ = cos  θ 2  0 + sin  θ 2  e  −iφ  1  Single Qubit Operations: Rotations α = rotation vector  0  ψ  α  Direction of α is the rotation axis Magnitude of α is the rotation angle  1  ψ = cos  θ 2  0 + sin  θ 2  e  −iφ  1  Single Qubit Operations: Rotations α = rotation vector  0  α  ψ  i α ⋅σ Uα = exp 2  ψ  Uα Uα ψ  Uα ψ 1  Single Qubit Operations: Rotations 2π  −2 π  The set of all single qubit rotations lives in a solid sphere of 2π radius 2π.  α  −2 π  ψ  Uα Uα ψ  Single Qubit Operations: Rotations General rule: Braiding inside an oval does not change the total topological charge of the enclosed particles. Important consequence: As long as we braid within a qubit, there is no leakage error.  1 Can we do arbitrary single qubit rotations this way?  1  2π  σ 22  σ12  −2 π  2π  −2 π  σ1-2  σ2-2  N=1  N=2  N=3  N=4  N=5  N=6  N=7  N=8  N=9  N = 10  N = 11  Brute Force Search 0  = i 0   i 0 0  0  0  + O (10 − 3 ) 1   Brute Force Search 0  = i 0   For brute force search: Braid Length  | ln ε |  i 0 0  0  0  + O (10 − 3 ) 1  “error” ε  |lnε |  Braid Length L. Hormozi, G. Zikos, NEB, S.H. Simon, PRB ‘07  Brute Force Search 0  = i 0   i 0 0  0  0  + O (10 − 3 ) 1  “error” ε  Brute force searching rapidly becomes infeasible as braids get longer. Fortunately, a clever algorithm due to Solovay and Kitaev allows for systematic improvement of the braid given a sufficiently dense covering of SU(2).  Solovay-Kitaev Construction 0  i 0   Braid Length  | ln ε |  c  ,  c≈4  i 0 0  0  0  + O (10 − 4 ) 1  “error” ε  What About Two Qubit Gates? a  ? 1  b 1  ?  ?  ?  Problems: 1. We are pulling quasiparticles out of qubits: Leakage error! 2. 87 dimensional search space (as opposed to 3 for threeparticle braids). Straightforward “brute force” search is problematic.  Two Qubit Controlled Gates 1 b  Control qubit  a  Target qubit  1  Goal: Find a braid in which some rotation is performed on the target qubit only if the control qubit is in the state 1. (b=1)  “Weaving” a Two Qubit Gate Weave a pair of anyons from the control qubit between anyons in the target qubit. 1 b  control pair  a 1  Important Rule: Braiding a q-spin 0 object does not induce transitions. Target qubit is only affected if control qubit is in state 1 (b = 1)  “Weaving” a Two Qubit Gate Only nontrivial case is when the control pair has q-spin 1. 1  control pair a 1  We’ve reduced the problem to weaving one anyon around three others. Still too hard for brute force approach!  Try Weaving Around Just Two Anyons We’re back to B3, so this is numerically feasible. 1  control pair a 1  Question: Can we find a weave which does not lead to leakage errors?  A Trick: Effective Braiding Actual Weaving  Effective Braiding  The effect of weaving the blue anyon through the two green anyons has approximately the same effect as braiding the two green anyons twice.  Controlled-“Knot” Gate  Effective braiding is all within the target qubit  No leakage!  Not a CNOT, but sufficient for universal quantum computation.  SK Improved Controlled-“Knot” Gate  Another Trick: Injection Weaving  ≈ Step 1: Inject the control pair into the target qubit. control pair  1 0 0    0 1 0 0 0 1    Another Trick: Injection Weaving control pair  Step 2: Weave the control pair inside the injected target qubit. control pair  ≈  0 i 0    i 0 0 0 0 1    Another Trick: Injection Weaving Step 3: Extract the control pair from the target using the inverse of the injection weave.  Putting it all together we have a CNOT gate:  Injection  Rotation  Extraction  SK Improved Controlled-NOT Gate  Universal Set of Gates Single qubit rotations:  ψ  Uφ  Uφ ψ  Controlled NOT:  NEB, L. Hormozi, G. Zikos, S.H. Simon, Phys. Rev. Lett. 95 140503 (2005)  Quantum Circuit  U  U  What braid corresponds to this circuit?  Quantum Circuit  U  U  Braid  Turning any Braid into a Weave  Turning any Braid into a Weave  I  Turning any Braid into a Weave  I I  Turning any Braid into a Weave  I -1  I I  Turning any Braid into a Weave  -1  I  I -1  I I  We know it is possible to carry out universal quantum computation by moving only a single particle.  Can we find an efficient CNOT construction in which only a single particle is woven through the other particles?  Another Useful Braid: The F-Braid F-Matrix:  τ τ 0  τ 0 −τ 0 0  1  0 1 1  0  1 1 0  =  1  1  1 1  0  F-Braid:  a c  a c  ≈  i  τ τ  τ −τ  0  0  0  1  0  Single Particle Weave Gate: Part 1 1 a  b 1  Single Particle Weave Gate: Part 1 1 a  b 1  F-Braid  Single Particle Weave Gate: Part 1 1 a a c c b 1  F-Braid  Single Particle Weave Gate: Part 1 1 a  a  b  0 0 1 1  0 1 0 1  a  b  b  1  Intermediate State  b 1 1 0 1  Single Particle Weave Gate: Part 1 1 a  a  b  0 0 1 1  0 1 0 1  a  b 1  b  b 1 1 0 1  Single Particle Weave Gate: Part 2  a  ≈ b  Phase Braid  a  b  0 0 1 1  0 1 0 1  b’ = 1  b’ = 0  −1 0  0  0  −1 0  0  0  1  b 1 1 0 1  Phase  -1 -1 +1 -1  Single Particle Weave Gate: Part 2  a  b  a  b  a  b  0 0 1 1  0 1 0 1  b 1 1 0 1  -1 -1 +1 -1  Single Particle Weave Gate: Part 2  a  b  a  b  a  b  0 0 1 1  0 1 0 1  b 1 1 0 1  Phase  -1 -1 +1 -1  Single Particle Weave Gate: Part 3 a  b  0 1 0 1 a 1  0 1 0 1  a b b 1  b 1 1 0 1  Phase  -1 -1 +1 -1  Controlled-Phase Gate a  1  b  1  F-Braid  Phase-Braid  Inverse of F-Braid  1  a  1  b  Final result a  Intermediate state b’  U=−  1 0 0 0  0 0 1 0 0 −1 0 0  0 0 0 1  + O(10-3)  SK Improved Controlled-Phase Gate  Universal “One-Particle Weave” Gates Single qubit rotations:  ψ  Uφ  Uφ ψ  Controlled-Phase gate: -Z  L. Hormozi, G. Zikos, NEB, and S.H. Simon, Phys. Rev. B 75, 165310 (2007).  How Big is Shor’s Braid? How many elementary braids are required to factor a K-bit number N using Shor’s algorithm? Bottleneck: Modular Exponentiation requires ~ K3 gates.  U mod exp a  i  0  o  = a  i  x a (mod N )  o  Specific requirements: ~ 3K  Qubits  ~ 40 K3  NOT gates  ~ 28 K3  CNOT gates  ~ 92 K3  CCNOT (Toffoli) gates  Beckman, Chari, Devabhaktuni, Preskill, PRA 54, 1034 (1996).  Quantum Gates for Modular Exp NOT Gate: =  0 1   1 0    Length (measured in elementary braids) grows logarithmically with decreasing error:  LNOT ≈ 18 log10 ε Roughly same scaling seen for all “three-weaves”  G. Zikos, et al., Int. J. Mod. Phys. B 23, 2727 (2009).  Quantum Gates for Modular Exp CNOT Gate:  LCNOT ≈ 5 LNOT ≈ 90 log10 ε  = -Z  = R(π/2 y )  -Z  R(-π/2 y )  CNOT is constructed using 3 three-weaves plus 2 single qubit rotations for a total of 5 three-weaves.  LCNOT ≈ 5 LNOT ≈ 90 log10 ε  Quantum Gates for Modular Exp CCNOT (Toffoli) Gate:  (from http://www.cl.cam.ac.uk/teaching/0607/QuantComp/lecture4.pdf )  =  CCNOT can be constructed using 6 CNOTs (up to single qubit rotations on the target) and 9 single qubit rotations. So 6x3 = 18 “CNOT” threeweaves + 9 “single qubit rotation” three-weaves = 27 three-weaves.  LCCNOT ≈ 27 LNOT ≈ 486 log10 ε  Number of Elementary Braids Total number of elementary braids:  LShor ≈ 50,000 | log10 ε | K 3 For a finite probability that no error occurs, we require:  1 |ε | ~ 3 50,000 K 2  To factor a 128-bit number: Number of Fibonacci anyons  ε ~ 3 ×10 −6  ≈ 1000  Number of elementary braids ≈  6 ×1011  M. Baraban, NEB, and S. H. Simon, PRA 81, 062317 (2010)  

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