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Topological Quantum Computation 2010

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 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). Main original sources: 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 Some excellent reviews: 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). Also: Nick Bonesteel,   Florida State University  Stone  = 0  = 1  = 1 The iStone   The iStone: 1 bit  The iStone 4: ~ 20 bits  The iPhone 4: ~ 2.6 x 1011 bits  The iPod: ~ 1.4 x 1012 bits  http://en.wikipedia.org/wiki/Hard_disk_drive  “spin up” “spin down” A spin-1/2 particle:  Many spin-1/2 particles: “spin up” “spin down” A spin-1/2 particle:  Magnetic Order “spin up” “spin down” A spin-1/2 particle:  Magnetic Order “spin up” “spin down” A spin-1/2 particle:  Magnetic Order = 0 “spin up” “spin down” A spin-1/2 particle:  Magnetic Order = 1 “spin up” “spin down” A spin-1/2 particle:  Magnetic Order = 1 “spin up” “spin down” A spin-1/2 particle: Terrific for storing classical information, but useless for quantum Information. ( )↓↑−↑↓= 2 1  A valence bond: ( )↓↑−↑↓= 2 1  A valence bond: ( )↓↑−↑↓= 2 1  A valence bond: Many spin-1/2 particles: ( )↓↑−↑↓= 2 1  A valence bond: Use periodic boundary conditions ( )↓↑−↑↓= 2 1  3 1 1 1 A valence bond: ( )↓↑−↑↓= 2 1  3 1 1 1 Odd A valence bond: ( )↓↑−↑↓= 2 1  3 1 3 1 Odd A valence bond:  |0 |0 |0 |0 |1 |0 |0 |1 |0 |1 |1 |1 |1 |1 |1 |0 Controlled-Not Any N qubit operation can be carried out using these two gates. φ U Single Qubit Rotation ψ ψφ U           MMM M aa aa    1 111 iΨ=fΨ Universal Quantum Gates |0 |0 |0 |0 |1 |0 |0 |1 |0 |1 |1 |1 |1 |1 |1 |0 Controlled Not Any N qubit operation can be carried out using these two gates. φ U Single Qubit Rotation ψ ψφ U           MMM M aa aa    1 111 iΨ=fΨ One way to go… Loss and DiVincenzo, ‘98 | 0    = |1    = 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. Topologically Ordered States:  Multiple ground states on topologically nontrivial surfaces with no locally observable order parameter. 2 1 +== zSm odd odd even odd odd even even even magnetization 2 1 −== zSm magnetization Nature’s classical error correcting codes ! Nature’s quantum error correcting codes ?  U U What braid corresponds to this circuit? ( )↓↑−↑↓= 2 1  3 1 3 1 Odd A valence bond: ( )↓↑−↑↓= 2 1  3 1 3 1 Odd A valence bond: ( )↓↑−↑↓= 2 1  3 1 3 1 Odd A valence bond: ( )↓↑−↑↓= 2 1  3 1 1 1 Odd A valence bond: ( )↓↑−↑↓= 2 1  Odd A valence bond: Quantum superposition of many valence-bond states:  A “spin liquid.” ( )↓↑−↑↓= 2 1  2 2 0 2 Even A valence bond: ( )↓↑−↑↓= 2 1  2 0 0 2 Even A valence bond: ( )↓↑−↑↓= 2 1  Even A valence bond: ( )↓↑−↑↓= 2 1  3 1 3 1 Odd |0 A valence bond: ( )↓↑−↑↓= 2 1  Odd A valence bond: |0 ( )↓↑−↑↓= 2 1  2 0 0 2 Even |1 A valence bond: ( )↓↑−↑↓= 2 1  Even A valence bond: |1      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 even odd odd even even even Nature’s quantum error correcting codes ?    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 A two dimensional gas of electrons in a strong magnetic field B. Electrons  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 1 3 9 For the ν = 1/3 state: … 3N 1 2 N  Fractionally charged quasiparticles 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. Essential features: A perfect place to hide quantum information!  ( ) ( )2112 ,, rrrr ψλψ = ( )21 , rrψ ( ) ( )12221 ,, rrrr ψλψ = Two exchanges = Identity λ2 = 1 λ = +1    Bosons λ = −1    Fermions One exchange A second exchange r1 r2 Photons, He4 atoms, Gluons… Electrons, Protons, Neutrons…  2 space dimensions 1 time dimension Particle “world-lines” form braids in 2+1 (=3) dimensions Particle “world-lines” form braids in 2+1 (=3) dimensions Clockwise exchange Counterclockwise exchange   iψ i i f e ψψ ϑ= Phase θ = 0    Bosons θ = π Fermions θ = π/3    ν=1/3 quasiparticles Only possible for particles in 2 space dimensions. Anyons  10 ψβψαψ +=i       β α 10 ~~ ψβψαψ +=f             =      β α β α 2221 1211~ ~ aa aa degenerate states             =      = β α β α ψ 2221 1211~ ~ aa aa f       = β α ψ i Matrix! Matrices form a non-Abelian representation of the braid group. (Related to the Jones Polynomial, TQFT (Witten), Conformal Field Theory (Moore, Seiberg), etc.)              MMM M aa aa    1 111 iΨ=fΨ iΨ fΨ   time   iΨ fΨ           MMM M aa aa    1 111 iΨ=fΨ   time   iΨ fΨ           MMM M aa aa    1 111 iΨ=fΨ   time Matrix depends only on the topology of the braid swept out by anyon world lines! Robust quantum computation?  |0 |0 |0 |0 |1 |0 |0 |1 |0 |1 |1 |1 |1 |1 |1 |0 Controlled-Not Any N qubit operation can be carried out using these two gates. φ U Single Qubit Rotation ψ ψφ U           MMM M aa aa    1 111 iΨ=fΨ  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   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 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 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 1σ= 2σ= 3σ= 1 3 4 2 σ1 −1 σ2 σ1 −1 σ2 σ3 σi `s and their inverses generate the braid group Braid Relations 111 +++ = iiiiii σσσσσσ = 2||, ≥−= jiijji σσσσ Matrix Rep of B3 from Fib Anyons a 1 time ?1 =σ 1 3 2 Matrix Rep of B3 from Fib Anyons a 1 time         == − 5/3 5/4 1 0 0 pi pi σ i i e e R 1 3 2 Matrix Rep of B3 from Fib Anyons a 1 time         == − 5/3 5/4 1 0 0 pi pi σ i i e e R 1 3 2 a 1 1 3 2 ?2 =σ Matrix Rep of B3 from Fib Anyons a 1 time         == − 5/3 5/4 1 0 0 pi pi σ i i e e R 1 3 2 a 1 1 3 2         − − == − −− ττ ττ σ pi pipi 5/3 5/35/ 2 i ii e eeFRF Elementary Braid  Matrices a 1     − −− ττ pipi 5/35/ ii ee         = − 5/3 5/4 1 0 0 pi pi σ i i e e     − == − ττ σσ pi 5/312 ie FF σ1 −1 σ2 σ1 −1 σ2     =   M i Ψ fΨ M T= iΨ a 1 1 1 Qubit Encoding 0 1 Non-Computational StateQubit States 1 0 =0 State of qubit is determined by q-spin of two leftmost particles Transitions to this state are leakage errors 1 1 =1 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 1 0 Initializing a Qubit Pull two quasiparticle-quasihole pairs out of the “vacuum”. 0 1 0 Measuring a Qubit Try to fuse the leftmost quasiparticle-quasihole pair. 10 βα + ? 1 Measuring a Qubit If they fuse back into the “vacuum” the result of the measurement is 0. 1 0 0 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 10 i−+ 1 φ 22 10 i 1 2 sin0 2 cos φθθψ ie−+= Single Qubit Operations: Rotations 0 ψα α = rotation vector Direction of α is the rotation axis 1 2 sin0 2 cos φθθψ ie−+=1 Magnitude of α is the rotation angle Single Qubit Operations: Rotations 0 ψ α = rotation vector α exp σαα   ⋅ = iU 1 ψαU ψ ψαU αU 2 Single Qubit Operations: Rotations 2 pi 2 pi−2 pi The set of all single qubit rotations lives in a solid sphere of α −2 pi radius 2pi. ψ ψαU αU Single Qubit Operations: Rotations Important consequence:  As long as we braid within a qubit, there is no leakage error. General rule:  Braiding inside an oval does not change the total topological charge of the enclosed particles. Can we do arbitrary single qubit rotations this way? 1 1 2 pi 2 pi−2 pi σ1 2 σ2 2 −2 pi 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 )10( 100 00 00 3−+           Oi i = Brute Force Search )10( 100 00 00 3−+           Oi i = ε“error” Braid Length Braid Length |ln| ε For brute force search: |lnε | L. Hormozi, G. Zikos, NEB, S.H. Simon, PRB ‘07 Brute Force Search )10( 100 00 00 3−+           Oi i = ε“error” 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 )10( 100 00 00 4−+           Oi i ε“error” Braid Length c|ln| ε 4≈c, 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 three- particle braids).  Straightforward “brute force” search is problematic. Two Qubit Controlled Gates b 1 Control qubit a 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) Target qubit “Weaving” a Two Qubit Gate Weave a pair of anyons from the control qubit between anyons in the target qubit. control pair b 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) a 1 “Weaving” a Two Qubit Gate Only nontrivial case is when the control pair has q-spin 1. control pair 1 a We’ve reduced the problem to weaving one anyon around three others.   Still too hard for brute force approach! 1 Try Weaving Around Just Two Anyons control pair We’re back to B3, so this is numerically feasible. 1 Question:  Can we find a weave which does not lead to leakage errors? a 1 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           100 010 001 ≈ control pair Step 1:  Inject the control pair into the target qubit. Another Trick: Injection Weaving control pair           100 00 00 i i ≈ Step 2: Weave the control pair inside the injected target qubit. control pair 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 φUψ ψφUSingle qubit rotations: 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 I I-1 Turning any Braid into a Weave I I-1 I I-1 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 1 0 1 1 1 0 1 0 = 100 0 0 ττ ττ − F-Matrix: F-Braid: 1 1 0 1 100 0 0 ττ ττ −i a a c c ≈ Single Particle Weave Gate: Part 1 1 a 1 b Single Particle Weave Gate: Part 1 1 a F-Braid 1 b Single Particle Weave Gate: Part 1 1 a a F-Braid 1 b c c Single Particle Weave Gate: Part 1 1 a a b 0 0 0 1 b a 1 1 1 1 0 1 0 1 1 b b Intermediate State Single Particle Weave Gate: Part 1 1 a a a b 0 0 0 1 b 1 1 1 1 0 1 0 1 1 b b Single Particle Weave Gate: Part 2 a a b 0 0 0 1 b 1 1 1 1 0 1 0 1 Phase -1 -1 +1 -1 100 010 001 − − b ≈ Phase Braid b’ = 1 b’ = 0 Single Particle Weave Gate: Part 2 a a a b 0 0 0 1 b 1 1 1 1 0 1 0 1 -1 -1 -1 +1 b b Single Particle Weave Gate: Part 2 a a 0 0 0 1 b 1 1 1 1 0 1 0 1 Phase -1 -1 +1 -1 a b b b Single Particle Weave Gate: Part 3 a 1 a 0 0 0 1 b 1 1 1 1 0 1 0 1 Phase -1 -1 +1 -1 a b b 1 b Controlled-Phase Gate F-Braid Inverse of F-BraidPhase-Braid a a11 1000 0100 0010 0001 − −=U + O(10-3)b’ a b b11 Intermediate state Final result SK Improved Controlled-Phase Gate Universal “One-Particle Weave” Gates φUψ ψφUSingle qubit rotations: Controlled-Phase gate: L. Hormozi, G. Zikos, NEB, and S.H. Simon, Phys. Rev. B 75, 165310 (2007). -Z 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. o a ioi NxaaU )(mod0expmod = Specific requirements: ~ 40 K3 NOT gates ~ 28 K3 CNOT gates ~ 92 K3 CCNOT (Toffoli) gates Beckman, Chari, Devabhaktuni, Preskill, PRA 54, 1034 (1996). ~   3 K    Qubits Quantum Gates for Modular Exp NOT Gate: Length (measured in elementary braids) grows logarithmically with       01 10 = decreasing error: ε10log18≈NOTL 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: ε10log905 ≈≈ NOTCNOT LL -Z = CNOT is constructed using 3 three-weaves plus 2 single qubit rotations for a total of 5 three-weaves. -Z = R(pi/2 y ) R(-pi/2 y ) ε10log905 ≈≈ NOTCNOT LL 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” three- weaves + 9 “single qubit rotation” three-weaves = 27 three-weaves. ε10log48627 ≈≈ NOTCCNOT LL 3 10 |log|000,50 KLShor ε≈ Number of Elementary Braids Total number of elementary braids: 2 1 ~|| ε For a finite probability that no error occurs, we require: 3000,50 K To factor a 128-bit number: 6103~ −×ε 11106 ×≈ Number of Fibonacci anyons 1000≈ Number of elementary braids M. Baraban, NEB, and S. H. Simon, PRA 81, 062317 (2010)

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