Perimeter Institute Talk 2009

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  • Perimeter Institute Talk 2009

    1. 1. Adiabatic Gate Teleportation & Its Applications Dave Bacon Department of Computer Science & Engineering University of Washington, Seattle, WA USA [joint work with Steve Flammia (Perimeter), Alice Neels (Washington), Andrew Landahl (Sandia Labs)]
    2. 2. Oops, I forgot to strap myself in!
    3. 3. Look over there! A jackalope!
    4. 4. I’ll bet most states are too entangled to be used as computational resources
    5. 5. ENIAC, 1946 17,468 vacuum tubes 7,200 crystal diodes 1,500 relays 70,000 resistors 10,000 capacitors ~5,000,000 hand-soldered joints
    6. 6. ENIAC, 1946 17,468 vacuum tubes 7,200 crystal diodes 1,500 relays 70,000 resistors 10,000 capacitors ~5,000,000 hand-soldered joints ~ one vacuum tube failure / day (equivalent to ~ one transistor failure /second today)
    7. 7. ENIAC, 1946 17,468 vacuum tubes 7,200 crystal diodes 1,500 relays 70,000 resistors 10,000 capacitors ~5,000,000 hand-soldered joints ~ one vacuum tube failure / day (equivalent to ~ one transistor failure /second today)
    8. 8. Alternative History Imagine: no transistors, no integrated circuits...
    9. 9. Alternative History Imagine: no transistors, no integrated circuits...
    10. 10. Alternative History Imagine: no transistors, no integrated circuits...
    11. 11. Alternative History Imagine: no transistors, no integrated circuits...
    12. 12. Alternative History Imagine: no transistors, no integrated circuits... Could we build a computer out of unreliable components? J. von Neumann, “Probabilistic Logics and the Synthesis of Reliable Organism from Unreliable Components” 1956
    13. 13. Von Neumann’s Solution 1x 1x 2x 1y M 3x 1z 4x 2x M 5x 2y M 1y 2z M 2y 3x 3 My 3y M π M 4y 3z 5y 4x M 4y M 1z 4z 2z M 3z 5x 4z 5y M 5z 5z gate error correction Theorem (von Neumann): A circuit of g gates can be made to fail with probability ε using O(g log (g/ε) ) gates which fail with probability p, assuming p<pth (vN estimated pth ~ 1%)
    14. 14. Engineering? We Don’t Need Your Stickin’ Engineering There are distinct physical reasons why robust classical computation is possible
    15. 15. Kitaev’s Idea “Magnetism arise from spins of individual atoms. Each spin is quite sensitive to thermal fluctuations. But the spins interact with each other and tend to be oriented in the same direction. If some spin flips to the opposite direction, the interaction forces it to flip back to the direction of other spins. This process is quite similar to the standard error correction procedure for the repetition code. We may say that errors are being corrected at the physical level. Can we propose something similar in the quantum case? Yes, but it is not so simple.” Alexei Kitaev Kitaev, “Fault-tolerant quantum computation by anyons” (1997, published 2003)
    16. 16. Topological Quantum Computing non-abelian Anyons used to build a universal quantum computer Kitaev, “Fault-tolerant quantum computation by anyons” (1997, published 2003) Freedman, “Quantum computation and the localization of modular functors” (2001)
    17. 17. Topological Quantum Computing non-abelian Anyons used to build a universal quantum computer √1 √1 H= √1 2 2 1 − √2 quantum computing 2 using representations of braid group Kitaev, “Fault-tolerant quantum computation by anyons” (1997, published 2003) Freedman, “Quantum computation and the localization of modular functors” (2001)
    18. 18. Kitaev’s Toric System plaquette operator Sp = Ze e∈δP vertex operator Sv = Xe e|v∈e [Sp , Sv ] = 0 Sp = Sv 2 2 =I periodic boundary conditions (torus)
    19. 19. Kitaev’s Toric System plaquette operator Sp = Ze e∈δP vertex operator Sv = Xe e|v∈e [Sp , Sv ] = 0 Sp = Sv 2 2 =I periodic boundary Kitaev Hamiltonian: conditions (torus) ∆ H=− Sp + Sv 4 p v
    20. 20. Controversy Is topological quantum computing fault-tolerant? gate 2x 1x 1x 1y M error correction 3x 1z 4x 2x M 5x 2y M 1y 2z M 2y 3x 3 My 3y M π M 4y 3z 5y 4x M 4y M 1z 4z 2z M 3z 5x 4z 5y M 5z 5z Theorem (von Neumann): A circuit of g gates can be made to fail with probability ε using O(g log (g/ε) ) gates which fail with probability p, assuming p<pth (vN estimated pth ~ 1%)
    21. 21. Controversy Is topological quantum computing fault-tolerant? gate 2x 1x 1x 1y M error correction 3x 1z If we use this definition of fault-tolerant, M 4x 2x 5x 2y M 1y 2z M then I believe the answer is NO! 2y 3x 3 My 3y M π M 4y 3z 5y 4x M 4y M 1z 4z 2z M 3z 5x 4z 5y M 5z 5z Theorem (von Neumann): A circuit of g gates can be made to fail with probability ε using O(g log (g/ε) ) gates which fail with probability p, assuming p<pth (vN estimated pth ~ 1%)
    22. 22. Controversy Is topological quantum computing fault-tolerant? gate 2x 1x 1x 1y M error correction 3x 1z If we use this definition of fault-tolerant, M 4x 2x 5x 2y M 1y 2z M then I believe the answer is NO! 2y 3x 3 My 3y M π M 4y 3z 5y 4x M 4y M 1z 4z 2z M 3z 5x 4z 5y M 5z 5z Theorem (von Neumann): A circuit of g gates can be made to fail with probability ε using But I also think this is the wrong definition O(g log (g/ε) ) gates which fail with probability p, assuming p<pth (vN estimated pth ~ 1%)
    23. 23. Cartoon Guide to Classical Physical Fault-Tolerance Energy E = −J si sj i,j neighbors total magnetization M= si i +1 spin “0” as M > N0 −1 si ∈ {+1, −1} “1” as M < −N0
    24. 24. Cartoon Guide to Classical Physical Fault-Tolerance Simple noise model
    25. 25. Cartoon Guide to Classical Physical Fault-Tolerance Simple noise model 1. Pick spin at random
    26. 26. Cartoon Guide to Classical Physical Fault-Tolerance Simple noise model 1. Pick spin at random 2. If flipping spin would decrease energy a. flip spin
    27. 27. Cartoon Guide to Classical Physical Fault-Tolerance Simple noise model 1. Pick spin at random 2. If flipping spin would decrease energy a. flip spin 3. If flipping spin would increase energy: “bare” noise time step a. flip spin with probability Nr runs of 1-3 exp(−β∆E) β=T −1
    28. 28. β=T −1 Memory 1D 2D m versus time 1.0 0.5 0.0 m 0.5 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 time rate ~ exp(−cβJ) m versus temperature 1.0 0.5 0.0 m 0.5 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 temperature
    29. 29. β=T −1 Memory 1D 2D m versus time m versus time m versus time 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 m m m 0.5 0.5 0.5 1.0 1.0 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 time time time rate ~ exp(−cβJ) T < TC T > TC m versus temperature 1.0 m versus temperature 1.0 0.5 0.5 0.0 m 0.0 m 0.5 0.5 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 temperature temperature
    30. 30. Fault-Tolerance 1D 2D m versus time m versus time 1.0 1.0 0.5 0.5 0.0 m 0.0 m 0.5 0.5 1.0 1.0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 time time Resilience to “gate” of flipping spins: self-correcting
    31. 31. Toy Model / Strawman plaquette operator Sp = Ze e∈δP vertex operator Sv = Xe e|v∈e [Sp , Sv ] = 0 Sp = Sv 2 2 =I periodic boundary conditions (torus)
    32. 32. Toy Model / Strawman plaquette operator Sp = Ze e∈δP vertex operator Sv = Xe e|v∈e [Sp , Sv ] = 0 Sp = Sv 2 2 =I periodic boundary Kitaev Toric Code Hamiltonian: conditions (torus) ∆ H=− Sp + Sv 4 p v
    33. 33. A Problem? Toric code is like 1D Ising: m versus time 1.0 “On Thermalization in Kitaev’s 2D 0.5 model” R. Alicki, M. Fannes, and M. 0.0 Horodecki, arXiv:0810.4584 m 0.5 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 time Decay rate for information encoded into ground state is independent of system size, but proportional to exp(−cβJ) Contrast with 4D model “On thermal stability of topological qubit in Kitaev's 4D model” R. Alicki, (M.+P. +R.) Horodecki, arXiv:0811.0033
    34. 34. Not a Problem? Factor a 1000 bit number: Naive algorithm: ~109 gates Gap to temp ratio: ~20 Seems likely that exponential suppression as function of temp well worth working for
    35. 35. Not a Problem? Factor a 1000 bit number: 18819881292060796383869723946165043 98071635633794173827007633564229888 59715234665485319060606504743045317 38801130339671619969232120573403187 9550656996221305168759307650257059 Naive algorithm: ~109 gates Gap to temp ratio: ~20 Seems likely that exponential suppression as function of temp well worth working for
    36. 36. A Problem? What about during computing? Creation and manipulation of excited states require operators like: How to do this? not How do you “create” and “move” anyons without creating errors = creating extra anyons
    37. 37. TQC: Gates “We now introduce into the system’s Hamiltonian a scalar potential composed of many local “traps,” each sufficient to capture exactly one quasiparticle. These traps may be created by impurities, by small gates, or by the potential created by tips of scanning microscopes. The quasiparticle’s charge screens the potential introduced by the trap and the quasiparticle-tip combination cannot be observed by local measurements from far away. We denote the positions of these traps by R1,...,Rk, and assume that these positions are well spaced from each other compared to the microscopic length scales. A state with quasiparticles at these positions can be viewed as an excited state of the Hamiltonian without the trap potential or, alternatively, as the ground state in the presence of the trap potential....” C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma “Non-Abelian anyons and topological quantum computation” Rev. Mod. Phys. 80, 1083 (2008).
    38. 38. TQC Problems? How does one guarantee that one can trap one and only one anyon? Every operation the previous paragraph needs to be demonstrated that it can be done with EXCEPTIONALLY small error if this is to be called physical fault-tolerance.
    39. 39. Adiabatic Dragging Energy Quantum Quantum error error correcting correcting codespace codespace Time Require: degenerate through entire evolution “Open loop” holonomic evolution
    40. 40. Adiabatic Teleportation Initial Code Final Code
    41. 41. Adiabatic Teleportation Initial Code Final Code Initial Stabilizer Final Stabilizer
    42. 42. Adiabatic Teleportation Initial Code Final Code Initial Hamiltonian Final Hamiltonian Evolution:
    43. 43. Adiabatic Teleportation Initial Hamiltonian Final Hamiltonian Energy 1.0 0.5 0.2 0.4 0.6 0.8 1.0s 0.5 1.0
    44. 44. Adiabatic Teleportation Initial Hamiltonian Final Hamiltonian
    45. 45. Adiabatic Gate Teleportation Note: spectrum unchanged by unitary Example (Hadamard gate):
    46. 46. Let’s Build a Computer One qubit gates Two qubit gates Problem: initial Hamiltonian requires 3 qubit interactions
    47. 47. Example 1 2 3 4 5 6 To get 2 qubit interactions: perturbation theory gadgets
    48. 48. Gadgets 1 2 3a 3b 1 2 3a 3b 4 5 6a 6b 4 5 6a 6b Replace 3,6 by encoded qubits: [S. D. Bartlett and T. Rudolph, Phys. Rev. A 74, 040302 (2006)]
    49. 49. Gadgets 1 2 3a 3b 1 2 3a 3b 4 5 6a 6b 4 5 6a 6b Initial ground Minimum gap: state fidelity:
    50. 50. Variations on a Theme Exchange interactions: Adiabatically preparing rotated Hamiltonians: Not all U’s allow a gap gap: gap:
    51. 51. “Impossibility” Theorem Is it possible to adiabatically SWAP with only one adiabatic interpolation? Diagonal in same basis so cannot transfer amplitude to perform SWAP
    52. 52. Architectures Sequence of Hamiltonians: Sequence of single qubit gates applied: Generalizing to more than 1 qubit: Universal QC by cycling between only 3 Hamiltonians
    53. 53. Architectures n qubit circuit 4n or 5n qubits
    54. 54. Comparison Holonomic quantum computing, but open loop open loop holomic QC: [D. Kult, J. Aberg, E. Sjoqvist, Phys. Rev. A 74, 022106 (2006)] Adiabatic universal quantum computing, but with piecewise evolution D. Aharonov et al., in 45th Annual IEEE FOCS (2004)] Path robustness in holonomy is subsystem dependent compare: [O. Oreshkov, T. A. Brun, D. A. Lidar, Phys. Rev. Lett. 102, 070502 (2009)] Piecewise keeps gap constant: compare spin-1 chain transmission of: [K. Eckert, O. Romero-Isart, and A. Sanpera, New J. Phys. 9, 155 (2007)] Details: arXiv:0905.0901
    55. 55. Back to TQC!
    56. 56. Encoded Qubits Smooth qubit Rough qubit
    57. 57. Back to TQC! Smooth qubit
    58. 58. Prepare Smooth Puncture Turn off 2 plaquette operators adjacent to a qubit where X is turned on.
    59. 59. Prepare Smooth Puncture
    60. 60. Prepare Smooth Puncture commutes with H(t) eigenspace of end up in +1 e.v. of smooth qubit encoded Z
    61. 61. Prepare Smooth Puncture Smooth qubit in +1 eigenvalue of encoded Z
    62. 62. Deform Smooth Puncture Turn on While turning off + relevant vertex operators Rough qubits: Ditto Rough qubit prepare +1 e.v. of encoded
    63. 63. Toolkit 1. Create small smooth qubit in +1 e.v. of Z 2. Create small rough qubit in +1 e.v. of X 3. Enlarge, move both punctures 4. Braid puncture: this give a CX target = rough These things all commute with each other, as of course these are Abelian anyons.
    64. 64. Universal Toolkit 1. Create small smooth qubit in +1 e.v. of Z 2. Create small rough qubit in +1 e.v. of X 3. Enlarge, move both punctures 4. Braid puncture: this give a CX target = rough 5. Measure encoded X,Z of qubits 6. Create magic states Requires distillation.
    65. 65. Measurement Simple, right? But how to perform why Hamiltonian still on? Note: destructive Pauli measurements are tolerant to errors because (1) complementary errors are large and (2) we can remove region being measured
    66. 66. Universal Toolkit 1. Create small smooth qubit in +1 e.v. of Z 2. Create small rough qubit in +1 e.v. of X 3. Enlarge, move both punctures 4. Braid puncture: this give a CX target = rough 5. Measure encoded X,Z of qubits 6. Create magic states 5+6 not tolerant to errors in this model
    67. 67. Summary Adiabatically deforming between different Hamiltonians which have stabilizer codewords as energy eigenstates yields interesting protocols like AGT Using AGT as a motivator you can understand how topological quantum computing can or cannot robustly compute (and thus can provide rigorous error bounds.)
    68. 68. Adiabatic Gate Teleportation Dave Bacon University of Washington, Seattle, WA USA [joint work with Steve Flammia (Perimeter), Alice Neels (Washington), Andrew Landahl (Sandia Labs)] 8th Symposium on Topological Quantum Computing, Zurich 29th-31st August 2009
    69. 69. Open Questions Like measurement based methods, but without having to perform correction afterwards: Can we (1) do cluster state computation in this manner? (2) other measurement based models?
    70. 70. Self-Correcting Calculate: free energy change to flip all spins = +1 = −1 1D 2D E ∝ Constant E ∝ Perimeter
    71. 71. Self-Correcting Calculate: free energy change to flip all spins = +1 = −1 1D 2D E ∝ Constant E ∝ Perimeter entropy wins
    72. 72. Self-Correcting Calculate: free energy change to flip all spins = +1 = −1 1D 2D E ∝ Constant E ∝ Perimeter entropy wins T < TC energy wins T > TC entropy wins
    73. 73. Von Neumann’s Model x xy f full adder y S f 00 1 c 01 1 1 S NAND gate 10 1 s 11 0 a M M xyz f 1 S x y 000 0 M cout M f b z 001 0 CS time 010 0 majority gate 011 1 suppose each gate fails 100 0 (outputs wrong answer) 101 1 110 1 independently and with 111 1 probability p
    74. 74. Von Neumann’s Conundrum Observation: If all gates fail with probability p, then an output bit of any circuit most fail with at least probability p Rest of M output input Circuit
    75. 75. Von Neumann’s Conundrum Observation: If all gates fail with probability p, then an output bit of any circuit most fail with at least probability p Rest of M output input Circuit “Loophole”: replace wire by “bundle” of N wires ≤βN 0s decode as “1” N ≥(1-β)N 0s decode as “0” 0<β<1/2
    76. 76. Von Neumann’s Solution 1 x Simulate the majority gate redundantly: y M f z 1x 1x 2x 1y M x 3x 1z 4x 2x 5x 2y M 1y 2z 1f M 2y 3x 2f y 4y 3y 3z 3y M 4f 3f f 5y 4x 5f 1z 4z 4y M N=5 2z z 3z 5x 4z 5y M 5z 5z
    77. 77. Von Neumann’s Solution 1 x Simulate the majority gate redundantly: y M f z 1x 1x 2x 1y M x 3x 1z 4x 2x 5x 2y M 1y 2z 1f M 2y 3x 2f y 4y 3y 3z 3y M 4f 3f f 5y 4x 5f 1z 4z 4y M N=5 2z z 3z 5x 4z 5y M 5z 5z 1. M gates may fail
    78. 78. Von Neumann’s Solution 1 x Simulate the majority gate redundantly: y M f z 1x 1x 2x 1y M x 3x 1z 4x 2x 5x 2y M 1y 2z 1f M 2y 3x 2f y 4y 3y 3z 3y M 4f 3f f 5y 4x 5f 1z 4z 4y M N=5 2z z 3z 5x 4z 5y M 5z 5z 2. incoming errors may add 1. M gates may fail
    79. 79. Von Neumann’s Solution 1 x Simulate the majority gate redundantly: y M f z 1x 1x 2x 1y M x 3x 1z 4x 2x 5x 2y M 1y 2z 1f M 2y 3x 2f y 4y 3y 3z 3y M 4f 3f f 5y 4x 5f 1z 4z 4y M N=5 2z z 3z 5x 4z 5y M 5z 5z 2. incoming errors may add 1. M gates may fail
    80. 80. Von Neumann’s Solution 2 “Restorative organ”: M Α versus Α 1.0 M 0.8 π 0.6 M 0.4 Α M 0.2 Α 0.0 0.0 0.2 0.4 0.6 0.8 1.0 M Α random permutation αN “0s” [3α2(1-α)+α3]N “0s” = α*N “0”s

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