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Quantum error correction for quantum memories

Barbara M. Terhal

Barbara M. Terhal

  • JARA Institute for Quantum Information, RWTH Aachen University, 52056 Aachen, Germany

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Rev. Mod. Phys. 87, 307 – Published 7 April, 2015

DOI: https://doi.org/10.1103/RevModPhys.87.307

Abstract

Active quantum error correction using qubit stabilizer codes has emerged as a promising, but experimentally challenging, engineering program for building a universal quantum computer. In this review the formalism of qubit stabilizer and subsystem stabilizer codes and their possible use in protecting quantum information in a quantum memory are considered. The theory of fault tolerance and quantum error correction is reviewed, and examples of various codes and code constructions, the general quantum error-correction conditions, the noise threshold, the special role played by Clifford gates, and the route toward fault-tolerant universal quantum computation are discussed. The second part of the review is focused on providing an overview of quantum error correction using two-dimensional (topological) codes, in particular, the surface code architecture. The complexity of decoding and the notion of passive or self-correcting quantum memories are discussed. The review does not focus on a particular technology but discusses topics that will be relevant for various quantum technologies.

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References (160)

  1. Aharonov, D., and M. Ben-Or, 1997, “Fault-tolerant quantum computation with constant error,” in Proceedings of the 29th STOC, pp. 176–188 [http://arxiv.org/abs/quant-ph/9611025].
  2. Aharonov, D., and L. Eldar, 2011, “On the complexity of commuting local Hamiltonians, and tight conditions for topological order in such systems,” in Proceedings of FOCS 2011 (IEEE), pp. 334–343 [http://arxiv.org/abs/1102.0770].
  3. Aharonov, D., A. Kitaev, and J. Preskill, 2006, “Fault-Tolerant Quantum Computation with Long-Range Correlated Noise,” Phys. Rev. Lett. 96, 050504.
  4. Ahn, C., A. C. Doherty, and A. J. Landahl, 2002, “Continuous quantum error correction via quantum feedback control,” Phys. Rev. A 65, 042301.
  5. Alicea, J., 2012, “New directions in the pursuit of Majorana fermions in solid state systems,” Rep. Prog. Phys. 75, 076501.
  6. Alicki, R., M. Horodecki, P. Horodecki, and R. Horodecki, 2010, “On thermal stability of topological qubit in Kitaev’s 4D model,” Open Syst. Inf. Dyn. 17, 1.
  7. Aliferis, P., 2007, “Level Reduction and the Quantum Threshold Theorem”, Ph.D. thesis (CalTech) [http://arxiv.org/abs/quant-ph/0703230].
  8. Aliferis, P., and A. Cross, 2007, “Subsystem fault-tolerance with the Bacon-Shor code,” Phys. Rev. Lett. 98, 220502.
  9. Aliferis, P., D. Gottesman, and J. Preskill, 2006, “Quantum accuracy threshold for concatenated distance-3 codes,” Quantum Inf. Comput. 6, 97.
  10. Aliferis, P., D. Gottesman, and J. Preskill, 2008, “Accuracy threshold for postselected quantum computation,” Quantum Inf. Comput. 8, 181.
  11. Aliferis, P., and J. Preskill, 2009, “Fibonacci scheme for fault-tolerant quantum computation,” Phys. Rev. A 79, 012332.
  12. Aliferis, P., and B. M. Terhal, 2007, “Fault-tolerant quantum computation for local leakage faults,” Quantum Inf. Comput. 7, 139.
  13. Aoki, T., G. Takahashi, T. Kajiya, J.-i. Yoshikawa, S. L. Braunstein, P. van Loock, and A. Furusawa, 2009, “Quantum error correction beyond qubits,” Nat. Phys. 5, 541.
  14. Bacon, D., 2006, “Operator quantum error correcting subsystems for self-correcting quantum memories,” Phys. Rev. A 73, 012340.
  15. Barends, R., et al., 2013, “Coherent Josephson Qubit Suitable for Scalable Quantum Integrated Circuits,” Phys. Rev. Lett. 111, 080502.
  16. Barends, R., et al., 2014, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500.
  17. Barreiro, J. T., M. Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, 2011, “An open-system quantum simulator with trapped ions,” Nature (London) 470, 486.
  18. Bell, B. A., D. A. Herrera-Martí, M. S. Tame, D. Markham, W. J. Wadsworth, and J. G. Rarity, 2014, “Experimental demonstration of a graph state quantum error-correction code,” Nat. Commun. 5, 3658.
  19. Bennett, C. H., D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, 1996, “Mixed state entanglement and quantum error correction,” Phys. Rev. A 54, 3824.
  20. Bény, C., and O. Oreshkov, 2010, “General Conditions for Approximate Quantum Error Correction and Near-Optimal Recovery Channels,” Phys. Rev. Lett. 104, 120501.
  21. Bombin, H., 2010a, “Topological Order with a Twist: Ising Anyons from an Abelian Model,” Phys. Rev. Lett. 105, 030403.
  22. Bombin, H. 2010b, “Topological subsystem codes,” Phys. Rev. A 81, 032301.
  23. Bombin, H., 2013, “Gauge Color Codes,” arXiv:1311.0879.
  24. Bombin, H., R. S. Andrist, M. Ohzeki, H. G. Katzgraber, and M. A. Martin-Delgado, 2012, “Strong Resilience of Topological Codes to Depolarization,” Phys. Rev. X 2, 021004.
  25. Bombin, H., G. Duclos-Cianci, and D. Poulin, 2012, “Universal topological phase of two-dimensional stabilizer codes,” New J. Phys. 14, 073048.
  26. Bombin, H., and M. A. Martin-Delgado, 2006, “Topological Quantum Distillation,” Phys. Rev. Lett. 97, 180501.
  27. Bombin, H., and M. A. Martin-Delgado, 2007, “Topological Computation without Braiding,” Phys. Rev. Lett. 98, 160502.
  28. Bombin, H., and M. A. Martin-Delgado, 2009, “Quantum measurements and gates by code deformation,” J. Phys. A 42, 095302.
  29. Braunstein, S., 1998, “Error correction for continuous quantum variables,” Phys. Rev. Lett. 80, 4084.
  30. Bravyi, S., G. Duclos-Cianci, D. Poulin, and M. Suchara, 2013, “Subsystem surface codes with three-qubit check operators,” Quantum Inf. Comput. 13, 0963.
  31. Bravyi, S., and J. Haah, 2011, “Energy Landscape of 3D Spin Hamiltonians with Topological Order,” Phys. Rev. Lett. 107, 150504.
  32. Bravyi, S., and J. Haah, 2013, “Analytic and numerical demonstration of quantum self-correction in the 3D Cubic Code,” Phys. Rev. Lett. 111, 200501.
  33. Bravyi, S., M. B. Hastings, and S. Michalakis, 2010, “Topological quantum order: Stability under local perturbations,” J. Math. Phys. (N.Y.) 51, 093512.
  34. Bravyi, S., and A. Kitaev, 2005, “Universal quantum computation with ideal Clifford gates and noisy ancillas,” Phys. Rev. A 71, 022316.
  35. Bravyi, S., and R. Koenig, 2013, “Classification of topologically protected gates for local stabilizer codes,” Phys. Rev. Lett. 110, 170503.
  36. Bravyi, S., B. Leemhuis, and B. M. Terhal, 2011, “Topological order in an exactly solvable 3D spin model,” Ann. Phys. (Amsterdam) 326, 839.
  37. Bravyi, S., D. Poulin, and B. M. Terhal, 2010, “Tradeoffs for Reliable Quantum Information Storage in 2D Systems,” Phys. Rev. Lett. 104, 050503.
  38. Bravyi, S., and B. M. Terhal, 2009, “A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes,” New J. Phys. 11, 043029.
  39. Bravyi, S. B., and A. Yu. Kitaev, 1998, “Quantum codes on a lattice with boundary,” arXiv:quant-ph/9811052.
  40. Brell, C. G., S. Burton, G. Dauphinais, S. T. Flammia, and D. Poulin, 2014, “Thermalization, Error Correction, and Memory Lifetime for Ising Anyon Systems,” Phys. Rev. X 4, 031058.
  41. Brooks, P., and J. Preskill, 2013, “Fault-tolerant quantum computation with asymmetric Bacon-Shor codes,” Phys. Rev. A 87, 032310.
  42. Bullock, S. S., and G. K. Brennen, 2007, “Qudit surface codes and gauge theory with finite cyclic groups,” J. Phys. A 40, 3481.
  43. Castelnovo, C., and C. Chamon, 2008, “Topological order in a three-dimensional toric code at finite temperature,” Phys. Rev. B 78, 155120.
  44. Chase, B. A., A. J. Landahl, and J. Geremia, 2008, “Efficient feedback controllers for continuous-time quantum error correction,” Phys. Rev. A 77, 032304.
  45. Cross, A., G. Smith, J. A. Smolin, and Bei Zeng, 2009, “Codeword stabilized quantum codes,” IEEE Trans. Inf. Theory 55, 433.
  46. Cross, A. W., D. P. DiVincenzo, and B. M. Terhal, 2009, “A comparative code study for quantum fault tolerance,” Quantum Inf. Comput. 9, 541.
  47. Dengis, J., R. König, and F. Pastawski, 2014, “An optimal dissipative encoder for the toric code,” New J. Phys. 16, 013023.
  48. Dennis, E., A. Kitaev, A. Landahl, and J. Preskill, 2002, “Topological quantum memory,” J. Math. Phys. (N.Y.) 43, 4452.
  49. DiVincenzo, D. P., 2009, “Fault-tolerant architectures for superconducting qubits,” Phys. Scr. T137, 014020.
  50. DiVincenzo, D. P., and P. Aliferis, 2007, “Effective fault-tolerant quantum computation with slow measurements,” Phys. Rev. Lett. 98, 020501.
  51. DiVincenzo, D. P., and F. Solgun, 2013, “Multi-qubit parity measurement in circuit quantum electrodynamics,” New J. Phys. 15, 075001.
  52. Douçot, B., and L. Ioffe, 2012, “Physical implementation of protected qubits,” Rep. Prog. Phys. 75, 072001.
  53. Duclos-Cianci, G., and D. Poulin, 2010, “A renormalization group decoding algorithm for topological quantum codes,” in Proceedings of Information Theory Workshop (ITW) (IEEE, New York), pp. 1–5 [http://arxiv.org/abs/1006.1362].
  54. Duclos-Cianci, G., and D. Poulin, 2014, “Fault-tolerant renormalization group decoder for Abelian topological codes,” Quantum Inf. Comput. 14, 721.
  55. Eastin, B., and E. Knill, 2009, “Restrictions on transversal encoded quantum gate sets,” Phys. Rev. Lett. 102, 110502.
  56. Edmonds, Jack, 1965, “Paths, trees, and flowers,” Can. J. Math. 17, 449.
  57. Elzerman, J. M., R. Hanson, J. S. Greidanus, L. H. Willems van Beveren, S. De Franceschi, L. M. K. Vandersypen, S. Tarucha, and L. P. Kouwenhoven, 2003, “Few-electron quantum dot circuit with integrated charge read out,” Phys. Rev. B 67, 161308.
  58. Fowler, A., 2015, “Minimum weight perfect matching of fault-tolerant topological quantum error correction in average O(1) parallel time,” Quantum Inf. Comput. 15, 0145.
  59. Fowler, A. G., 2013a, “Accurate simulations of planar topological codes cannot use cyclic boundaries,” Phys. Rev. A 87, 062320.
  60. Fowler, A. G., 2013b, “Optimal complexity correction of correlated errors in the surface code,” arXiv:1310.0863.
  61. Fowler, A. G., M. Mariantoni, J. M. Martinis, and A. N. Cleland, 2012, “Surface codes: Towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324.
  62. Fowler, A. G., A. M. Stephens, and P. Groszkowski, 2009, “High-threshold universal quantum computation on the surface code,” Phys. Rev. A 80, 052312.
  63. Fowler, A. G., A. C. Whiteside, A. L. McInnes, and A. Rabbani, 2012, “Topological code Autotune,” Phys. Rev. X 2, 041003.
  64. Fowler, Austin G., Adam C. Whiteside, and Lloyd C. L. Hollenberg, 2012, “Towards practical classical processing for the surface code: Timing analysis,” Phys. Rev. A 86, 042313.
  65. Freedman, M. H., and M. B. Hastings, 2013, “Quantum Systems on non-k-hyperfinite complexes: a generalization of classical statistical mechanics on expander graphs,” Quantum Inf. Comput. 14, 144.
  66. Freedman, M. H., and D. A. Meyer, 2001, “Projective plane and planar quantum codes,” Found. Comput. Math. 1, 325.
  67. Fujii, K., M. Negoro, N. Imoto, and M. Kitagawa, 2014, “Measurement-free topological protection using dissipative feedback,” Phys. Rev. X 4, 041039.
  68. Ghosh, J., and A. G. Fowler, 2015, “A leakage-resilient scheme for the measurement of stabilizer operators in superconducting quantum circuits,” Phys. Rev. A 91, 020302(R).
  69. Ghosh, J., A. G. Fowler, and M. R. Geller, 2012, “Surface code with decoherence: An analysis of three superconducting architectures,” Phys. Rev. A 86, 062318.
  70. Gladchenko, S., D. Olaya, E. Dupont-Ferrier, B. Douçot, L. B. Ioffe, and M. E. Gershenson, 2009, “Superconducting nanocircuits for topologically protected qubits,” Nat. Phys. 5, 48.
  71. Glancy, S., and E. Knill, 2006, “Error analysis for encoding a qubit in an oscillator,” Phys. Rev. A 73, 012325.
  72. Gottesman, D., 1997, “Stabilizer Codes and Quantum Error Correction,” Ph.D. thesis (CalTech) [http://arxiv.org/abs/quant-ph/9705052].
  73. Gottesman, D., 1999a, “Fault-tolerant quantum computation with higher-dimensional systems,” Chaos Solitons Fractals 10, 1749.
  74. Gottesman, D., 1999b, “The Heisenberg representation of quantum computers,” in Group22: Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, edited by S. P. Corney, R. Delbourgo, and P. D. Jarvis, pp. 32–43 (International Press, Cambridge, MA) [http://arxiv.org/abs/quant-ph/9807006].
  75. Gottesman, D., 2000, “Fault-tolerant quantum computation with local gates,” J. Mod. Opt. 47, 333.
  76. Gottesman, D., 2009, “An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation,” arXiv:0904.2557.
  77. Gottesman, D., 2014, “Fault-Tolerant Quantum Computation with Constant Overhead,” Quantum Inf. Comput. 14, 1338.
  78. Gottesman, D., and I. L. Chuang, 1999, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations,” Nature (London) 402, 390.
  79. Gottesman, D., A. Yu. Kitaev, and J. Preskill, 2001, “Encoding a qubit in an oscillator,” Phys. Rev. A 64, 012310.
  80. Guth, L., and A. Lubotzky, 2014, “Quantum error-correcting codes and 4-dimensional arithmetic hyperbolic manifolds,” J. Math. Phys. (N.Y.) 55, 082202.
  81. Haah, J., 2011, “Local stabilizer codes in three dimensions without string logical operators,” Phys. Rev. A 83, 042330.
  82. Haroche, S., M. Brune, and J.-M. Raimond, 2007, “Measuring the photon number parity in a cavity: from light quantum jumps to the tomography of non-classical field states,” J. Mod. Opt. 54, 2101.
  83. Haroche, S., and J.-M. Raimond, 2006, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University Press, Oxford).
  84. Harrington, J., 2004, “Analysis of quantum error-correcting codes: symplectic lattice codes and toric codes,” Ph.D. thesis (CalTech) [http://thesis.library.caltech.edu/1747/].
  85. Hastings, M. B., 2013, “Decoding in hyperbolic spaces: LDPC Codes with linear rate and efficient error correction,” arXiv:1312.2546.
  86. Hastings, M. B., and A. Geller, 2014, “Reduced space-time and time costs using dislocation codes and arbitrary ancillas,” arXiv:1408.3379.
  87. Herold, M., E. T. Campbell, J. Eisert, and M. J. Kastoryano, 2014, “Cellular-automaton decoders for topological quantum memories,” arXiv:1406.2338.
  88. Horsman, C., A. G. Fowler, S. Devitt, and R. Van Meter, 2012, “Surface code quantum computing by lattice surgery,” New J. Phys. 14, 123011.
  89. Jones, C., 2013a, “Multilevel distillation of magic states for quantum computing,” Phys. Rev. A 87, 042305.
  90. Jones, C., 2013b, “Logic Synthesis for Fault-Tolerant Quantum Computers,” Ph.D. thesis (Stanford) [http://arxiv.org/abs/1310.7290].
  91. Katzgraber, H. G., and R. S. Andrist, 2013, “Stability of topologically-protected quantum computing proposals as seen through spin glasses,” J. Phys. Conf. Ser. 473, 012019.
  92. Kerckhoff, J., L. Bouten, A. Silberfarb, and H. Mabuchi, 2009, “Physical model of continuous two-qubit parity measurement in a cavity-QED network,” Phys. Rev. A 79, 024305.
  93. Kerckhoff, J., H. I. Nurdin, D. S. Pavlichin, and H. Mabuchi, 2010, “Designing quantum memories with embedded control: photonic circuits for autonomous quantum error correction,” Phys. Rev. Lett. 105, 040502.
  94. Kerckhoff, J., D. S. Pavlichin, H. Chalabi, and H. Mabuchi, 2011, “Design of nanophotonic circuits for autonomous subsystem quantum error correction,” New J. Phys. 13, 055022.
  95. Kitaev, A., 2003, “Fault-tolerant quantum computation by anyons,” Ann. Phys. (Amsterdam) 303, 2.
  96. Kitaev, A., 2006, “Protected qubit based on a superconducting current mirror,” arXiv:cond-mat/0609441.
  97. Kitaev, Alexei, 2006, “Anyons in an exactly solved model and beyond,” Ann. Phys. (Amsterdam) 321, 2.
  98. Kitaev, A. Yu., 1997, “Quantum computations: algorithms and error correction,” Russ. Math. Surv. 52, 1191.
  99. Knill, E., 2005, “Quantum computing with realistically noisy devices,” Nature (London) 434, 39.
  100. Knill, E., and R. Laflamme, 1997, “A theory of quantum error-correcting codes,” Phys. Rev. A 55, 900.
  101. Knill, E., R. Laflamme, and W. Zurek, 1998, “Resilient quantum computation,” Science 279, 342.
  102. Koch, J., M. Yu. Terri, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, 2007, “Charge-insensitive qubit design derived from the Cooper pair box,” Phys. Rev. A 76, 042319.
  103. Koenig, R., G. Kuperberg, and B. W. Reichardt, 2010, “Quantum computation with Turaev-Viro codes,” Ann. Phys. (Amsterdam) 325, 2707.
  104. Kovalev, A. A., and L. P. Pryadko, 2013, “Fault-tolerance of quantum low-density parity check codes with sublinear distance scaling,” Phys. Rev. A 87, 020304.
  105. Kribs, D., R. Laflamme, and D. Poulin, 2005, “Unified and generalized approach to quantum error correction,” Phys. Rev. Lett. 94, 180501.
  106. Lalumière, K., J. M. Gambetta, and A. Blais, 2010, “Tunable joint measurements in the dispersive regime of cavity QED,” Phys. Rev. A 81, 040301.
  107. Landahl, A. J., J. T. Anderson, and P. R. Rice, 2011, “Fault-tolerant quantum computing with color codes,” arXiv:1108.5738.
  108. Leghtas, Z., G. Kirchmair, B. Vlastakis, R. J. Schoelkopf, M. H. Devoret, and M. Mirrahimi, 2013, “Hardware-efficient autonomous quantum memory protection,” Phys. Rev. Lett. 111, 120501.
  109. Leung, D. W., M. A. Nielsen, I. L. Chuang, and Y. Yamamoto, 1997, “Approximate quantum error correction can lead to better codes,” Phys. Rev. A 56, 2567.
  110. Levin, M. A., and X.-G. Wen, 2005, “String-net condensation: A physical mechanism for topological phases,” Phys. Rev. B 71, 045110.
  111. Levy, J. E., A. Ganti, C. A. Phillips, B. R. Hamlet, A. J. Landahl, T. M. Gurrieri, R. D. Carr, and M. S. Carroll, 2009, “The impact of classical electronics constraints on a solid-state logical qubit memory,” in Proceedings of the 21st annual symposium on Parallelism in algorithms and architectures (ACM, New York), pp. 166–168 [http://arxiv.org/abs/0904.0003].
  112. Lidar, D. A., 2014, “Review of decoherence free subspaces, noiseless subsystems, and dynamical decoupling,” Adv. Chem. Phys. 154, 295.
  113. Lidar, D. A., and T. A. Brun, 2013, Eds., Quantum Error Correction (Cambridge University Press, Cambridge, England).
  114. Lloyd, S., and J.-J. Slotine, 1998, “Analog quantum error correction,” Phys. Rev. Lett. 80, 4088.
  115. Menicucci, N. C., 2014, “Fault-tolerant measurement-based quantum computing with continuous-variable cluster states,” Phys. Rev. Lett. 112, 120504.
  116. Metcalfe, M., E. Boaknin, V. Manucharyan, R. Vijay, I. Siddiqi, C. Rigetti, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, 2007, “Measuring the decoherence of a quantronium qubit with the cavity bifurcation amplifier,” Phys. Rev. B 76, 174516.
  117. Michnicki, K. P., 2014, “3D Topological quantum memory with a power-law energy barrier,” Phys. Rev. Lett. 113, 130501.
  118. Mirrahimi, M., Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, 2013, “Dynamically protected cat-qubits: a new paradigm for universal quantum computation,” arXiv:1312.2017.
  119. Müller, M., K. Hammerer, Y. L. Zhou, C. F. Roos, and P. Zoller, 2011, “Simulating open quantum systems: from many-body interactions to stabilizer pumping,” New J. Phys. 13, 085007.
  120. Napp, J., and J. Preskill, 2013, “Optimal Bacon-Shor codes,” Quantum Inf. Comput. 13, 490.
  121. Ng, H. K., and P. Mandayam, 2010, “Simple approach to approximate quantum error correction based on the transpose channel,” Phys. Rev. A 81, 062342.
  122. Nielsen, A. E. B., 2010, “Fighting decoherence in a continuous two-qubit odd- or even-parity measurement with a closed-loop setup,” Phys. Rev. A 81, 012307.
  123. Nielsen, M. A., and I. L. Chuang, 2000, Quantum computation and quantum information (Cambridge University Press, Cambridge, England).
  124. Nielsen, M. A., and D. Poulin, 2007, “Algebraic and information-theoretic conditions for operator quantum error-correction,” Phys. Rev. A 75, 064304(R).
  125. Nigg, D., M. Müller, E. A. Martinez, P. Schindler, M. Hennrich, T. Monz, M. A. Martin-Delgado, and R. Blatt, 2014, “Quantum computations on a topologically encoded qubit,” Science 345, 302.
  126. Nigg, S. E., and S. M. Girvin, 2013, “Stabilizer quantum error correction toolbox for superconducting qubits,” Phys. Rev. Lett. 110, 243604.
  127. Nussinov, Z., and G. Ortiz, 2009, “Symmetry and Topological Order,” Proc. Natl. Acad. Sci. U.S.A. 106, 16944.
  128. Paetznick, A., and B. W. Reichardt, 2013, “Universal fault-tolerant quantum computation with only transversal gates and error correction,” Phys. Rev. Lett. 111, 090505.
  129. Poulin, D., 2005, “Stabilizer formalism for operator quantum error correction,” Phys. Rev. Lett. 95, 230504.
  130. Poulin, D., 2006, “Optimal and efficient decoding of concatenated quantum block codes,” Phys. Rev. A 74, 052333.
  131. Preskill, J., 1998, “Fault-tolerant quantum computation,” in Introduction to Quantum Computation (World Scientific, Singapore), pp. 213–269.
  132. Raussendorf, R., and J. Harrington, 2007, “Fault-tolerant quantum computation with high threshold in two dimensions,” Phys. Rev. Lett. 98, 190504.
  133. Raussendorf, R., J. Harrington, and K. Goyal, 2007, “Topological fault-tolerance in cluster state quantum computation,” New J. Phys. 9, 199.
  134. Reichardt, Ben W., 2005, “Quantum universality by distilling certain one- and two-qubit states with stabilizer operations,” Quantum Inf. Process. 4, 251.
  135. Shor, P. W., 1996, “Fault-tolerant quantum computation,” in Proceedings of 37th FOCS (IEEE Computer Society, Washington, DC), pp. 56–65.
  136. Steane, A., 1999, “Quantum Reed-Muller codes,” IEEE Trans. Inf. Theory 45, 1701.
  137. Steane, A., 2003, “Overhead and noise threshold of fault-tolerant quantum error correction,” Phys. Rev. A 68, 042322.
  138. Steane, A. M., 1997, “Active stabilization, quantum computation, and quantum state synthesis,” Phys. Rev. Lett. 78, 2252.
  139. Suchara, M., S. Bravyi, and B. Terhal, 2011, “Constructions and noise threshold of topological subsystem codes,” J. Phys. A 44, 155301.
  140. Suchara, M., A. W. Cross, and J. M. Gambetta, 2014, “Leakage suppression in the toric code,” arXiv:1410.8562.
  141. Suchara, M., A. Faruque, C.-Y. Lai, G. Paz, F. T. Chong, and J. Kubiatowicz, 2013, “Comparing the overhead of topological and concatenated quantum error correction,” arXiv:1312.2316.
  142. Sun, L., et al., 2014, “Tracking photon jumps with repeated quantum non-demolition parity measurements,” Nature (London) 511, 444.
  143. Svore, K. M., A. W. Cross, I. L. Chuang, and A. V. Aho, 2006, “A flow-map model for analyzing pseudothresholds in fault-tolerant quantum computing,” Quantum Inf. Comput. 6, 193.
  144. Svore, K. M., D. P. DiVincenzo, and B. M. Terhal, 2007, “Noise threshold for a fault-tolerant two-dimensional lattice architecture,” Quantum Inf. Comput. 7, 297.
  145. Terhal, B. M., F. Hassler, and D. P. DiVincenzo, 2012, “From Majorana fermions to topological order,” Phys. Rev. Lett. 108, 260504.
  146. Tillich, Jean-Pierre, and Gilles Zémor, 2014, “Quantum LDPC codes with positive rate and minimum distance proportional to n12,” IEEE Trans. Inf. Theory 60, 1193.
  147. Vandersypen, L. M. K., and I. L. Chuang, 2005, “NMR techniques for quantum control and computation,” Rev. Mod. Phys. 76, 1037.
  148. van Handel, R., and H. Mabuchi, 2005, “Optimal error tracking via quantum coding and continuous syndrome measurement,” arXiv:quant-ph/0511221.
  149. Vasconcelos, H., L. Sanz, and S. Glancy, 2010, “All-optical generation of states for “Encoding a qubit in an oscillator”,” Opt. Lett. 35, 3261.
  150. Wang, C., J. Harrington, and J. Preskill, 2003, “Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory,” Ann. Phys. (Amsterdam) 303, 31.
  151. Wang, D. S., A. G. Fowler, and L. C. L. Hollenberg, 2011, “Quantum computing with nearest neighbor interactions and error rates over 1%,” Phys. Rev. A 83, 020302(R).
  152. Weissman, M. B., 1988, “1/f noise and other slow, nonexponential kinetics in condensed matter,” Rev. Mod. Phys. 60, 537.
  153. Wiseman, H., and G. J. Milburn, 2010, Quantum Measurement and Control (Cambridge University Press, Cambridge).
  154. Wootton, J. R., 2012, “Quantum memories and error correction,” J. Mod. Opt. 59, 1717.
  155. Wootton, J. R., J. Burri, S. Iblisdir, and D. Loss, 2014, “Decoding non-Abelian topological quantum memories,” Phys. Rev. X 4, 011051.
  156. Yao, X.-C., et al., 2012, “Experimental demonstration of topological error correction,” Nature (London) 482, 489.
  157. Yoshida, Beni, 2011, “Feasibility of self-correcting quantum memory and thermal stability of topological order,” Ann. Phys. (Amsterdam) 326, 2566.
  158. Zeng, Bei, Andrew W. Cross, and Isaac L. Chuang, 2011, “Transversality versus universality for additive quantum codes,” IEEE Trans. Inf. Theory 57, 6272.
  159. Zhang, J., C. Xie, K. Peng, and P. van Loock, 2008, “Anyon statistics with continuous variables,” Phys. Rev. A 78, 052121.
  160. Zhou, X., D. W. Leung, and I. L. Chuang, 2000, “Methodology for quantum logic gate construction,” Phys. Rev. A 62, 052316.

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Rev. Mod. Phys. 87, 307– Published 7 April, 2015
  • Received 17 March 2014

© 2015 American Physical Society

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