Quantum error correction for quantum memories

Reviews of Modern Physics

Volumn 87, Issue 2, 2015, Pages 307-346

  Terhal, Barbara M ^a  

^a RWTH AACHEN UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

CODES (SYMBOLS); ERROR CORRECTION; FAULT TOLERANCE; QUANTUM COMPUTERS; QUANTUM OPTICS;

CODE CONSTRUCTION; ENGINEERING PROGRAM; NOISE THRESHOLDS; QUANTUM ERROR CORRECTIONS; QUANTUM INFORMATION; QUANTUM MEMORY; QUANTUM TECHNOLOGIES; STABILIZER CODES;

QUANTUM NOISE;

EID: 84930198343     PISSN: 00346861     EISSN: 15390756     Source Type: Journal    
DOI: 10.1103/RevModPhys.87.307     Document Type: Review

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. PRLTAO 0031-9007 10.1103/PhysRevLett.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. PLRAAN 1050-2947 10.1103/PhysRevA.65.042301
5. Alicea, J., 2012, " New directions in the pursuit of Majorana fermions in solid state systems," Rep. Prog. Phys. 75, 076501. RPPHAG 0034-4885 10.1088/0034-4885/75/7/076501
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. OSIDFU 1230-1612 10.1142/S1230161210000023
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. PRLTAO 0031-9007 10.1103/PhysRevLett.98.220502
9. Aliferis, P., D. Gottesman, and J. Preskill, 2006, " Quantum accuracy threshold for concatenated distance-3 codes," Quantum Inf. Comput. 6, 97. QICUAW 1533-7146
10. Aliferis, P., D. Gottesman, and J. Preskill, 2008, " Accuracy threshold for postselected quantum computation," Quantum Inf. Comput. 8, 181. QICUAW 1533-7146
11. Aliferis, P., and J. Preskill, 2009, " Fibonacci scheme for fault-tolerant quantum computation," Phys. Rev. A 79, 012332. PLRAAN 1050-2947 10.1103/PhysRevA.79.012332
12. Aliferis, P., and B. M. Terhal, 2007, " Fault-tolerant quantum computation for local leakage faults," Quantum Inf. Comput. 7, 139. QICUAW 1533-7146
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. NPAHAX 1745-2473 10.1038/nphys1309
14. Bacon, D., 2006, " Operator quantum error correcting subsystems for self-correcting quantum memories," Phys. Rev. A 73, 012340. PLRAAN 1050-2947 10.1103/PhysRevA.73.012340
15. Barends, R., 2013, " Coherent Josephson Qubit Suitable for Scalable Quantum Integrated Circuits," Phys. Rev. Lett. 111, 080502. PRLTAO 0031-9007 10.1103/PhysRevLett.111.080502
16. Barends, R., 2014, " Superconducting quantum circuits at the surface code threshold for fault tolerance," Nature (London) 508, 500. NATUAS 0028-0836 10.1038/nature13171
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. NATUAS 0028-0836 10.1038/nature09801
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. NCAOBW 2041-1723
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. PLRAAN 1050-2947 10.1103/PhysRevA.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. PRLTAO 0031-9007 10.1103/PhysRevLett.104.120501
21. Bombin, H., 2010 a, " Topological Order with a Twist: Ising Anyons from an Abelian Model," Phys. Rev. Lett. 105, 030403. PRLTAO 0031-9007 10.1103/PhysRevLett.105.030403
22. Bombin, H. 2010 b, " Topological subsystem codes," Phys. Rev. A 81, 032301. PLRAAN 1050-2947 10.1103/PhysRevA.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. PRXHAE 2160-3308 10.1103/PhysRevX.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. NJOPFM 1367-2630 10.1088/1367-2630/14/7/073048
26. Bombin, H., and M .A. Martin-Delgado, 2006, " Topological Quantum Distillation," Phys. Rev. Lett. 97, 180501. PRLTAO 0031-9007 10.1103/PhysRevLett.97.180501
27. Bombin, H., and M. A. Martin-Delgado, 2007, " Topological Computation without Braiding," Phys. Rev. Lett. 98, 160502. PRLTAO 0031-9007 10.1103/PhysRevLett.98.160502
28. Bombin, H., and M. A. Martin-Delgado, 2009, " Quantum measurements and gates by code deformation," J. Phys. A 42, 095302. JPAMB5 1751-8113 10.1088/1751-8113/42/9/095302
29. Braunstein, S., 1998, " Error correction for continuous quantum variables," Phys. Rev. Lett. 80, 4084. PRLTAO 0031-9007 10.1103/PhysRevLett.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. QICUAW 1533-7146
31. Bravyi, S., and J. Haah, 2011, " Energy Landscape of 3D Spin Hamiltonians with Topological Order," Phys. Rev. Lett. 107, 150504. PRLTAO 0031-9007 10.1103/PhysRevLett.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. PRLTAO 0031-9007 10.1103/PhysRevLett.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. JMAPAQ 0022-2488 10.1063/1.3490195
34. Bravyi, S., and A. Kitaev, 2005, " Universal quantum computation with ideal Clifford gates and noisy ancillas," Phys. Rev. A 71, 022316. PLRAAN 1050-2947 10.1103/PhysRevA.71.022316
35. Bravyi, S., and R. Koenig, 2013, " Classification of topologically protected gates for local stabilizer codes," Phys. Rev. Lett. 110, 170503. PRLTAO 0031-9007 10.1103/PhysRevLett.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. APNYA6 0003-4916 10.1016/j.aop.2010.11.002
37. Bravyi, S., D. Poulin, and B. M. Terhal, 2010, " Tradeoffs for Reliable Quantum Information Storage in 2D Systems," Phys. Rev. Lett. 104, 050503. PRLTAO 0031-9007 10.1103/PhysRevLett.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. NJOPFM 1367-2630 10.1088/1367-2630/11/4/043029
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. PRXHAE 2160-3308 10.1103/PhysRevX.4.031058
41. Brooks, P., and J. Preskill, 2013, " Fault-tolerant quantum computation with asymmetric Bacon-Shor codes," Phys. Rev. A 87, 032310. PLRAAN 1050-2947 10.1103/PhysRevA.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. JPAMB5 1751-8113 10.1088/1751-8113/40/13/013
43. Castelnovo, C., and C. Chamon, 2008, " Topological order in a three-dimensional toric code at finite temperature," Phys. Rev. B 78, 155120. PRBMDO 1098-0121 10.1103/PhysRevB.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. PLRAAN 1050-2947 10.1103/PhysRevA.77.032304
45. Cross, A., G. Smith, J. A. Smolin, and Bei Zeng, 2009, " Codeword stabilized quantum codes," IEEE Trans. Inf. Theory 55, 433. IETTAW 0018-9448 10.1109/TIT.2008.2008136
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. QICUAW 1533-7146
47. Dengis, J., R. König, and F. Pastawski, 2014, " An optimal dissipative encoder for the toric code," New J. Phys. 16, 013023. NJOPFM 1367-2630 10.1088/1367-2630/16/1/013023
48. Dennis, E., A. Kitaev, A. Landahl, and J. Preskill, 2002, " Topological quantum memory," J. Math. Phys. (N.Y.) 43, 4452. JMAPAQ 0022-2488 10.1063/1.1499754
49. DiVincenzo, D. P., 2009, " Fault-tolerant architectures for superconducting qubits," Phys. Scr. T137, 014020. 10.1088/0031-8949/2009/T137/014020 PHSTBO 0031-8949
50. DiVincenzo, D. P., and P. Aliferis, 2007, " Effective fault-tolerant quantum computation with slow measurements," Phys. Rev. Lett. 98, 020501. PRLTAO 0031-9007 10.1103/PhysRevLett.98.020501
51. DiVincenzo, D. P., and F. Solgun, 2013, " Multi-qubit parity measurement in circuit quantum electrodynamics," New J. Phys. 15, 075001. NJOPFM 1367-2630 10.1088/1367-2630/15/7/075001
52. Douçot, B., and L. Ioffe, 2012, " Physical implementation of protected qubits," Rep. Prog. Phys. 75, 072001. RPPHAG 0034-4885 10.1088/0034-4885/75/7/072001
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. QICUAW 1533-7146
55. Eastin, B., and E. Knill, 2009, " Restrictions on transversal encoded quantum gate sets," Phys. Rev. Lett. 102, 110502. PRLTAO 0031-9007 10.1103/PhysRevLett.102.110502
56. Edmonds, Jack, 1965, " Paths, trees, and flowers," Can. J. Math. 17, 449. CJMAAB 0008-414X 10.4153/CJM-1965-045-4
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. PRBMDO 0163-1829 10.1103/PhysRevB.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. QICUAW 1533-7146
59. Fowler, A. G., 2013 a, " Accurate simulations of planar topological codes cannot use cyclic boundaries," Phys. Rev. A 87, 062320. PLRAAN 1050-2947 10.1103/PhysRevA.87.062320
60. Fowler, A. G., 2013 b, " 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. PLRAAN 1050-2947 10.1103/PhysRevA.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. PLRAAN 1050-2947 10.1103/PhysRevA.80.052312
63. Fowler, A. G., A. C. Whiteside, A. L. McInnes, and A. Rabbani, 2012, " Topological code Autotune," Phys. Rev. X 2, 041003. PRXHAE 2160-3308 10.1103/PhysRevX.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. PLRAAN 1050-2947 10.1103/PhysRevA.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. QICUAW 1533-7146
66. Freedman, M. H., and D. A. Meyer, 2001, " Projective plane and planar quantum codes," Found. Comput. Math. 1, 325. 1615-3375 10.1007/s102080010013
67. Fujii, K., M. Negoro, N. Imoto, and M. Kitagawa, 2014, " Measurement-free topological protection using dissipative feedback," Phys. Rev. X 4, 041039. PRXHAE 2160-3308 10.1103/PhysRevX.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). PLRAAN 1050-2947 10.1103/PhysRevA.91.020302
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. PLRAAN 1050-2947 10.1103/PhysRevA.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. NPAHAX 1745-2473 10.1038/nphys1151
71. Glancy, S., and E. Knill, 2006, " Error analysis for encoding a qubit in an oscillator," Phys. Rev. A 73, 012325. PLRAAN 1050-2947 10.1103/PhysRevA.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., 1999 a, " Fault-tolerant quantum computation with higher-dimensional systems," Chaos Solitons Fractals 10, 1749. CSFOEH 0960-0779 10.1016/S0960-0779(98)00218-5
74. Gottesman, D., 1999 b, " 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. JMOPEW 0950-0340 10.1080/09500340008244046
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. QICUAW 1533-7146
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. NATUAS 0028-0836 10.1038/46503
79. Gottesman, D., A. Yu. Kitaev, and J. Preskill, 2001, " Encoding a qubit in an oscillator," Phys. Rev. A 64, 012310. PLRAAN 1050-2947 10.1103/PhysRevA.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. JMAPAQ 0022-2488 10.1063/1.4891487
81. Haah, J., 2011, " Local stabilizer codes in three dimensions without string logical operators," Phys. Rev. A 83, 042330. PLRAAN 1050-2947 10.1103/PhysRevA.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. JMOPEW 0950-0340 10.1080/09500340701391118
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. NJOPFM 1367-2630 10.1088/1367-2630/14/12/123011
89. Jones, C., 2013 a, " Multilevel distillation of magic states for quantum computing," Phys. Rev. A 87, 042305. PLRAAN 1050-2947 10.1103/PhysRevA.87.042305
90. Jones, C., 2013 b, " 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. JPCSDZ 1742-6588 10.1088/1742-6596/473/1/012019
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. PLRAAN 1050-2947 10.1103/PhysRevA.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. PRLTAO 0031-9007 10.1103/PhysRevLett.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. NJOPFM 1367-2630 10.1088/1367-2630/13/5/055022
95. Kitaev, A., 2003, " Fault-tolerant quantum computation by anyons," Ann. Phys. (Amsterdam) 303, 2. 10.1016/S0003-4916(02)00018-0 APNYA6 0003-4916
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. APNYA6 0003-4916 10.1016/j.aop.2005.10.005
98. Kitaev, A. Yu., 1997, " Quantum computations: algorithms and error correction," Russ. Math. Surv. 52, 1191. 0036-0279 10.1070/RM1997v052n06ABEH002155
99. Knill, E., 2005, " Quantum computing with realistically noisy devices," Nature (London) 434, 39. NATUAS 0028-0836 10.1038/nature03350
100. Knill, E., and R. Laflamme, 1997, " A theory of quantum error-correcting codes," Phys. Rev. A 55, 900. PLRAAN 1050-2947 10.1103/PhysRevA.55.900
101. Knill, E., R. Laflamme, and W. Zurek, 1998, " Resilient quantum computation," Science 279, 342. SCIEAS 0036-8075 10.1126/science.279.5349.342
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. PLRAAN 1050-2947 10.1103/PhysRevA.76.042319
103. Koenig, R., G. Kuperberg, and B. W. Reichardt, 2010, " Quantum computation with Turaev-Viro codes," Ann. Phys. (Amsterdam) 325, 2707. APNYA6 0003-4916 10.1016/j.aop.2010.08.001
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. PLRAAN 1050-2947 10.1103/PhysRevA.87.020304
105. Kribs, D., R. Laflamme, and D. Poulin, 2005, " Unified and generalized approach to quantum error correction," Phys. Rev. Lett. 94, 180501. PRLTAO 0031-9007 10.1103/PhysRevLett.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. PLRAAN 1050-2947 10.1103/PhysRevA.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. PRLTAO 0031-9007 10.1103/PhysRevLett.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. PLRAAN 1050-2947 10.1103/PhysRevA.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. PRBMDO 1098-0121 10.1103/PhysRevB.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. ADCPAA 0065-2385 10.1002/9781118742631.ch11
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. PRLTAO 0031-9007 10.1103/PhysRevLett.80.4088
115. Menicucci, N. C., 2014, " Fault-tolerant measurement-based quantum computing with continuous-variable cluster states," Phys. Rev. Lett. 112, 120504. PRLTAO 0031-9007 10.1103/PhysRevLett.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. PRBMDO 1098-0121 10.1103/PhysRevB.76.174516
117. Michnicki, K. P., 2014, " 3D Topological quantum memory with a power-law energy barrier," Phys. Rev. Lett. 113, 130501. PRLTAO 0031-9007 10.1103/PhysRevLett.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. NJOPFM 1367-2630 10.1088/1367-2630/13/8/085007
120. Napp, J., and J. Preskill, 2013, " Optimal Bacon-Shor codes," Quantum Inf. Comput. 13, 490. QICUAW 1533-7146
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. PLRAAN 1050-2947 10.1103/PhysRevA.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. PLRAAN 1050-2947 10.1103/PhysRevA.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). PLRAAN 1050-2947 10.1103/PhysRevA.75.064304
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. SCIEAS 0036-8075 10.1126/science.1253742
126. Nigg, S. E., and S. M. Girvin, 2013, " Stabilizer quantum error correction toolbox for superconducting qubits," Phys. Rev. Lett. 110, 243604. PRLTAO 0031-9007 10.1103/PhysRevLett.110.243604
127. Nussinov, Z., and G. Ortiz, 2009, " Symmetry and Topological Order," Proc. Natl. Acad. Sci. U.S.A. 106, 16944. PNASA6 0027-8424 10.1073/pnas.0803726105
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. PRLTAO 0031-9007 10.1103/PhysRevLett.111.090505
129. Poulin, D., 2005, " Stabilizer formalism for operator quantum error correction," Phys. Rev. Lett. 95, 230504. PRLTAO 0031-9007 10.1103/PhysRevLett.95.230504
130. Poulin, D., 2006, " Optimal and efficient decoding of concatenated quantum block codes," Phys. Rev. A 74, 052333. PLRAAN 1050-2947 10.1103/PhysRevA.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. PRLTAO 0031-9007 10.1103/PhysRevLett.98.190504
133. Raussendorf, R., J. Harrington, and K. Goyal, 2007, " Topological fault-tolerance in cluster state quantum computation," New J. Phys. 9, 199. NJOPFM 1367-2630 10.1088/1367-2630/9/6/199
134. Reichardt, Ben W., 2005, " Quantum universality by distilling certain one- and two-qubit states with stabilizer operations," Quantum Inf. Process. 4, 251. QIPUAT 1570-0755 10.1007/s11128-005-7654-8
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. IETTAW 0018-9448 10.1109/18.771249
137. Steane, A., 2003, " Overhead and noise threshold of fault-tolerant quantum error correction," Phys. Rev. A 68, 042322. PLRAAN 1050-2947 10.1103/PhysRevA.68.042322
138. Steane, A. M., 1997, " Active stabilization, quantum computation, and quantum state synthesis," Phys. Rev. Lett. 78, 2252. PRLTAO 0031-9007 10.1103/PhysRevLett.78.2252
139. Suchara, M., S. Bravyi, and B. Terhal, 2011, " Constructions and noise threshold of topological subsystem codes," J. Phys. A 44, 155301. JPAMB5 1751-8113 10.1088/1751-8113/44/15/155301
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., 2014, " Tracking photon jumps with repeated quantum non-demolition parity measurements," Nature (London) 511, 444. NATUAS 0028-0836 10.1038/nature13436
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. QICUAW 1533-7146
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. QICUAW 1533-7146
145. Terhal, B. M., F. Hassler, and D. P. DiVincenzo, 2012, " From Majorana fermions to topological order," Phys. Rev. Lett. 108, 260504. PRLTAO 0031-9007 10.1103/PhysRevLett.108.260504
146. Tillich, Jean-Pierre, and Gilles Zémor, 2014, " Quantum LDPC codes with positive rate and minimum distance proportional to (Equation presented)," IEEE Trans. Inf. Theory 60, 1193. IETTAW 0018-9448 10.1109/TIT.2013.2292061
147. Vandersypen, L. M. K., and I. L. Chuang, 2005, " NMR techniques for quantum control and computation," Rev. Mod. Phys. 76, 1037. RMPHAT 0034-6861 10.1103/RevModPhys.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. OPLEDP 0146-9592 10.1364/OL.35.003261
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. APNYA6 0003-4916 10.1016/S0003-4916(02)00019-2
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). PLRAAN 1050-2947 10.1103/PhysRevA.83.020302
152. Weissman, M. B., 1988, " (Equation presented) noise and other slow, nonexponential kinetics in condensed matter," Rev. Mod. Phys. 60, 537. RMPHAT 0034-6861 10.1103/RevModPhys.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. JMOPEW 0950-0340 10.1080/09500340.2012.737937
155. Wootton, J. R., J. Burri, S. Iblisdir, and D. Loss, 2014, " Decoding non-Abelian topological quantum memories," Phys. Rev. X 4, 011051. PRXHAE 2160-3308 10.1103/PhysRevX.4.011051
156. Yao, X.-C., 2012, " Experimental demonstration of topological error correction," Nature (London) 482, 489. NATUAS 0028-0836 10.1038/nature10770
157. Yoshida, Beni, 2011, " Feasibility of self-correcting quantum memory and thermal stability of topological order," Ann. Phys. (Amsterdam) 326, 2566. APNYA6 0003-4916 10.1016/j.aop.2011.06.001
158. Zeng, Bei, Andrew W. Cross, and Isaac L. Chuang, 2011, " Transversality versus universality for additive quantum codes," IEEE Trans. Inf. Theory 57, 6272. IETTAW 0018-9448 10.1109/TIT.2011.2161917
159. Zhang, J., C. Xie, K. Peng, and P. van Loock, 2008, " Anyon statistics with continuous variables," Phys. Rev. A 78, 052121. PLRAAN 1050-2947 10.1103/PhysRevA.78.052121
160. Zhou, X., D. W. Leung, and I. L. Chuang, 2000, " Methodology for quantum logic gate construction," Phys. Rev. A 62, 052316. PLRAAN 1050-2947 10.1103/PhysRevA.62.052316