Error tolerance of topological codes with independent bit-flip and measurement errors

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Topological quantum error correction codes are currently among the most promising candidates for efficiently dealing with the decoherence effects inherently present in quantum devices. Numerically, their theoretical error threshold can be calculated by mapping the underlying quantum problem to a related classical statistical-mechanical spin system with quenched disorder. Here, we present results for the general fault-tolerant regime, where we consider both qubit and measurement errors. However, unlike in previous studies, here we vary the strength of the different error sources independently. Our results highlight peculiar differences between toric and color codes. This study complements previous results published in New J. Phys. 13, 083006 (2011).
©2016 American Physical Society. The authors thank ETH Zurich for CPU time on the Brutus cluster, the Santa Fe Institute for CPU time on the Scoville cluster, and the Centro de Supercomputacion y Visualizacion de Madrid (CeSViMa) for access to the Magerit cluster. M.A.M.-D. and H.B. acknowledge financial support from the Spanish MINECO Grant No. FIS2012-33152, the Spanish MINECO Grant No. FIS2015-67411, and the CAM research consortium QUITEMAD+, Grant No. S2013/ICE-2801. The research of M.A.M.-D. has been supported in part by the U.S. Army Research Office through Grant No. W911NF-14-1-0103. H.G.K. acknowledges support from the National Science Foundation (Grant No. DMR-1151387) and the Swiss National Science Foundation (Grant No. PP002-114713). Part of the research of H.G.K. is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via MIT Lincoln Laboratory Air Force Contract No. FA8721-05-C-0002. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purpose notwithstanding any copyright annotation thereon.
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1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2000). 2. A. Galindo and M. A. Martin-Delgado, Information and computation: Classical and quantum aspects, Rev. Mod. Phys. 74, 347 (2002). 3. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys.43, 4452 (2002). 4. T. Ohno, G. Arakawa, I. Ichinose, and T. Matsui, Phase structure of the random-plaquette Z2gauge model: Accuracy threshold for a toric quantum memory, Nucl. Phys. B 697, 462 (2004). 5. C. Wang, J. Harrington, and J. Preskill, Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory, Ann. Phys. (NY) 303, 31 (2003). 6. R. S. Andrist, H. G. Katzgraber, H. Bombin, and M. A. Martin-Delgado, Tricolored lattice gauge theory with randomness: Fault-tolerance in topological color codes, New J. Phys. 13, 083006 (2011). 7. H. G. Katzgraber, H. Bombin, and M. A. Martin-Delgado, Error Threshold for Color Codes and Random 3-Body Ising Models, Phys. Rev. Lett. 103, 090501 (2009). 8. M. Ohzeki, Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices, Phys. Rev. E 80, 011141 (2009). 9. R. Raussendorf and J. Harrington, Fault-Tolerant Quantum Computation with High Threshold in Two Dimensions, Phys. Rev. Lett. 98, 190504 (2007). 10. A. G. Fowler, D. S. Wang, and L. C. L. Hollenberg, Surface code quantum error correction incorporating accurate error propagation, Quantum Inf. Comput. 11, 8 (2011). 11. A. J. Landahl, J. T. Anderson, and P. R. Rice, Fault-tolerant quantum computing with color codes, arXiv:1108.5738. 12. P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493 (1995). 13. A. M. Steane, Error Correcting Codes in Quantum Theory, Phys. Rev. Lett. 77, 793 (1996). 14. A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54, 1098 (1996). 15. A. Steane, Multiple-particle interference and quantum error correction, Proc. R. Soc. A 452, 2551 (1996). 16. D. Gottesman, Class of quantum error-correcting codes saturating the quantum Hamming bound, Phys. Rev. A 54, 1862 (1996). 17. C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Mixed-state entanglement and quantum error correction, Phys. Rev. A 54, 3824 (1996). 18. B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015). 19. P. W. Shor, in Proceedings of the 37th Symposium on the Foundations of Computer Science (IEEE Computer Society, New York, 1996), p. 56. 20. E. Knill, R. Laflamme, and W. Zurek, Threshold accuracy for quantum computation,arXiv:quant-ph/9610011. 21. D. Aharonov and M. Ben-Or, in Proceedings of the 29th Annual ACM Symposium on Theory of Computation (ACM, El Paso, TX, 1997), p. 188. 22. A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. (NY) 303, 2 (2003). 23. H. Bombin and M. A. Martin-Delgado, Topological Quantum Distillation, Phys. Rev. Lett. 97, 180501 (2006). 24. H. Bombin and M. A. Martin-Delgado, Topological Computation Without Braiding, Phys. Rev. Lett. 98, 160502 (2007). 25. H. Bombin and M. A. Martin-Delgado, Exact topological quantum order in D=3 and beyond: Branyons and brane-net condensates, Phys. Rev. B 75, 075103 (2007). 26. 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). 27. D. Nigg, M. Mueller, E. A. Martinez, P. Schindler, M. Hennrich, T. Monz, M. A. Martin-Delgado, and R. Blatt, Experimental quantum computations on a topologically encoded qubit, Science 345, 302 (2014). 28. M. Niedermayr, K. Lakhmanskiy, M. Kumph, S. Partel, J. Edlinger, M. Brownnutt, and R. Blatt, Cryogenic surface ion trap based on intrinsic silicon, New J. Phys. 16, 113068 (2014). 29. R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell et al., Superconducting quantum circuits at the surface code threshold for fault tolerance, Nature (London) 508, 500 (2014). 30. A. D. Córcoles, E. Magesan, S. J. Srinivasan, A. W. Cross, M. Steffen, J. M. Gambetta, and J. M. Chow, Demonstration of a quantum error detection code using a square lattice of four superconducting qubits, Nat. Commun. 6, 6979 (2015). 31. R. Kueng, D. M. Long, A. C. Doherty, and S. T. Flammia, Comparing Experiments to the Fault-Tolerance Threshold, arXiv:1510.05653. 32. M. Müller, A. Rivas, E. A. Martínez, D. Nigg, P. Schindler, T. Monz, R. Blatt, and M. A. Martin-Delgado, Iterative Phase Optimisation of Elementary Quantum Error Correcting Codes,arXiv:1603.00402. 33. H. Bombin and M. A. Martin-Delgado, Homological error correction: Classical and quantum codes, J. Math. Phys. 48, 052105 (2007). 34. H. Nishimori, Internal energy, specific heat and correlation function of the bond-random Ising model, Prog. Theor. Phys. 66, 1169 (1981). 35. R. S. Andrist, H. Bombin, H. G. Katzgraber, and M. A. Martin-Delgado, Optimal error correction in topological subsystem codes, Phys. Rev. A 85, 050302 (2012). 36. K. Hukushima and K. Nemoto, Exchange Monte Carlo method and application to spin glass simulations, J. Phys. Soc. Jpn. 65, 1604 (1996).