Quantum data-syndrome codes: subsystem and impure code constructions. (English) Zbl 1542.81313
Summary: Quantum error correction requires the use of error syndromes derived from measurements that may be unreliable. Recently, quantum data-syndrome (QDS) codes have been proposed as a possible approach to protect against both data and syndrome errors, in which a set of linearly dependent stabilizer measurements are performed to increase redundancy. Motivated by wanting to reduce the total number of measurements performed, we introduce QDS subsystem codes, and show that they can outperform similar QDS stabilizer codes derived from them. We also give a construction of single-error-correcting QDS stabilizer codes from impure stabilizer codes and show that any such code must satisfy a variant of the quantum Hamming bound for QDS codes. Finally, we use this bound to prove a new bound that applies to impure, but not pure, stabilizer codes that may be of independent interest.
Software:
Code TablesReferences:
| [1] | Shor, P.W.: Fault-tolerant quantum computation. In: Proceedings of the 37th IEEE Symposium on Foundations of Computer Science (FOCS), Burlington, Vermont, USA, pp. 56-67 (1996) |
| [2] | DiVincenzo, DP; Shor, PW, Fault-tolerant error correction with efficient quantum codes, Phys. Rev. Lett., 77, 15, 3260-3263 (1996) · doi:10.1103/PhysRevLett.77.3260 |
| [3] | Steane, A., Active stabilization, quantum computation, and quantum state synthesis, Phys. Rev. Lett., 78, 11, 2252-2255 (1997) · doi:10.1103/PhysRevLett.78.2252 |
| [4] | Chao, R.; Reichardt, BW, Quantum error correction with only two extra qubits, Phys. Rev. Lett., 121, 5 (2018) · doi:10.1103/PhysRevLett.121.050502 |
| [5] | Gottesman, D.: Stabilizer Codes and Quantum Error Correction. PhD thesis, California Institute of Technology, Pasadena, CA (1997). arXiv:quant-ph/9705052 |
| [6] | Calderbank, AR; Rains, EM; Shor, PW; Sloane, NJA, Quantum error correction via codes over GF(4), IEEE Trans. Inf. Theory, 44, 4, 1369-1387 (1998) · Zbl 0982.94029 · doi:10.1109/18.681315 |
| [7] | Fujiwara, Y., Ability of stabilizer quantum error correction to protect itself from its own imperfection, Phys. Rev. A, 90, 6 (2014) · doi:10.1103/PhysRevA.90.062304 |
| [8] | Delfosse, N., Reichardt, B.W.: Short Shor-style syndrome sequences (2020). arXiv:2008.05051 [quant-ph] |
| [9] | Tansuwannont, T., Brown, K.R.: Adaptive syndrome measurements for Shor-style error correction (2022). arXiv:2208.05601 [quant-ph] |
| [10] | Ashikhmin, A., Lai, C.-Y., Brun, T.A.: Robust quantum error syndrome extraction by classical coding. In: Proceedings of the 2014 IEEE International Symposium on Information Theory (ISIT), Honolulu, Hawaii, USA, pp. 546-550 (2014) |
| [11] | Ashikhmin, A., Lai, C.-Y., Brun, T.A.: Correction of data and syndrome errors by stabilizer codes. In: Proceedings of the 2016 IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain, pp. 2274-2278 (2016) |
| [12] | Ashikhmin, A.; Lai, C-Y; Brun, TA, Quantum data syndrome codes, IEEE J. Sel. Areas Commun., 38, 3, 449-462 (2020) · doi:10.1109/JSAC.2020.2968997 |
| [13] | Fujiwara, Y.: Global stabilizer quantum error correction with combinatorial arrays. In: Proceedings of the 2015 IEEE International Symposium on Information Theory (ISIT), Hong Kong, China, pp. 1114-1118 (2015) |
| [14] | Kuo, K.-Y., Chern, I.-C., Lai, C.-Y.: Decoding of quantum data-syndrome codes via belief propagation. In: Proceedings of the 2021 IEEE International Symposium on Information Theory (ISIT), Melbourne, Australia, pp. 1552-1557 (2021) |
| [15] | Raveendran, N., Rengaswamy, N., Pradhan, A.K., Vasić, B.: Soft syndrome decoding of quantum LDPC codes for joint correction of data and syndrome errors. In: Proceedings of the 2022 IEEE International Conference on Quantum Computing and Engineering (QCE), Broomfield, Colorado, USA, pp. 275-281 (2022) |
| [16] | Premakumar, VN; Sha, H.; Crow, D.; Bach, E.; Joynt, R., 2-designs and redundant syndrome extraction for quantum error correction, Quantum Inf. Process., 20, 3, 84 (2021) · Zbl 1509.81343 · doi:10.1007/s11128-021-03015-1 |
| [17] | Zeng, W., Ashikhmin, A., Woolls, M., Pryadko, L.P.: Quantum convolutional data-syndrome codes. In: Proceedings of the IEEE 20th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Cannes, France (2019) |
| [18] | Wagner, T., Kampermann, H., Bruß, D., Kliesch, M.: Learning logical quantum noise in quantum error correction (2022). arXiv:2209.09267 [quant-ph] |
| [19] | Bombín, H., Single-shot fault-tolerant quantum error correction, Phys. Rev. X, 5, 3 (2015) |
| [20] | Brown, BJ; Nickerson, NH; Browne, DE, Fault-tolerant error correction with the gauge color code, Nat. Commun., 7, 1, 12302 (2016) · doi:10.1038/ncomms12302 |
| [21] | Campbell, ET, A theory of single-shot error correction for adversarial noise, Quantum Sci. Technol., 4, 2 (2019) · doi:10.1088/2058-9565/aafc8f |
| [22] | Delfosse, N.; Reichardt, BW; Svore, KM, Beyond single-shot fault-tolerant quantum error correction, IEEE Trans. Inf. Theory, 68, 1, 287-301 (2022) · Zbl 1489.81016 · doi:10.1109/TIT.2021.3120685 |
| [23] | Poulin, D., Stabilizer formalism for operator quantum error correction, Phys. Rev. Lett., 95, 23 (2005) · doi:10.1103/PhysRevLett.95.230504 |
| [24] | Kribs, D.; Poulin, D.; Lidar, DA; Brun, TA, Operator quantum error correction, Quantum Error Correction (2013), New York: Cambridge University Press, New York · doi:10.1017/CBO9781139034807.008 |
| [25] | Shor, PW, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A, 52, 4, 2493-2496 (1995) · doi:10.1103/PhysRevA.52.R2493 |
| [26] | Bacon, D., Operator quantum error-correcting subsystems for self-correcting quantum memories, Phys. Rev. A, 73, 1 (2006) · doi:10.1103/PhysRevA.73.012340 |
| [27] | Cordaro, JT; Wagner, TJ, Optimum \(\left(n,2\right)\) codes for small values of channel error probability, IEEE Trans. Inf. Theory, 13, 2, 349-350 (1967) · Zbl 0149.15904 · doi:10.1109/TIT.1967.1053987 |
| [28] | Grassl, M.: Bounds on the Minimum Distance of Linear Codes and Quantum Codes. Accessed: Jun. 13, 2020. http://www.codetables.de/ |
| [29] | Steane, A., Multiple-particle interference and quantum error correction, Proc. R. Soc. Lond. A, 452, 1954, 2551-2577 (1996) · Zbl 0876.94002 · doi:10.1098/rspa.1996.0136 |
| [30] | Li, Z.; Xing, L., Classification of \(q\)-ary perfect quantum codes, IEEE Trans. Inf. Theory, 59, 1, 631-634 (2013) · Zbl 1364.81096 · doi:10.1109/TIT.2012.2217475 |
| [31] | Shaw, B.; Wilde, MM; Oreshkov, O.; Kremsky, I.; Lidar, DA, Encoding one logical qubit into six physical qubits, Phys. Rev. A, 78, 1 (2008) · doi:10.1103/PhysRevA.78.012337 |
| [32] | Bennett, CH; DiVincenzo, DP; Smolin, JA; Wootters, WK, Mixed-state entanglement and quantum error correction, Phys. Rev. A, 54, 5, 3824-3851 (1996) · Zbl 1371.81041 · doi:10.1103/PhysRevA.54.3824 |
| [33] | Laflamme, R.; Miquel, C.; Paz, JP; Zurek, WH, Perfect quantum error correcting code, Phys. Rev. Lett., 77, 1, 198-201 (1996) · doi:10.1103/PhysRevLett.77.198 |
| [34] | Cross, A.: private communication (2022) |
| [35] | Gottesman, D., Class of quantum error-correcting codes saturating the quantum Hamming bound, Phys. Rev. A, 54, 3, 1862-1868 (1996) · doi:10.1103/PhysRevA.54.1862 |
| [36] | Yu, S.; Bierbrauer, J.; Dong, Y.; Chen, Q.; Oh, CH, All the stabilizer codes of distance 3, IEEE Trans. Inf. Theory, 59, 8, 5179-5185 (2013) · Zbl 1364.81102 · doi:10.1109/TIT.2013.2259138 |
| [37] | Li, Z.; Xing, L., On a problem concerning the quantum hamming bound for impure quantum codes, IEEE Trans. Inf. Theory, 56, 9, 4731-4734 (2010) · Zbl 1366.81128 · doi:10.1109/TIT.2010.2054610 |
| [38] | Sarvepalli, P.; Klappenecker, A., Degenerate quantum codes and the quantum Hamming bound, Phys. Rev. A, 81, 3 (2010) · doi:10.1103/PhysRevA.81.032318 |
| [39] | Dallas, E., Andreadakis, F., Lidar, D.: \(No ((n,K,d<127))\) code can violate the quantum Hamming bound (2022). arXiv:2208.11800v2 [quant-ph] |
| [40] | Rains, EM, Nonbinary quantum codes, IEEE Trans. Inform. Theory, 45, 6, 1827-1832 (1999) · Zbl 0958.94038 · doi:10.1109/18.782103 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
