Abstract
In this work, we explore the accuracy of quantum error correction depending of the order of the implemented syndrome measurements. CSS codes require that bit-flip and phase-flip syndromes be measured separately. To comply with fault-tolerant demands and to maximize accuracy, this set of syndrome measurements should be repeated allowing for flexibility in the order of their implementation. We examine different possible orders of Shor-state and Steane-state syndrome measurements for the [[7,1,3]] quantum error correction code. We find that the best choice of syndrome order, determined by the fidelity of the state after noisy error correction, will depend on the error environment. We also compare the fidelity when syndrome measurements are done with Shor states versus Steane states and find that Steane states generally, but not always, lead to final states with higher fidelity. Together, these results allow a quantum computer programmer to choose the optimal syndrome measurement scheme based on the system’s error environment.
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References
Nielsen, M., Chuang, I.: Quantum Information and Computation. Cambridge University Press, Cambridge (2000)
Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493–R2496 (1995)
Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54, 1098–1105 (1996)
Steane, A.M.: Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793–797 (1996)
Preskill, J.: Reliable quantum computers. Proc. Roy. Soc. Lond. A 454, 385–410 (1998)
Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. In Proceedings of the 35th Annual Symposium on Fundamentals of Computer Science, IEEE Press, Los Alamitos (1996)
Gottesman, D.: Theory of fault-tolerant quantum computation. Phys. Rev. A 57, 127–137 (1998)
Aleferis, P., Gottesman, D., Preskill, J.: Quantum accuracy threshold for concatenated distance-3 code. Quant. Inf. Comput. 6, 97–165 (2006)
Steane, A.: Multiple particle interference and quantum error correction. Proc. Roy. Soc. Lond. A 452, 2551–2577 (1996)
Weinstein, Y.S., Buchbinder, S.D.: Use of Shor states for the [7,1,3] quantum error-correcting code. Phys. Rev. A 86, 052336 (2012)
Weinstein, Y.S.: Syndrome measurement strategies for the [[7,1,3]] code. Quant. Inf. Proc. 14, 1841–1854 (2015)
Aggarwal, V., Calderbank, A.R., Gilbert, G., Weinstein, Y.S.: Volume thresholds for quantum fault tolerance. Quant. Inf. Proc. 9, 541 (2010)
Aliferis, P., Preskill, J.: Fault-tolerant quantum computation against biased noise. Phys. Rev. A 78, 052331 (2008)
Weinstein, Y.S.: Non-fault-tolerant T gates for the [7,1,3] quantum error-correction code. Phys. Rev. A 87, 032320 (2013)
Weinstein, Y.S.: Quantum-error-correction implementation after multiple gates. Phys. Rev. A 88, 012325 (2013)
Weinstein, Y.S.: Quantum error correction during 50 gates. Phys. Rev. A 89, 020301(R) (2014)
Weinstein, Y.S.: Fidelity of an encoded [7,1,3] logical zero. Phys. Rev. A 84, 012323 (2011)
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This research is supported under the MITRE Innovation Program.
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Weinstein, Y.S. Syndrome measurement order for the [[7,1,3]] quantum error correction code. Quantum Inf Process 15, 1263–1271 (2016). https://doi.org/10.1007/s11128-015-1068-z
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DOI: https://doi.org/10.1007/s11128-015-1068-z