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Markov Chains and Hitting Times for Error Accumulation in Quantum Circuits

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Performance Evaluation Methodologies and Tools (VALUETOOLS 2021)

Abstract

We study a classical model for the accumulation of errors in multi-qubit quantum computations. By modeling the error process in a quantum computation using two coupled Markov chains, we are able to capture a weak form of time-dependency between errors in the past and future. By subsequently using techniques from the field of discrete probability theory, we calculate the probability that error quantities such as the fidelity and trace distance exceed a threshold analytically. The formulae cover fairly generic error distributions, cover multi-qubit scenarios, and are applicable to the randomized benchmarking protocol. To combat the numerical challenge that may occur when evaluating our expressions, we additionally provide an analytical bound on the error probabilities that is of lower numerical complexity. Besides this, we study a model describing continuous errors accumulating in a single qubit. Finally, taking inspiration from the field of operations research, we illustrate how our expressions can be used to decide how many gates one can apply before too many errors accumulate with high probability, and how one can lower the rate of error accumulation in existing circuits through simulated annealing.

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References

  1. Aaronson, S., Gottesman, D.: Improved simulation of stabilizer circuits. Phys. Rev. A 70(5), 052328 (2004)

    Google Scholar 

  2. Amy, M.: Formal methods in quantum circuit design (2019)

    Google Scholar 

  3. Ball, H., Stace, T.M., Flammia, S.T., Biercuk, M.J.: Effect of noise correlations on randomized benchmarking. Phys. Rev. A 93(2), 022303 (2016)

    Google Scholar 

  4. Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70(13), 1895 (1993)

    Article  MathSciNet  Google Scholar 

  5. Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69(20), 2881 (1992)

    Article  MathSciNet  Google Scholar 

  6. Bhatia, R.: Matrix Analysis, vol. 169. Springer Science & Business Media (2013). https://doi.org/10.1007/978-1-4612-0653-8

  7. Bravyi, S., Englbrecht, M., König, R., Peard, N.: Correcting coherent errors with surface codes. npj Quantum Inf. 4(1), 55 (2018)

    Google Scholar 

  8. Brémaud, P.: Discrete Probability Models and Methods, vol. 78. Springer (2017). https://doi.org/10.1007/978-3-319-43476-6

  9. Brown, W.G., Eastin, B.: Randomized benchmarking with restricted gate sets. Phys. Rev. A 97(6), 062323 (2018)

    Google Scholar 

  10. Carignan-Dugas, A., Boone, K., Wallman, J.J., Emerson, J.: From randomized benchmarking experiments to gate-set circuit fidelity: how to interpret randomized benchmarking decay parameters. New J. Phys. 20(9), 092001 (2018)

    Google Scholar 

  11. Cleve, R., Ekert, A., Macchiavello, C., Mosca, M.: Quantum algorithms revisited. Proc. Royal Soc. London. Ser. A: Math. Phys. Eng. Sci. 454(1969), 339–354 (1998)

    Google Scholar 

  12. Cramer, J., et al.: Repeated quantum error correction on a continuously encoded qubit by real-time feedback. Nat. Commun. 7, 11526 (2016)

    Article  Google Scholar 

  13. Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proc. Royal Soc. London. Ser. A: Math. Phys. Sci. 439(1907), 553–558 (1992)

    Google Scholar 

  14. Epstein, J.M., Cross, A.W., Magesan, E., Gambetta, J.M.: Investigating the limits of randomized benchmarking protocols. Phys. Rev. A 89(6), 062321 (2014)

    Google Scholar 

  15. Fong, B.H., Merkel, S.T.: Randomized benchmarking, correlated noise, and ising models. arXiv preprint arXiv:1703.09747 (2017)

  16. Fowler, A.G., Hollenberg, L.C.: Scalability of Shor’s algorithm with a limited set of rotation gates. Phys. Rev. A 70(3), 032329 (2004)

    Google Scholar 

  17. França, D.S., Hashagen, A.: Approximate randomized benchmarking for finite groups. J. Phys. A: Math. Theoret. 51(39), 395302 (2018)

    Google Scholar 

  18. Fujii, K.: Stabilizer formalism and its applications. In: Quantum Computation with Topological Codes, pp. 24–55. Springer (2015). https://doi.org/10.1007/978-981-287-996-7

  19. Gottesman, D.: The Heisenberg representation of quantum computers. arXiv preprint quant-ph/9807006 (1998)

    Google Scholar 

  20. Gottesman, D.: Efficient fault tolerance. Nature 540, 44 (2016)

    Article  Google Scholar 

  21. Greenbaum, D., Dutton, Z.: Modeling coherent errors in quantum error correction. Quantum Sci. Technol. 3(1), 015007 (2017)

    Google Scholar 

  22. Greenberger, D.M., Horne, M.A., Zeilinger, A.: Going beyond Bell’s theorem. In: Bell’s Theorem, Quantum Theory and Conceptions of the Universe, pp. 69–72. Springer (1989). https://doi.org/10.1007/978-94-017-0849-4_10

  23. Gutiérrez, M., Svec, L., Vargo, A., Brown, K.R.: Approximation of realistic errors by Clifford channels and Pauli measurements. Phys. Rev. A 87(3), 030302 (2013)

    Google Scholar 

  24. Gutmann, H.: Description and control of decoherence in quantum bit systems. Ph.D. thesis, lmu (2005)

    Google Scholar 

  25. Harper, R., Hincks, I., Ferrie, C., Flammia, S.T., Wallman, J.J.: Statistical analysis of randomized benchmarking. Phys. Rev. A 99(5), 052350 (2019)

    Google Scholar 

  26. Huang, E., Doherty, A.C., Flammia, S.: Performance of quantum error correction with coherent errors. Phys. Rev. A 99(2), 022313 (2019)

    Google Scholar 

  27. Janardan, S., Tomita, Yu., Gutiérrez, M., Brown, K.R.: Analytical error analysis of Clifford gates by the fault-path tracer method. Quantum Inf. Process. 15(8), 3065–3079 (2016). https://doi.org/10.1007/s11128-016-1330-z

    Article  MathSciNet  MATH  Google Scholar 

  28. Kliuchnikov, V., Maslov, D.: Optimization of Clifford circuits. Phys. Rev. A 88(5), 052307 (2013)

    Google Scholar 

  29. Knill, E.: Quantum computing with realistically noisy devices. Nature 434(7029), 39 (2005)

    Article  Google Scholar 

  30. Koenig, R., Smolin, J.A.: How to efficiently select an arbitrary Clifford group element. J. Math. Phys. 55(12), 122202 (2014)

    Google Scholar 

  31. Linke, N.M., et al.: Fault-tolerant quantum error detection. Sci. Adv. 3(10), e1701074 (2017)

    Google Scholar 

  32. Ma, L., Sanders, J.: Markov chains and hitting times for error accumulation in quantum circuits. arXiv preprint arXiv:1909.04432 (2021)

  33. Magesan, E., Gambetta, J.M., Emerson, J.: Scalable and robust randomized benchmarking of quantum processes. Phys. Rev. Lett. 106(18), 180504 (2011)

    Google Scholar 

  34. Magesan, E., Puzzuoli, D., Granade, C.E., Cory, D.G.: Modeling quantum noise for efficient testing of fault-tolerant circuits. Phys. Rev. A 87(1), 012324 (2013)

    Google Scholar 

  35. Maslov, D., Dueck, G.W., Miller, D.M., Negrevergne, C.: Quantum circuit simplification and level compaction. IEEE Trans. Comput.-Aided Des. Integrated Circ. Syst. 27(3), 436–444 (2008)

    Article  Google Scholar 

  36. Moll, N., et al.: Quantum optimization using variational algorithms on near-term quantum devices. Quantum Sci. Technol. 3(3), 030503 (2018)

    Google Scholar 

  37. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information: 10th Anniversary Edition, 10th edn. Cambridge University Press, New York (2011)

    MATH  Google Scholar 

  38. Preskill, J.: Quantum computing: pro and con. Proc. Royal Soc. London. Ser. A: Math. Phys. Eng. Sci. 454(1969), 469–486 (1998)

    Google Scholar 

  39. Proctor, T., Rudinger, K., Young, K., Sarovar, M., Blume-Kohout, R.: What randomized benchmarking actually measures. Phys. Rev. Lett. 119(13), 130502 (2017)

    Google Scholar 

  40. Roberts, P.H., Ursell, H.D.: Random walk on a sphere and on a Riemannian manifold. Philos. Trans. Royal Soc. London. Ser. A, Math. Phys. Sci. 252(1012), 317–356 (1960)

    Google Scholar 

  41. Ruskai, M.B.: Pauli exchange errors in quantum computation. Phys. Rev. Lett. 85(1), 194 (2000)

    Article  Google Scholar 

  42. Selinger, P.: Generators and relations for n-qubit Clifford operators. Logical Meth. Comput. Sci. 11 (2013)

    Google Scholar 

  43. Wallman, J.J.: Randomized benchmarking with gate-dependent noise. Quantum 2, 47 (2018). https://doi.org/10.22331/q-2018-01-29-47

  44. Wallman, J.J., Barnhill, M., Emerson, J.: Robust characterization of loss rates. Phys. Rev. Lett. 115(6), 060501 (2015)

    Google Scholar 

  45. Wood, C.J., Gambetta, J.M.: Quantification and characterization of leakage errors. Phys. Rev. A 97(3), 032306 (2018)

    Google Scholar 

  46. Xia, T., et al.: Randomized benchmarking of single-qubit gates in a 2D array of neutral-atom qubits. Phys. Rev. Lett. 114(10), 100503 (2015)

    Google Scholar 

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Acknowledgments

We are grateful to Bart van Schooten, who contributed the code on TU/e’s GitLab server. Finally, this research received financial support from the Chinese Scholarship Council (CSC) in the form of a CSC Scholarship.

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Authors and Affiliations

  1. Faculty of Electrical Engineering, Mathematics, and Computer Science, Delft University of Technology, Delft, Netherlands

    Long Ma

  2. Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, Netherlands

    Jaron Sanders

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  1. Long Ma
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Correspondence to Long Ma .

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Editors and Affiliations

  1. Tsinghua University, Beijing, China

    Qianchuan Zhao

  2. Sun Yat-sen University, Guangzhou, China

    Li Xia

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Ma, L., Sanders, J. (2021). Markov Chains and Hitting Times for Error Accumulation in Quantum Circuits. In: Zhao, Q., Xia, L. (eds) Performance Evaluation Methodologies and Tools. VALUETOOLS 2021. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 404. Springer, Cham. https://doi.org/10.1007/978-3-030-92511-6_3

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  • DOI: https://doi.org/10.1007/978-3-030-92511-6_3

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  • Online ISBN: 978-3-030-92511-6

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