A Logarithmic Bayesian Approach to Quantum Error Detection

Ian Convy and K. Birgitta Whaley

Department of Chemistry, University of California, Berkeley, CA 94720, USA
Berkeley Quantum Information and Computation Center, University of California, Berkeley, CA 94720, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


We consider the problem of continuous quantum error correction from a Bayesian perspective, proposing a pair of digital filters using logarithmic probabilities that are able to achieve near-optimal performance on a three-qubit bit-flip code, while still being reasonable to implement on low-latency hardware. These practical filters are approximations of an optimal filter that we derive explicitly for finite time steps, in contrast with previous work that has relied on stochastic differential equations such as the Wonham filter. By utilizing logarithmic probabilities, we are able to eliminate the need for explicit normalization and can reduce the Gaussian noise distribution to a simple quadratic expression. The state transitions induced by the bit-flip errors are modeled using a Markov chain, which for log-probabilties must be evaluated using a LogSumExp function. We develop the two versions of our filter by constraining this LogSumExp to have either one or two inputs, which favors either simplicity or accuracy, respectively. Using simulated data, we demonstrate that the single-term and two-term filters are able to significantly outperform both a double threshold scheme and a linearized version of the Wonham filter in tests of error detection under a wide variety of error rates and time steps.

► BibTeX data

► References

[1] Peter W. Shor ``Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer'' SIAM Journal on Computing 26, 1484-1509 (1997).

[2] ``Quantum Error Correction'' Cambridge University Press (2013).

[3] W. M. Wonham ``Some Applications of Stochastic Differential Equations to Optimal Nonlinear Filtering'' Journal of the Society for Industrial and Applied Mathematics Series A Control 2, 347–369 (1964).

[4] Juan Atalaya, Mohammad Bahrami, Leonid P. Pryadko, and Alexander N. Korotkov, ``Bacon-Shor code with continuous measurement of noncommuting operators'' Physical Review A 95, 032317 (2017).

[5] Razieh Mohseninia, Jing Yang, Irfan Siddiqi, Andrew N. Jordan, and Justin Dressel, ``Always-On Quantum Error Tracking with Continuous Parity Measurements'' Quantum 4, 358 (2020).

[6] J. Atalaya, S. Zhang, M. Y. Niu, A. Babakhani, H. C. H. Chan, J. M. Epstein, and K. B. Whaley, ``Continuous quantum error correction for evolution under time-dependent Hamiltonians'' Physical Review A 103, 042406 (2021).

[7] Satoshi Moritaand Hidetoshi Nishimori ``Mathematical foundation of quantum annealing'' Journal of Mathematical Physics 49, 125210 (2008).

[8] Kurt Jacobsand Daniel A. Steck ``A straightforward introduction to continuous quantum measurement'' Contemporary Physics 47, 279–303 (2006).

[9] Alexander N. Korotkov ``Quantum Bayesian approach to circuit QED measurement with moderate bandwidth'' Physical Review A 94, 042326 (2016).

[10] Howard M. Wisemanand Gerard J. Milburn ``Quantum Measurement and Control'' Cambridge University Press (2009).

[11] Vic Barnett ``Comparative statistical inference'' Wiley (1999).

[12] Alexander N. Korotkov ``Continuous quantum measurement of a double dot'' Physical Review B 60, 5737–5742 (1999).

[13] Bruno De Finetti, Antonio Machì, Adrian F. M. Smith, and Bruno De Finetti, ``Theory of probability: a critical introductory treatment'' John Wiley & Sons (2017).

[14] D. S. Siviaand J. Skilling ``Data analysis: a Bayesian tutorial'' Oxford University Press (2006).

[15] J. R. Norris ``Markov Chains'' Cambridge University Press (1997).

[16] Aiken Pangand Peter Membrey ``Beginning FPGA: Programming Metal'' Apress (2017).

[17] Hideo Mabuchi ``Continuous quantum error correction as classical hybrid control'' New Journal of Physics 11, 105044 (2009).

[18] A. V. Borisov ``L1-Optimal Filtering of Markov Jump Processes. I. Exact Solution and Numerical Implementation Schemes'' Automation and Remote Control 81, 1945–1962 (2020).

[19] Yin Gang George, Qing Zhang, and Yuanjin Liu, ``Discrete-time approximation of Wonham filters'' Journal of Control Theory and Applications 2, 1–10 (2004).

[20] G. M. Hieftje ``Signal-to-noise enhancement through instrumental techniques. II. Signal averaging, boxcar integration, and correlation techniques'' Analytical Chemistry 44, 69A–78a (1972).

[21] Pierre Blanchard, Desmond J Higham, and Nicholas J Higham, ``Accurately computing the log-sum-exp and softmax functions'' IMA Journal of Numerical Analysis draa038 (2020).

[22] M. Haselman, M. Beauchamp, A. Wood, S. Hauck, K. Underwood, and K.S. Hemmert, ``A Comparison of Floating Point and Logarithmic Number Systems for FPGAs'' 13th Annual IEEE Symposium on Field-Programmable Custom Computing Machines (FCCM’05) 181–190 (2005).

[23] Peter W. Shor ``Scheme for reducing decoherence in quantum computer memory'' Physical Review A 52, R2493–R2496 (1995).

[24] A. M. Steane ``Error Correcting Codes in Quantum Theory'' Physical Review Letters 77, 793–797 (1996).

Cited by

On Crossref's cited-by service no data on citing works was found (last attempt 2022-05-28 18:48:44). On SAO/NASA ADS no data on citing works was found (last attempt 2022-05-28 18:48:44).