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.
 Peter W. Shor ``Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer'' SIAM Journal on Computing 26, 1484-1509 (1997).
 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).
 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).
 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).
 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).
 Bruno De Finetti, Antonio Machì, Adrian F. M. Smith, and Bruno De Finetti, ``Theory of probability: a critical introductory treatment'' John Wiley & Sons (2017).
 D. S. Siviaand J. Skilling ``Data analysis: a Bayesian tutorial'' Oxford University Press (2006).
 Hideo Mabuchi ``Continuous quantum error correction as classical hybrid control'' New Journal of Physics 11, 105044 (2009).
 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).
 Yin Gang George, Qing Zhang, and Yuanjin Liu, ``Discrete-time approximation of Wonham filters'' Journal of Control Theory and Applications 2, 1–10 (2004).
 G. M. Hieftje ``Signal-to-noise enhancement through instrumental techniques. II. Signal averaging, boxcar integration, and correlation techniques'' Analytical Chemistry 44, 69A–78a (1972).
 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).
 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).
 Ian Convy, Haoran Liao, Song Zhang, Sahil Patel, William P Livingston, Ho Nam Nguyen, Irfan Siddiqi, and K Birgitta Whaley, "Machine learning for continuous quantum error correction on superconducting qubits", New Journal of Physics 24 6, 063019 (2022).
The above citations are from Crossref's cited-by service (last updated successfully 2023-11-28 22:46:29) and SAO/NASA ADS (last updated successfully 2023-11-28 22:46:29). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.