A Logarithmic Bayesian Approach to Quantum Error Detection

TL;DR

This paper introduces a Bayesian digital filtering method using logarithmic probabilities for quantum error detection, achieving near-optimal performance with low-latency implementation suitable for quantum hardware.

Contribution

It proposes a novel logarithmic Bayesian filtering approach for quantum error correction, simplifying computations and improving performance over previous stochastic differential equation methods.

Findings

Significantly outperforms threshold schemes and linearized filters in simulations.

Effective for various error rates and time steps.

Provides practical filters suitable for low-latency quantum hardware.

Abstract

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.