# Always-On Quantum Error Tracking with Continuous Parity Measurements

## TL;DR

This paper explores continuous quantum error correction using parity measurements, proposing practical filtering methods that improve real-time error tracking efficiency and reduce overhead compared to traditional approaches.

## Contribution

It introduces and compares practical filtering techniques, including an optimal Bayesian filter, for continuous error correction in quantum systems, enhancing real-time error tracking.

## Key findings

- Continuous parity measurements enable real-time error tracking.
- Practical filters outperform simple thresholding in fidelity decay.
- Optimal Bayesian filter offers computational advantages.

## Abstract

We investigate quantum error correction using continuous parity measurements to correct bit-flip errors with the three-qubit code. Continuous monitoring of errors brings the benefit of a continuous stream of information, which facilitates passive error tracking in real time. It reduces overhead from the standard gate-based approach that periodically entangles and measures additional ancilla qubits. However, the noisy analog signals from continuous parity measurements mandate more complicated signal processing to interpret syndromes accurately. We analyze the performance of several practical filtering methods for continuous error correction and demonstrate that they are viable alternatives to the standard ancilla-based approach. As an optimal filter, we discuss an unnormalized (linear) Bayesian filter, with improved computational efficiency compared to the related Wonham filter introduced by Mabuchi [New J. Phys. 11, 105044 (2009)]. We compare this optimal continuous filter to two practical variations of the simplest periodic boxcar-averaging-and-thresholding filter, targeting real-time hardware implementations with low-latency circuitry. As variations, we introduce a non-Markovian ``half-boxcar'' filter and a Markovian filter with a second adjustable threshold; these filters eliminate the dominant source of error in the boxcar filter, and compare favorably to the optimal filter. For each filter, we derive analytic results for the decay in average fidelity and verify them with numerical simulations.