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.
Highlights
Quantum error correction (QEC) is essential to building a scalable and fault-tolerant quantum computer [1, 2]. the theory of QEC has been developing since the 1990s and is well established for the circuit model of quantum computation, the practical implemention of QEC in realistic hardware raises additional nuance that prompts more detailed investigation
We show that for passive error tracking the benefits of continuous measurements can outweigh the disadvantages, enabling high-fidelity decoding of the logical qubit without the need for active feedback
We have analyzed the 3-qubit bit-flip code to assess the performance of direct methods for measuring the syndromes using time-continuous parity measurements
Summary
Quantum error correction (QEC) is essential to building a scalable and fault-tolerant quantum computer [1, 2]. A possible route to perform QEC without the overhead of ancilla qubits is to directly monitor the error syndromes continuously in time [22, 23] With this variation, the code subspaces for the error syndromes are directly coupled to a continuous readout device [24,25,26,27,28,29,30], avoiding the need for periodic entangling gates and additional ancilla measurements. We assess the performance of implementing continuous QEC for the simplest three-qubit bit-flip code, assuming a simplified model of modern superconducting hardware, and develop practical filters to interpret the stochastic time-continuous signals. We include an Appendix that contains a complementary analysis of an ancilla-based projective measurement implementation of the three-qubit bit-flip code
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