The realistic coherent errors could induce very different behaviors compared with their stochastic counterparts in the quantum error correction (QEC) and fault tolerant quantum computation. Their impacts are believed to be very subtle, more detrimental and hard to analyze compared to those ideal stochastic errors. In this paper, we study the independent coherent error due to the imperfect unitary rotation on each physical qubit of the toric code. We find that the surface code under coherent error satisfies generalized Knill-Laflamme (K-L) criterion and falls into the category of approximate QEC. The extra term in the generalized K-L criterion corresponds to the coherent part of the error channel at logical level, and then show that the generalized K-L criterion approaches the normal K-L criterion when the code distance becomes large. In addition, we also find that if the code with a fixed distance d is
ε-correctable, the value of
ε describing the accuracy of the approximate QEC cannot be smaller than a lower bound. We then study the success probability of QEC under such coherent errors, and confirm that the exact success probability under coherent error is smaller than the results using Pauli twirling approximation at physical level.
Abstract Quantum error correction (QEC) is essential for scalable quantum computing, but its realistic evaluation is hindered by the difficulty of simulating coherent noise in large codes. We propose a unified framework for the rotated surface code that integrates a qubit-reduction strategy preserving error dynamics, a coherent noise model with spatial and temporal correlations, and an effective error model validated through simulations up to d=7. GPU-accelerated simulations using QULACS show that coherent errors in the d=5 code predominantly interfere destructively, reducing their effective impact by about 13%. Based on these results, we establish conditions for break-even QEC, requiring a gate-time-to-coherence-time ratio below 0.005–0.007. Our findings indicate that current superconducting platforms operate close to this regime, with residual coupling-induced coherent errors as the main limitation, and provide a practical tool for optimizing surface-code QEC.
Studies of quantum error correction (QEC) typically focus on stochastic Pauli errors because the existence of a threshold error rate below which stochastic Pauli errors can be corrected implies that there exists a threshold below which generic errors can be corrected. However, rigorous estimates of the threshold for generic errors are typically orders of magnitude worse than the threshold for stochastic Pauli errors. Specifically, coherent errors have a particularly harmful impact on the encoded space because they can map encoded states to superpositions of logical and error states. Further, coherent errors can add up and interfere over multiple rounds of error correction or between syndrome measurements, which may result in significantly worse errors than expected under a stochastic Pauli error model. In this paper, we present an algorithm which decoheres noise at the logical level, projecting the state of the system onto a logical state with a well-defined error. The algorithm does not significantly increase the depth of the logical circuit (and usually does not lead to any increase in depth), and applies generally to most fault-tolerant gadgets and error correction steps.