The design and performance analysis of quantum error correction (QEC) codes are often based on incoherent and independent noise models since it is easy to simulate. However, these models fail to capture realistic hardware noise sources, such as correlated errors (crosstalk), which can significantly impact QEC code performance, especially when they occur between data and ancillary qubits. In this paper, we systematically study various types of crosstalk noise and quantify their effects on surface codes through memory and stability experiments. Based on our findings, we introduce crosstalk-robust implementations of QEC via flag qubit designs and redundant stabilizer checks. We perform both numerical and analytical studies to demonstrate the efficacy of these strategies. In addition, we analyze logical crosstalk in an
[[n,k>1,d]]code block and establish analytical conditions under which physical crosstalk does not lead to logical crosstalk. Together, our analytical and numerical results shed light on designing QEC codes that are robust against hardware realistic crosstalk noise, paving the way for reliable experimental realization of fault-tolerant quantum computing.
Quantum error correction (QEC), the lynchpin of fault-tolerant quantum computing (FTQC), is designed and validated against well-behaved Pauli stochastic error models. But in real-world deployment, QEC protocols encounter a vast array of other errors -- coherent and non-Pauli errors -- whose impacts on quantum circuits are vastly different than those of stochastic Pauli errors. The impacts of these errors on QEC and FTQC protocols have been largely unpredictable to date due to exponential classical simulation cost. Here, we show how to accurately and efficiently model the effects of coherent and non-Pauli errors on FTQC, and we study the effects of such errors on syndrome extraction for surface and bivariate bicycle codes, and on magic state cultivation. Our analysis suggests that coherent error can shift fault-tolerance thresholds, increase the space-time cost of magic state cultivation, and can increase logical error rates by an order of magnitude compared to equivalent stochastic errors. These analyses are enabled by a new technique for mapping any Markovian circuit-level error model with sufficiently small error rates onto a detector error model (DEM) for an FTQC circuit. The resulting DEM enables Monte Carlo estimation of logical error rates and noise-adapted decoding, and its parameters can be analytically related to the underlying physical noise parameters to enable approximate strong simulation.
We propose a sampling-based simulation for fault-tolerant quantum error correction under coherent noise. A mixture of incoherent and coherent noise, possibly due to over-rotation, is decomposed into Clifford channels with a quasiprobability distribution. Then, an unbiased estimator of the logical error probability is constructed by sampling Clifford channels with an appropriate postprocessing. We characterize the sampling cost via the channel robustness and find that the proposed sampling-based method is feasible even for planar surface codes with relatively large code distances intractable for full state-vector simulations. As a demonstration, we simulate repetitive faulty syndrome measurements on the planar surface code of distance 5 with 81 qubits. We find that the coherent error increases the logical error rate. This is a practical application of the quasiprobability simulation for a meaningful task and would be useful to explore experimental quantum error correction on the near-term quantum devices.
Studying the threshold behavior of surface codes under biased noise models is an active area of research. Previous work Tuckett et al. (2018), using an optimal tensor-network decoder, demonstrated that replacing
Z-type stabilizers with
Y-type stabilizers significantly improves the surface code threshold under code-capacity level dephasing noise. In this work, we construct and study a
ZY surface code by replacing the
X-type stabilizers with
Y-type stabilizers. We compare it with the standard
ZX surface code under circuit-level Pauli-
X biased noise, with and without an additional gate-based
XX crosstalk noise. We find that for the
ZX surface code, the
X-memory threshold increases monotonically with bias while the
Z-memory threshold decreases and saturates. For the
ZY surface code, the
Y-memory threshold is nearly constant across all bias values. The
Z-memory thresholds of the
ZX and
ZY codes are consistent within the uncertainty. Adding
XX crosstalk reduces the
Z-memory threshold beyond the fitting uncertainty while leaving the
X memory threshold largely unaffected. The choice of CNOT ordering redistributes threshold performance between the two logical memories. Our work extends prior observations from code-capacity level noise to circuit-level noise. It also indicates the need for decoders capable of jointly reasoning over correlated syndrome information so that tailored stabilizer structures could be fully utilized for quantum error correction.
Abstract Surface codes are building blocks of quantum computing platforms based on 2D arrays of qubits responsible for detecting and correcting errors. The error suppression achieved by the surface code is usually estimated by simulating toy noise models describing random Pauli errors. However, Pauli noise models fail to capture coherent processes such as systematic unitary errors caused by imperfect control pulses. Here we report the first large-scale simulation of quantum error correction protocols based on the surface code in the presence of coherent noise. We observe that the standard Pauli approximation provides an accurate estimate of the error threshold but underestimates the logical error rate in the sub-threshold regime. We find that for large code size the logical-level noise is well approximated by random Pauli errors even though the physical-level noise is coherent. Our work demonstrates that coherent effects do not significantly change the error correcting threshold of surface codes. This gives more confidence in the viability of the fault-tolerance architecture pursued by several experimental groups.
Fault-tolerant quantum computation critically depends on the ability to detect and correct physical qubit errors faster than they accumulate. The surface code (SC) has emerged as the most experimentally promising quantum error correction (QEC) framework due to its high threshold and local stabilizer structure, yet its practical performance is bottlenecked by decoder efficiency. Traditional decoding approaches such as minimum-weight perfect matching (MWPM) achieve near-optimal accuracy under independent noise but struggle with correlated errors, circuit-level noise, and real-time latency constraints. This paper investigates how neural network (NN)-based decoders reduce logical error rates in SC quantum systems by learning complex syndrome-to-correction mappings from data. We propose a hybrid convolutional-recurrent neural architecture that processes stabilizer measurement syndromes across both spatial and temporal dimensions, enabling the decoder to capture correlated error patterns inherent in realistic hardware. Systematic simulations under depolarizing, biased, and circuit-level noise models demonstrate that the proposed neural decoder achieves a logical error rate reduction of up to 41.3% compared to MWPM at a physical error rate of p = 0.008, while maintaining sub-millisecond decoding latency compatible with superconducting qubit cycle times. These results establish that data-driven neural decoders represent a scalable and hardware-compatible pathway toward achieving the logical error rates required for practical fault-tolerant quantum computing.