Surface codes are promising for practical quantum error correction due to their high threshold and experimental feasibility. However, their performance under realistic noise conditions, particularly those involving correlated errors, requires further investigation. In this study, we systematically investigate the effect of several distinct types of correlated errors that could potentially arise from next-nearest-neighbor coupling in quantum systems. Our analytical results reveal that these correlated errors exhibit lower logical error rates due to their inherent structural symmetries. We further perform circuit-level simulations that incorporate realistic circuit level errors alongside correlated noise components, demonstrating that surface code performance is intimately related to the structural characteristics of the correlation patterns. This deepens our insights into the threshold of surface code and hence facilitates a more robust design of quantum circuits with a higher noise threshold.
The surface code represents a promising candidate for fault-tolerant quantum computation due to its high error threshold and experimental accessibility with nearest-neighbor interactions. However, current exact surface code threshold analyses are based on the assumption of independent and identically distributed (i.i.d.) errors. Though there are numerical studies for the threshold with correlated error, they are only the lower bound rather than the exact value, which offers potential for higher error thresholds. Here, we establish an error-edge map, which allows for the mapping of quantum error correction to a square-octagonal random bond Ising model. We then present the maximum-likelihood threshold under a realistic noise model that combines independent single-qubit errors with correlated errors between nearest-neighbor data qubits. Our method is applicable for any ratio of nearest-neighbor correlated errors to i.i.d. errors. We present analytical constraints for the error correction threshold of surface codes, and by applying these constraints, we numerically obtain the error threshold higher than the existing values. Our error threshold is both an upper bound and achievable.
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.
Abstract The realization of fault-tolerant quantum computers hinges on effective quantum error correction (QEC) protocols, whose performance significantly relies on the nature of the underlying noise. In this work, we directly study the structure of non-Markovian correlated errors and their impact on surface code memory performance. Specifically, we compare surface code performance under non-Markovian noise and independent circuit-level noise, while keeping marginal error rates constant. Our analysis shows that while not all temporally correlated structures are detrimental, certain structures, particularly multi-time ‘streaky’ correlations affecting syndrome qubits and two-qubit gates, can severely degrade logical error rate scaling. Furthermore, we discuss our results in the context of recent QEC experiments on physical devices. These findings underscore the importance of understanding and mitigating non-Markovian noise toward achieving practical, fault-tolerant quantum computing.
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.
For quantum computers to reach their full potential will require error correction. We study the surface code, one of the most promising quantum error correcting codes, in the context of predominantly dephasing (Z-biased) noise, as found in many quantum architectures. We find that the surface code is highly resilient to Y-biased noise, and tailor it to Z-biased noise, whilst retaining its practical features. We demonstrate ultrahigh thresholds for the tailored surface code: ~39% with a realistic bias of = 100, and ~50% with pure Z noise, far exceeding known thresholds for the standard surface code: ~11% with pure Z noise, and ~19% with depolarizing noise. Furthermore, we provide strong evidence that the threshold of the tailored surface code tracks the hashing bound for all biases. We reveal the hidden structure of the tailored surface code with pure Z noise that is responsible for these ultrahigh thresholds. As a consequence, we prove that its threshold with pure Z noise is 50%, and we show that its distance to Z errors, and the number of failure modes, can be tuned by modifying its boundary. For codes with appropriately modified boundaries, the distance to Z errors is O(n) compared to O(n1/2) for square codes, where n is the number of physical qubits. We demonstrate that these characteristics yield a significant improvement in logical error rate with pure Z and Z-biased noise. Finally, we introduce an efficient approach to decoding that exploits code symmetries with respect to a given noise model, and extends readily to the fault-tolerant context, where measurements are unreliable. We use this approach to define a decoder for the tailored surface code with Z-biased noise. Although the decoder is suboptimal, we observe exceptionally high fault-tolerant thresholds of ~5% with bias = 100 and exceeding 6% with pure Z noise. Our results open up many avenues of research and, recent developments in bias-preserving gates, highlight their direct relevance to experiment.
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.