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arXiv:2605.03054

Closed form logical error rate approximations for surface codes

Shaked Regev, Daniel Dilley, Ryan Bennink

quant-phcs.DMmath.COSubmitted May 4, 2026• Announced May 6, 2026• 0 citations• Revised Sep 11, 2026 (v3)
v1 · May 4, 2026v2 · Aug 6, 2026v3 · Sep 11, 2026
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Abstract

We propose a novel method to calculate logical error rates in surface codes, assuming independent and identically distributed physical errors. These results fit well known scaling laws for studied physical error rates and code distances, but break down in some other regimes. Increasing the code distance still reduces logical error rates, but returns diminish quickly beyond a certain point. We show how to use our method to analyze hypothetical quantum computers with various configurations and select designs with lower error rates. Currently, this requires expensive classical simulations of quantum decoders for various distances and physical error rates or inaccurate extrapolation from minimal experimental data. Instead, we use the symmetry of the problem to count the configurations that result in a logical error with our novel software. Given a physical error rate, we can deduce the probability of a logical error to provably good accuracy quickly. We include an analysis of measurement errors to allow for a more complete comparison of different surface code implementations. We extend this counting method to a particular locally-correlated error model and another globally-correlated error model.

Comments
8 pages, 7 figures

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We propose a novel method to calculate logical error rates in surface codes, assuming independent and identically distributed physical errors. These results fit well known scaling laws for studied physical error rates and code distances, but break down in some other regimes. Increasing the code distance still reduces logical error rates, but returns diminish quickly beyond a certain point. We show how to use our method to analyze hypothetical quantum computers with various configurations and select designs with lower error rates. Currently, this requires expensive classical simulations of quantum decoders for various distances and physical error rates or inaccurate extrapolation from minimal experimental data. Instead, we use the symmetry of the problem to count the configurations that result in a logical error with our novel software. Given a physical error rate, we can deduce the probability of a logical error to provably good accuracy quickly. We include an analysis of measurement errors to allow for a more complete comparison of different surface code implementations. We extend this counting method to a particular locally-correlated error model and another globally-correlated error model.
Practical quantum computing requires error rates much lower than those achievable with physical qubits. Surface codes provide a reliable method to implement logical qubits using physical qubits. However, when these physical qubits are transmitted within quantum networks, their error rates can vary due to differences in the routing paths. Our experimental results indicate that the geometric arrangement of these physical qubits within a surface code significantly impacts its logical error rate. In this paper, we address this issue by proposing disparity surface codes, which are a group of geometric configurations for surface codes that adapt them for transmission within quantum networks. For a distance-ddddisparity surface code, onlydddphysical qubits need to be transmitted with low error rates, and we identify multiple equivalent choices for selecting thesedddqubits. Additionally, we propose Di-MWPM, a new decoder designed to exploit physical error rate disparity for enhanced error correction. Results demonstrate that disparity surface codes maintain favorable performance in different network scenarios, even under high noise conditions. Our Di-MWPM decoder achieves improved Pauli-error thresholds.
Practical quantum computers will require resource-efficient error-correcting codes. The rotated surface code uses approximately half the number of qubits as the unrotated surface code to create a logical qubit with the same error-correcting distance. However, instead of distance, a more useful qubit-saving metric would be based on logical error rates. In this work we find the well-below-threshold scaling of logical to physical error rates under circuit-level noise for both codes at high odd and even distances and then compare the number of qubits used by each code to achieve equal logical error rates. We perform Monte Carlo sampling of memory experiment circuits with all valid CNOT orders using the stabilizer simulator Stim and the uncorrelated minimum-weight perfect matching decoder PyMatching 2. We find that the rotated code uses about 74 % the number of qubits used by the unrotated code to achieve a logical error rate of p L = 10 − 12 at the operational physical error rate of p = 10 − 3 . The ratio remains ≈ 75 % for p values within a factor of two of p = 10 − 3 for all useful p L . Our work finds the low- p L scaling of the surface code and clarifies the qubit savings provided by the rotated surface code, providing numerical justification for its use in future implementations of the surface code.
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Quantum error-correcting circuits are affected by multiple physical noise mechanisms, whose contributions to logical failure must be understood to evaluate code performance and guide improvements in hardware. The individual error budget contributions for an error type iii can be characterized by its logical sensitivity νi=∂pL∂piν_i = \frac{\partial p_L}{\partial p_i}νi​=∂pi​∂pL​​, which measures the response of the logical error rate pLp_LpL​ to each physical noise parameter pip_ipi​. Normally, νiν_iνi​ is measured using linear fits such as finite differences, where pLp_LpL​ is measured at two or more values of pip_ipi​ to calculate partial derivatives. In this paper, we develop a differentiable estimator to obtain all components of ∇ppL\nabla_{\mathbf p}p_L∇p​pL​ simultaneously from a single Monte Carlo data set at one noise configuration, using information about the underlying fault configurations. In surface code simulations with circuit-level noise, the estimator agrees with conventional finite differences while requiring one to two orders of magnitude fewer shots to achieve the same variance. We apply this technique to quantify error budgets, resolve sensitivities at the individual qubit level, and infer effective code distance. Finally, we incorporate the sensitivities into Newton root finding to locate and trace threshold contours in multidimensional noise models. These results provide an efficient method for extracting and applying logical sensitivity information from standard quantum error correction simulations.
Quantum Computing is a computing framework that takes advantage of unique quantum mechanical properties (such as superposition and entanglement) to perform calculations and implement algorithms that could offer exponential speed-ups over classical computing. However, in physical implementations of such quantum computers, qubits – the fundamental components of these systems – can accumulate errors that must be accounted for. In order to mitigate these errors, various quantum error correction (QEC) codes have been developed, including the repetition code and surface codes. In this experiment I implement and evaluate three types of QEC codes on the Qiskit simulator to compare their efficacy and applicability in correcting for different kinds of errors. I hypothesize that surface codes, with their more effective design and range of correction methods, should perform the best with much lower error thresholds and resultant logical error rates. The results support the hypothesis and suggest that surface codes are a viable method of implementing scalable error correction in quantum computers.