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arXiv:2511.20576

Dynamic local single-shot checks for toric codes

Yingjia Lin, Abhinav Anand, Kenneth R. Brown

quant-phSubmitted Nov 25, 2025• Announced Nov 26, 2025• 0 citations
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Abstract

Quantum error correction typically requires repeated syndrome extraction due to measurement noise, which results in substantial time overhead in fault-tolerant computation. Single-shot error correction aims to suppress errors using only one round of syndrome extraction. However, for most codes, it requires high-weight checks, which significantly degrade, and often eliminate, single-shot performance at the circuit level. In this work, we introduce local single-shot checks, where we impose constraints on check weights. Using a dynamic measurement scheme, we show that the number of required measurement rounds can be reduced by a factor determined by this constraint. As an example, we show through numerical simulation that our scheme can improve decoding performance compared to conventional checks when using sliding-window decoding with a reduced window size under circuit-level noise models for toric codes. Our work provides a new direction for constructing checks that can reduce time overhead in large-scale fault-tolerant quantum computation.

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Quantum error correction typically requires repeated syndrome extraction due to measurement noise, which results in substantial time overhead in fault-tolerant computation. Single-shot error correction aims to suppress errors using only one round of syndrome extraction. However, for most codes, it requires high-weight checks, which significantly degrade, and often eliminate, single-shot performance at the circuit level. In this work, we introduce local single-shot checks, where we impose constraints on check weights. Using a dynamic measurement scheme, we show that the number of required measurement rounds can be reduced by a factor determined by this constraint. As an example, we show through numerical simulation that our scheme can improve decoding performance compared to conventional checks when using sliding-window decoding with a reduced window size under circuit-level noise models for toric codes. Our work provides a new direction for constructing checks that can reduce time overhead in large-scale fault-tolerant quantum computation.
For quantum error correction codes the required number of measurement rounds typically increases with the code distance when measurements are faulty. Single-shot error correction allows for an error threshold with only one round of noisy syndrome measurements regardless of the code size. Here we implement single-shot check operators for toric codes. The single-shot checks are constructed by Gaussian elimination following Campbell [Campbell, Quantum Sci. Technol. 4, 025006 (2019)]. The single-shot check operators result in a sustainable threshold at 5.62% for an error model with noisy measurements, outperforming the conventional toric code-check operators with multiple rounds of noisy measurement. The cost of the transformation is nonlocal high-weight stabilizer generators. We then consider a gate-based error model that leads to increased measurement error with stabilizer weight. Here we find no single-shot threshold behavior and instead find the code family will have an optimal code size for a fixed error rate. For this error model the conventional check operators with multiple measurements yields a lower logical error rate.
Quantum error correction (QEC) with single-shot decoding enables reduction of errors after every single round of noisy stabilizer measurement, easing the time-overhead requirements for fault tolerance. Notably, several classes of quantum low-density-parity-check (qLDPC) codes are known which facilitate single-shot decoding, potentially giving them an additional overhead advantage. However, the perceived advantage of single-shot decoding is limited because it can significantly degrade the effective code distance. This degradation may be compensated for by using a much larger code size to achieve the desired target logical error rate, at the cost of increasing the amount of syndrome information to be processed, as well as, increasing complexity of logical operations. Alternatively, in this work we study sliding-window decoding, which corrects errors from previous syndrome measurement rounds while leaving the most recent errors for future correction. We observe that sliding-window decoding significantly improves the logical memory lifetime and hence the effective distance compared to single-shot decoding on hypergraph-product codes and lifted-product codes. Remarkably, we find that this improvement may not cost a larger decoding complexity. Thus, the sliding-window strategy can be more desirable for fast and accurate decoding for fault-tolerant quantum computing with qLDPC codes.
Quantum error correction will likely be essential for building a large-scale quantum computer, but it comes with significant requirements at the level of classical control software. In particular, a quantum error-correcting code must be supplemented with a fast and accurate classical decoding algorithm. Standard techniques for measuring the parity-check operators of a quantum error-correcting code involve repeated measurements, which both increases the amount of data that needs to be processed by the decoder, and changes the nature of the decoding problem. Knill error correction is a technique that replaces repeated syndrome measurements with a single round of measurements, but requires an auxiliary logical Bell state. Here, we provide a theoretical and numerical investigation into Knill error correction from the perspective of decoding. We give a self-contained description of the protocol, prove its fault tolerance under locally decaying (circuit-level) noise, and numerically benchmark its performance for quantum low-density parity-check codes. We show analytically and numerically that the time-constrained decoding problem for Knill error correction can be solved using the same decoder used for the simpler code-capacity noise model, illustrating that Knill error correction may alleviate the stringent requirements on classical control required for building a large-scale quantum computer.
Quantum error correction (QEC) with single-shot decoding enables the reduction of errors after every single round of noisy stabilizer measurement, easing the time-overhead requirements for fault tolerance. Notably, several classes of quantum low-density parity check (qLDPC) codes are known which facilitate single-shot decoding, potentially giving them an additional overhead advantage. However, the perceived advantage of single-shot decoding is limited because it can significantly degrade the effective code distance. This degradation may be compensated for by using a much larger code size to achieve the desired target logical error rate, at the cost of increasing the amount of syndrome information to be processed, as well as increasing complexity of logical operations. Alternatively, in this paper we study sliding-window decoding, which corrects errors from previous syndrome measurement rounds while leaving the most recent errors for future correction. We observe that sliding-window decoding significantly improves the logical memory lifetime and hence the effective distance compared to single-shot decoding on hypergraph-product codes and lifted-product codes. Remarkably, we find that this improvement may not cost a larger decoding complexity. Thus, the sliding-window strategy can be more desirable for fast and accurate decoding for fault-tolerant quantum computing with qLDPC codes.
Quantum low-density parity-check (qLDPC) codes have emerged as a promising approach for realizing low-overhead logical quantum memories. Recent theoretical developments have established shift automorphisms as a fundamental building block for completing the universal set of logical gates for qLDPC codes. However, practical challenges remain because the existing SWAP-based shift automorphism yields logical error rates that are orders of magnitude higher than those for fault-tolerant idle operations. In this work, we address this issue by dynamically varying the syndrome measurement circuits to implement the shift automorphisms without reducing the circuit distance. We benchmark our approach on both twisted and untwisted weight-6 generalized toric codes, including the gross code family. Our time-dynamic circuits for shift automorphisms achieve performance comparable to the idle operations under the circuit-level noise model (SI1000). Specifically, the dynamic circuits achieve more than an order of magnitude reduction in logical error rates relative to the SWAP-based scheme for the gross code at a physical error rate of 10−310^{-3}10−3, employing the BP-OSD decoder. Our findings improve both the error resilience and the time overhead of the shift automorphisms in qLDPC codes. Furthermore, our work can lead to alternative syndrome extraction circuit designs, such as leakage removal protocols, providing a practical pathway to utilizing dynamic circuits that extend beyond surface codes towards qLDPC codes.
Abstract Fault-tolerant protocols and quantum error correction (QEC) are essential to building reliable quantum computers from imperfect components that are vulnerable to errors. Optimizing the resource and time overheads needed to implement QEC is one of the most pressing challenges. Here, we introduce a new topological quantum error-correcting code, the three-dimensional subsystem toric code (3D STC). The 3D STC can be realized with geometrically-local parity checks of weight at most three on the cubic lattice with open boundary conditions. We prove that one round of parity-check measurements suffices to perform reliable QEC with the 3D STC even in the presence of measurement errors. We also propose an efficient single-shot QEC decoding strategy for the 3D STC and numerically estimate the resulting storage threshold against independent bit-flip, phase-flip and measurement errors to bepSTC ≈ 1.045%. Such a high threshold together with local parity-check measurements make the 3D STC particularly appealing for realizing fault-tolerant quantum computing.