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arXiv:2610.02137

One-Shot any Code

Andrew C. Yuan

quant-phmath-phSubmitted Oct 1, 2026• Announced Oct 2, 2026• 0 citations
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Abstract

We first provide a simple proof showing that any single-shot code is self-correcting. We then show that any $[[n,k]]$ CSS quantum low-density parity-check (QLDPC) code can be transformed into an $[[nm,k]]$ CSS QLDPC code with single-shot (SS) quantum error correction and an efficient local decoder which runs in $O(\log m)$ parallel time. Below a constant threshold $p_{\rm RG}>0$ for joint local stochastic physical and measurement noise, the logical failure probability over $T$ correction rounds is bounded by \begin{equation*} O(T n)\exp[-Ω(m^α)] \end{equation*} for a constant $α>0$; we say that the \textit{threshold} is $p_{\rm RG}$ and \textit{error suppression} is $Ω(m^α)$. Moreover, if the input code is equipped with a single-shot decoder with threshold $p_c>0$ and error suppression $Ω(n^β)$, then the output code \textit{enhances} the error suppression to $Ω(m^α n^β)$, while maintaining the decoder-independent threshold $p_{\rm RG}$, and thus \textit{enhanced} if $p_c < p_{\rm RG}$.

Comments
The simple proof (single-shot implies self-correction) is only 2 pages, and independent of everything else

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We first provide a simple proof showing that any single-shot code is self-correcting. We then show that any [[n,k]][[n,k]][[n,k]] CSS quantum low-density parity-check (QLDPC) code can be transformed into an [[nm,k]][[nm,k]][[nm,k]] CSS QLDPC code with single-shot (SS) quantum error correction and an efficient local decoder which runs in O(log⁡m)O(\log m)O(logm) parallel time. Below a constant threshold pRG>0p_{\rm RG}>0pRG​>0 for joint local stochastic physical and measurement noise, the logical failure probability over TTT correction rounds is bounded by \begin{equation*}
O(T n)\exp[-Ω(m^α)] \end{equation*} for a constant α>0α>0α>0; we say that the \textit{threshold} is pRGp_{\rm RG}pRG​ and \textit{error suppression} is Ω(mα)Ω(m^α)Ω(mα). Moreover, if the input code is equipped with a single-shot decoder with threshold pc>0p_c>0pc​>0 and error suppression Ω(nβ)Ω(n^β)Ω(nβ), then the output code \textit{enhances} the error suppression to Ω(mαnβ)Ω(m^α n^β)Ω(mαnβ), while maintaining the decoder-independent threshold pRGp_{\rm RG}pRG​, and thus \textit{enhanced} if pc<pRGp_c < p_{\rm RG}pc​<pRG​.
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Preparing logical codewords with a single round of noisy syndrome measurements can reduce the time overhead of fault-tolerant quantum computation. It remains unclear which structural properties of the code suffice to guarantee a threshold for single-shot state preparation. Here we prove that linear confinement suffices for a nonzero single-shot state-preparation threshold for CSS quantum low-density parity-check codes, in contrast to the common belief that soundness is necessary. When the distance is at least logarithmic, we prove that below nonzero local stochastic readout and data error thresholds, the logical error probability after a formal ideal final recovery is exponentially small and approaches zero when the code size approaches infinity. Under the stronger assumption of linear soundness, we show that the residual data error of the preparation protocol is local stochastic, directly allowing composition with other fault-tolerant gadgets. To investigate single-shot state-preparation threshold beyond the above assumptions, we further develop a statistical mechanical mapping that expresses the final logical error rate in terms of a classical partition function. We perform Monte Carlo simulations of this model for the three-dimensional toric code. The numerical results suggest that sublinear soundness can also support a single-shot state-preparation threshold.
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.
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