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arxiv: 2605.14637 · v1 · pith:XTOL6XTPnew · submitted 2026-05-14 · 🪐 quant-ph

Adaptive Window Decoding based on Spatiotemporal Complementary Gap

Moeto Mishima , Riki Toshio , Kaito Kishi , Jun Fujisaki , Hirotaka Oshima , Shintaro Sato , Keisuke Fujii This is my paper

Pith reviewed 2026-06-30 20:43 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum error correctionwindow decodingadaptive decodingsoft informationspatiotemporal complementary gapfault-tolerant quantum computingreal-time decodingbuffer size
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The pith

An adaptive window decoding scheme based on spatiotemporal complementary gap reduces average buffer size by about 40 percent while keeping the logical error rate unchanged.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an adaptive version of window decoding for real-time quantum error correction. It first attempts decoding with a small buffer region and computes a new confidence value called the spatiotemporal complementary gap. When that value indicates low confidence, the buffer is enlarged and decoding is repeated for that shot. Simulations show the average buffer size drops by roughly 40 percent because most shots succeed with the small buffer, while the overall logical error rate matches the fixed large-buffer baseline. This approach targets the latency bottleneck in fault-tolerant quantum computing by shrinking typical window sizes without performance loss.

Core claim

The paper establishes that the spatiotemporal complementary gap provides usable soft information for small-buffer window decoding, enabling an adaptive scheme that starts with a small buffer and enlarges it only when the gap signals insufficient confidence; numerical simulations confirm this cuts the average buffer size by approximately 40 percent while the logical error rate remains the same as in non-adaptive large-buffer decoding.

What carries the argument

The spatiotemporal complementary gap, a new soft-information measure designed for window decoding with small buffers that quantifies decoding confidence to decide whether buffer enlargement is required.

If this is right

  • Average decoding time falls because the small buffer handles the majority of shots.
  • Logical error rates stay equivalent to those achieved with fixed buffers at least as large as the code distance.
  • Window decoding becomes practical at smaller typical sizes without degrading fault tolerance.
  • Real-time decoding latency decreases for quantum error correction loops.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The gap measure could extend to adaptive strategies in other decoding architectures beyond fixed windows.
  • Hardware schedulers might allocate buffer resources dynamically based on per-shot gap values.
  • Further tests on varied code distances and noise models would clarify how the 40 percent saving scales.

Load-bearing premise

The spatiotemporal complementary gap reliably flags cases where a small buffer suffices versus cases needing enlargement, without any net increase in logical error rate.

What would settle it

A direct comparison simulation on the same code and noise model where the adaptive scheme produces a higher logical error rate than the fixed large-buffer decoder would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.14637 by Hirotaka Oshima, Jun Fujisaki, Kaito Kishi, Keisuke Fujii, Moeto Mishima, Riki Toshio, Shintaro Sato.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of the adaptive window decoding scheme. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic illustration of the boundary nodes used in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Examples of logical errors caused by a small buffer size. Red lines represent the actual error chains, and blue lines [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Difficulties in computing the complementary error [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a). Suppose that, in the minimum-weight error, this defect is matched to the right boundary with weight s, while in reality an error chain connecting the defect to the virtual boundary has occurred. In this case, tmin can be evaluated explicitly. The minimum-weight scenario is one in which the error chain appears at the round right after the committed window. The condition for tmin to lead to a logical er… view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Error chains that should be penalized and those that [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Procedure for computing the path-selected STCG. After the minimum-weight error is first obtained, the buffer boundary [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The path-selected STCG computed for each window in sliding window decoding of the repetition code under the [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Adaptive sliding window decoding applied to the repetition code under the phenomenological noise model. (a) Logical [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The path-selected STCG computed for each window in sliding window decoding of the surface code under the circuit [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Adaptive sliding window decoding applied to the surface code under the circuit-level noise model. (a) Logical error [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Conditional window-induced logical error rate for [PITH_FULL_IMAGE:figures/full_fig_p014_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. (a) Gap distribution in a single window and the per-shot minimum-gap distribution in multiple windows. (b) [PITH_FULL_IMAGE:figures/full_fig_p015_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Comparison of the data predicted by Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p015_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. (a) Logical error rate per window and (b) ratio of the logical error rate to that of the global decoder, at [PITH_FULL_IMAGE:figures/full_fig_p016_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. The STCG computed for each window in sliding window decoding of the repetition code under the phenomenological [PITH_FULL_IMAGE:figures/full_fig_p018_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. The distance-shifted STCG computed for each window in sliding window decoding of the repetition code under the [PITH_FULL_IMAGE:figures/full_fig_p018_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. The STCG computed for each window in sliding window decoding of the surface code under the circuit-level noise [PITH_FULL_IMAGE:figures/full_fig_p019_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Window used in computing the path-selected STCG [PITH_FULL_IMAGE:figures/full_fig_p019_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. The path-selected STCG computed for each window in parallel window decoding of the surface code. (a) Probability [PITH_FULL_IMAGE:figures/full_fig_p020_22.png] view at source ↗
read the original abstract

Real-time decoding plays a crucial role in practical fault-tolerant quantum computing. Window decoding, in which the decoding problem is divided into windows, is a promising approach. While reducing the window size is desirable for faster decoding, each window contains a buffer region whose size must typically be at least the code distance to avoid degrading the logical error rate, which limits how much the window can shrink. In this paper, we propose an adaptive decoding scheme in which window decoding is first performed with a small buffer size and a decoding confidence (soft information) is computed; if the confidence is low, the buffer size is enlarged and decoding is redone. This approach reduces the average decoding time, since most shots are decoded with a small buffer. A central challenge in realizing this scheme is that existing forms of soft information are not directly applicable to window decoding, especially with a small buffer. We address this challenge by introducing a new form of soft information, the spatiotemporal complementary gap, specifically designed for this setting. Numerical simulations demonstrate that the proposed scheme reduces the average buffer size by approximately 40% while maintaining the logical error rate.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper proposes an adaptive window decoding scheme for real-time decoding in fault-tolerant quantum computing. Window decoding is performed first with a small buffer; a new soft-information metric called the spatiotemporal complementary gap is computed, and the buffer is enlarged and decoding redone only if confidence is low. The central claim, supported by numerical simulations, is that this reduces average buffer size by approximately 40% while preserving the logical error rate.

Significance. If the empirical result holds under standard surface-code or similar models, the scheme would meaningfully lower average decoding latency without sacrificing error suppression, addressing a practical bottleneck in real-time FTQC. The introduction of a window-specific soft metric is a targeted contribution that could be reusable beyond the adaptive-buffer setting.

major comments (1)
  1. [Abstract] Abstract: the central empirical claim (approximately 40% average buffer-size reduction while maintaining logical error rate) is stated without any description of the underlying codes, noise models, decoder implementation, number of Monte Carlo shots, error bars, or statistical significance tests. This information is load-bearing for assessing whether the spatiotemporal complementary gap reliably identifies safe small-buffer cases.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comment on the abstract. We address it point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central empirical claim (approximately 40% average buffer-size reduction while maintaining logical error rate) is stated without any description of the underlying codes, noise models, decoder implementation, number of Monte Carlo shots, error bars, or statistical significance tests. This information is load-bearing for assessing whether the spatiotemporal complementary gap reliably identifies safe small-buffer cases.

    Authors: We agree that the abstract would benefit from a concise statement of the simulation parameters supporting the central claim. Although the full details of the codes, noise models, decoder, Monte Carlo statistics, and error analysis are provided in the numerical simulations section of the manuscript, we acknowledge that the abstract should be more self-contained for this key empirical result. In the revised manuscript we will expand the abstract with a brief description of these elements while respecting length constraints. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a new soft-information metric (spatiotemporal complementary gap) for adaptive window decoding and validates it via numerical simulations showing ~40% average buffer-size reduction at fixed logical error rate. No derivation chain, fitted-parameter prediction, or self-citation load-bearing step is present; the central claim is an empirical outcome directly falsifiable by the reported simulations. The work is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Review based solely on abstract; no explicit free parameters, axioms, or invented entities beyond the newly introduced soft-information metric can be identified.

invented entities (1)
  • spatiotemporal complementary gap no independent evidence
    purpose: soft information for decoding confidence in small-buffer window decoding
    Newly introduced metric described in the abstract as the key enabler of the adaptive scheme

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Reference graph

Works this paper leans on

59 extracted references · 21 canonical work pages · 6 internal anchors

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    In this paper, a detector whose value is 1 is referred to as a defect. For the standard syndrome extraction circuits of sur- face codes, even under circuit-level noise, elementary er- ror events can be decomposed in such a way that each error event flips at most two detectors. This allows us to define a graph whose nodes correspond to detectors and whose ...

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    Therefore, by rerunning MWPM withσ(v b,0) =n 0 ⊕1 andσ(v b,1) =n 1 ⊕1 while keeping all other defect con- figurations unchanged, one obtains the minimum-weight errorE comp(σ) in the different logical class. III. ADAPTIVE WINDOW DECODING A. Key Issue The method we propose in this paper for reducing the size of the decoding problem is as follows (see also F...

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    General Theory In this section, we introduce quantities that serve as indicators of the conditioned logical error rate, given a syndrome measurement outcome within a window under sliding window decoding. The underlying idea is anal- ogous to that of the complementary gap. As discussed in Sec. II C, the complementary gap provides a reliable estimate of the...

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    Spatiotemporal Complementary Gap As discussed in Sec. III B 1, the principal mechanism by which a logical error arises is that defects are incorrectly matched to the virtual boundary, leading to the commit- ment of incorrect errors (see Fig. 4). This observation motivates a natural choice ofE alt; namely, the error ob- tained by taking the complementary c...

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    III B 2, we considered the quantityw alt −w min while neglectingt min

    Distance-shifted STCG In Sec. III B 2, we considered the quantityw alt −w min while neglectingt min. As noted there, computing the exact value oft min for general error configurationE min andE alt is difficult. We therefore introduce a simple approximation fort min. We first consider a simple example in which only a single defect is present in the window,...

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    While this procedure yields anE alt of small weight, the correspondingt min may not be small

    Path-selected STCG In computing distance-shifted STCG, we first deter- mineE alt as the complementary error between the virtual and commit boundaries with respect toE min, and then computet min approximately. While this procedure yields anE alt of small weight, the correspondingt min may not be small. In other words, there may exist an alterna- tiveE alt ...

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    For practi- cal relevance, it is important to extend the soft-output schemes to the circuit-level noise model on the surface code

    Extension to Circuit-level Noise The discussion so far has assumed a phenomenologi- cal noise model with uniform edge weights. For practi- cal relevance, it is important to extend the soft-output schemes to the circuit-level noise model on the surface code. The main differences from the phenomenological case are twofold: diagonal edges appear in addition ...

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    We define the window-induced logical error rate as the prob- ability of events in which window decoding leads to a logical error while global decoding does not

    Properties of the Proposed Gaps We first compute the STCG, distance-shifted STCG, and path-selected STCG in sliding window decoding without switching, with the commit and buffer region sizes set to a common valuer com =r buf =⌊d/2⌋. We define the window-induced logical error rate as the prob- ability of events in which window decoding leads to a logical e...

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    Performance of Switching We next apply the adaptive sliding window decoding based on each of the three proposed gaps to the repe- tition code of the code distanced= 13 under the phe- nomenological noise model with the physical error rate p= 0.025. Herer buf denotes the default buffer size used before switching is invoked. Figure 11 (a) shows the logical e...

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    Properties of Proposed Gaps Figure 12 presents the probability distribution of the path-selected STCG and the window-induced logical er- ror rate conditioned on the per-shot minimum gap, evalu- ated for sliding window decoding on the surface code un- der uniform circuit-level noisep= 0.0025. As in the case of the repetition code, the probability distribut...

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