arxiv: 2604.22471 · v1 · submitted 2026-04-24 · 🪐 quant-ph

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Boundary-Aware Stabilizer Scheduling for Distributed Quantum Error Correction

Sanidhya Gupta , Sanidhay Bhambay , Narges Alavisamani , Neil Walton , Thirupathaiah Vasantam

Authors on Pith no claims yet

Pith reviewed 2026-05-08 12:08 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum error correctiondistributed quantum computingcolor codesstabilizer schedulinglogical error rateseam boundariesfault tolerancephotonic interconnects

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The pith

By scheduling seam stabilizer measurements less frequently, distributed quantum error correction can achieve lower logical error rates than measuring all seams every round.

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

Future modular quantum computers will link separate processors with photonic connections, creating boundaries where parity checks must use slow, probabilistic remote operations. In topological codes like color codes, these seam checks force data qubits to idle and pick up extra noise while waiting for Bell pairs. The paper develops two simple scheduling rules that skip some seam measurements and reuse the previous syndrome instead: a fixed-period skip and an adaptive version that picks the skip interval based on code size and entanglement success rate. Circuit simulations that include the extra idling noise show these rules cut the cost of remote operations and often produce better logical error rates than the obvious strategy of measuring every seam every round. In one range of entanglement rates the logical error rate still falls as the code grows larger, the expected sign of fault tolerance.

Core claim

We develop SS-τ and AST policies that integrate directly into standard syndrome-extraction circuits for triangular color codes. Under circuit-level noise in Stim that includes idling errors from Bell-pair generation delays, both policies reduce remote-operation overhead and can lower the logical error rate relative to the Measure-All baseline. For physical error rate p = 10^{-3} we identify an entanglement-generation-rate regime in which SS-τ and AST exhibit fault-tolerant scaling, with logical error rate decreasing as code distance increases, and both policies outperform Measure-All across the tested regimes.

What carries the argument

Boundary-aware scheduling policies (SS-τ and AST) that decide how often to perform seam parity checks while copying the most recent syndrome in skipped rounds.

If this is right

SS-τ and AST reduce remote-operation overhead relative to measuring every seam every round.
Both policies can produce lower logical error rates than the Measure-All baseline.
For physical error rate 10^{-3} and appropriate entanglement generation rates, logical error rate decreases as code distance increases under SS-τ and AST.
SS-τ and AST outperform Measure-All across the simulated regimes.

Where Pith is reading between the lines

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

The same skip-and-reuse idea could be applied to other topological codes that require remote operations at boundaries.
Real hardware may need dynamic adjustment of the skip interval round-by-round if Bell-pair generation times fluctuate more than the model assumes.
Combining the scheduler with improved decoder awareness of stale seam data might yield further reductions in logical error rate.

Load-bearing premise

The circuit-level noise model used in the simulations accurately captures the dominant error sources and timing in real distributed hardware with photonic interconnects.

What would settle it

An experiment on real photonic-interconnect hardware that measures logical error rates for triangular color codes under SS-τ or AST at physical error rate 10^{-3} and finds that the logical error rate does not decrease with increasing code distance in the claimed entanglement-generation-rate regime would falsify the fault-tolerant scaling result.

Figures

Figures reproduced from arXiv: 2604.22471 by Narges Alavisamani, Neil Walton, Sanidhay Bhambay, Sanidhya Gupta, Thirupathaiah Vasantam.

**Figure 1.** Figure 1: The [[7, 1, 3]] triangular color code. Data qubits (red circles) reside on the vertices, with the three triangular faces shaded red, green, and blue. An example X-type stabilizer, S X f , is shown on the top red face. The logical Z operator, ZL, is realized as the product of Z operators on the three qubits along any one colored boundary, for example, along the bottom red edge. a sequence of local operation… view at source ↗

**Figure 2.** Figure 2: Circuit for a remote CNOT. The gate is implemented using a shared EPR pair view at source ↗

**Figure 3.** Figure 3: Logical error rate (LER) versus physical error rate for triangular color codes view at source ↗

**Figure 4.** Figure 4: Distributed color-code architecture and syndrome-measurement schedules. (a) A distributed QEC setting with a central scheduler coordinating four QPUs connected by photonic links. Bulk and seam checks arise from locality: bulk checks are local to a single QPU, whereas seam checks cross QPU boundaries. (b) Syndrome-measurement schedules used in this work. BC = bulk check; SC = seam check. the scheduling po… view at source ↗

**Figure 5.** Figure 5: LER as a function of EGR for fixed seam-refresh intervals view at source ↗

**Figure 6.** Figure 6: Balanced four-QPU partition of a d = 9 triangular color code. Colored dots denote data qubits assigned to each QPU (QPU-0 blue, QPU-1 red, QPU-2 green, QPU-3 purple). The dark inner triangle indicates the central subpatch of distance d ′ = 5 mapped to QPU-0, while the three outer wedge-shaped regions are mapped to other QPUs. 5.1 Noise Model We adopt a circuit-level Pauli error model [38] that includes wai… view at source ↗

**Figure 7.** Figure 7: Logical error rate (LER) versus physical error rate for triangular color codes at view at source ↗

**Figure 8.** Figure 8: Logical error rate (LER) versus entanglement generation rate (EGR) at fixed view at source ↗

**Figure 9.** Figure 9: Logical error rate (LER) versus entanglement generation rate (EGR) at fixed view at source ↗

read the original abstract

Future quantum architectures are expected to be modular, with quantum processors connecting multiple quantum processing units (QPUs) via photonic interconnects. In topological quantum error correction, such as color codes, this creates seam boundaries where parity checks require remote CNOT operations using heralded Bell pairs. These non-local checks are slower and noisier than bulk local checks because entanglement generation is probabilistic, causing data qubits to accumulate idle noise while waiting for remote operations. A natural way to reduce this overhead is to skip some seam measurements; however, doing so makes seam syndrome information stale and can degrade decoding. The central scheduling problem is therefore to determine how frequently seam checks should be measured so as to balance remote-operation and waiting noise against syndrome staleness. To address this trade-off, we develop a scheduling module that integrates directly into standard syndrome-extraction circuits. We consider two policies: Skip-Seam-$\tau$ (SS-$\tau$), which measures all bulk checks every round while measuring seam checks once every $\tau$ rounds and copying the most recent syndrome in skipped rounds, and Adaptive Skip-$\tau$ (AST), which selects $\tau$ as a function of code distance and entanglement generation rate (EGR). We evaluate these policies on triangular color codes under circuit-level noise in Stim, including idling errors induced by Bell-pair generation delays. Our simulations show that SS-tau and AST reduce remote-operation overhead and can lower the logical error rate (LER) relative to the Measure-All (MA) baseline. For physical error rate $p = 10^{-3}$, we identify an EGR regime in which both SS-$\tau$ and AST exhibit behavior consistent with fault-tolerant scaling, with LER decreasing as code distance increases. Across these regimes, SS-$\tau$ and AST outperform MA.

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.

Desk Editor's Note private letter to a colleague

The paper introduces two concrete scheduling policies for seam checks in distributed color codes that cut remote overhead in Stim simulations and show LER improvement plus distance scaling in a narrow EGR window at p=10^{-3}.

read the letter

The core contribution is the SS-τ policy, which measures seam checks only every τ rounds while copying the prior syndrome, plus the AST variant that picks τ from code distance and entanglement generation rate. These plug straight into standard syndrome-extraction circuits for triangular color codes and are tested under circuit-level depolarizing noise that includes deterministic idle errors from Bell-pair delays. In the reported runs, both policies lower logical error rate relative to the always-measure baseline and recover the expected drop in LER with increasing distance inside a particular EGR range. That is useful engineering detail for anyone trying to scale modular architectures with photonic links. The work is new in the specific combination of skip interval, adaptive rule, and integration with idle-error modeling; it is not just a rehash of earlier scheduling papers. The simulations are run in Stim, which is reproducible, and the policies are defined explicitly without circular fitting. The main soft spot is the noise model itself. Fixed-delay idling plus depolarizing errors may miss variable heralding failures, photon-loss erasures, or seam-correlated noise that real hardware will introduce, and those omissions could erase the reported advantage. The abstract also gives no error bars, exact run counts, or data-exclusion rules, so the statistical reliability of the scaling claim is still preliminary. This paper is aimed at researchers working on distributed fault tolerance and modular QEC implementations. A reader who needs a ready-to-insert scheduling module and simulation baselines will find it directly usable. It is coherent on its own terms and addresses a genuine scaling issue, so it deserves a serious referee even though the hardware-model gap will need attention in revision.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes two boundary-aware scheduling policies for stabilizer measurements in distributed color-code quantum error correction: Skip-Seam-τ (SS-τ), which measures seam checks every τ rounds while copying prior syndromes, and Adaptive Skip-τ (AST), which selects τ based on code distance and entanglement generation rate (EGR). These are integrated into standard syndrome-extraction circuits for triangular color codes. Under Stim simulations with circuit-level depolarizing noise augmented by idling errors from Bell-pair delays, the policies are shown to reduce remote-operation overhead and logical error rate (LER) relative to the Measure-All (MA) baseline. At physical error rate p=10^{-3}, an EGR regime is identified where both policies exhibit fault-tolerant scaling (LER decreasing with distance) and outperform MA.

Significance. If the simulation results hold, the work provides concrete, implementable scheduling strategies that could improve the practicality of modular quantum architectures connected by photonic interconnects. Credit is due for the direct embedding of scheduling into existing QEC circuits, the explicit trade-off analysis between remote overhead and syndrome staleness, and the use of Stim for circuit-level simulations that allow reproducible evaluation of the EGR window for scaling. The identification of regimes where skipping seam checks is beneficial is a useful contribution to distributed QEC design.

major comments (2)

[§4.2] §4.2 and associated figures: the LER curves demonstrating outperformance of SS-τ and AST over MA, and the distance-dependent decrease at p=10^{-3}, are reported without error bars, Monte Carlo shot counts, or statistical tests; this undermines confidence in the claimed fault-tolerant scaling and relative gains, as small differences could be consistent with sampling noise.
[§3.1] §3.1, noise model description: the circuit-level model includes deterministic idling proportional to entanglement generation time but does not incorporate or sensitivity-test against probabilistic heralding failures or photon-loss erasures typical of photonic links; if these channels dominate, the identified EGR regime for scaling and the LER advantage may not persist, making the central empirical claims model-dependent without further justification.

minor comments (2)

[Abstract] Abstract: the phrase 'behavior consistent with fault-tolerant scaling' is used without a quantitative definition (e.g., LER scaling exponent or threshold crossing criterion), which should be stated explicitly for clarity.
[§2.3] §2.3: the definition of the adaptive rule in AST could include an explicit pseudocode or equation showing how τ is computed from distance and EGR to improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and indicate the revisions we will make to strengthen the work.

read point-by-point responses

Referee: [§4.2] §4.2 and associated figures: the LER curves demonstrating outperformance of SS-τ and AST over MA, and the distance-dependent decrease at p=10^{-3}, are reported without error bars, Monte Carlo shot counts, or statistical tests; this undermines confidence in the claimed fault-tolerant scaling and relative gains, as small differences could be consistent with sampling noise.

Authors: We agree that the lack of error bars, shot counts, and statistical tests in §4.2 reduces confidence in the reported LER differences and scaling claims. In the revised manuscript we will explicitly state the number of Monte Carlo shots used for each data point, add error bars (standard error of the mean) to all LER curves, and include a short discussion of statistical significance for the observed outperformance and distance-dependent trends. revision: yes
Referee: [§3.1] §3.1, noise model description: the circuit-level model includes deterministic idling proportional to entanglement generation time but does not incorporate or sensitivity-test against probabilistic heralding failures or photon-loss erasures typical of photonic links; if these channels dominate, the identified EGR regime for scaling and the LER advantage may not persist, making the central empirical claims model-dependent without further justification.

Authors: Our model in §3.1 focuses on idling noise from entanglement-generation delays under the assumption of successful heralding (with failed attempts retried), which is the dominant overhead in the photonic-link setting we consider. We will revise the noise-model section to state this assumption explicitly and add a qualitative discussion of how photon-loss erasures or heralding failures could shift the EGR window. A full quantitative sensitivity analysis incorporating erasure channels would require new simulations outside the current Stim framework and is planned for future work. revision: partial

Circularity Check

0 steps flagged

No load-bearing circularity; explicit policy definitions evaluated via independent Stim simulations

full rationale

The paper defines SS-τ and AST scheduling policies directly (measure seam checks every τ rounds or adaptively) and evaluates them by forward circuit-level simulation in Stim under a fixed noise model that includes idling from Bell-pair delays. The reported LER reductions and fault-tolerant scaling behavior are outputs of these simulations compared to the Measure-All baseline; no equations, fitted parameters, or self-citations reduce the performance claims to quantities constructed from the same data. The derivation chain consists of policy specification followed by independent Monte-Carlo evaluation and is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The work rests on standard circuit-level depolarizing noise assumptions and the Stim noise model for idling during probabilistic entanglement generation; no new free parameters are introduced beyond the tunable τ and the EGR value, which are simulation inputs rather than fitted constants.

free parameters (1)

τ (skip interval)
Chosen as a function of code distance and EGR in AST; fixed in SS-τ; controls the trade-off but is not fitted to the target LER.

axioms (1)

domain assumption Circuit-level noise model with idling errors during Bell-pair waits accurately represents distributed hardware
Invoked when including idle noise in the Stim simulations described in the abstract.

pith-pipeline@v0.9.0 · 5644 in / 1325 out tokens · 26523 ms · 2026-05-08T12:08:01.286744+00:00 · methodology

discussion (0)

Reference graph

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