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arxiv: 2501.12072 · v3 · submitted 2025-01-21 · 🪐 quant-ph · cs.IT· math.IT

Fault-tolerant syndrome extraction in [[n,1,3]] non-CSS code family generated using measurements on graph states

Harsh Gupta , Mainak Bhattacharyya , Ritik Jain , Ankur Raina This is my paper

Pith reviewed 2026-05-23 04:55 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords non-CSS quantum codesfault-tolerant syndrome extractionbare ancillagraph stateshook errorsquantum error correction[[n,1,3]] codesdepolarizing noise
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The pith

Non-CSS quantum codes from graph states achieve higher rates for distance three while staying fault tolerant to hook errors via bare-ancilla extraction.

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

The paper builds a family of non-CSS quantum error-correcting codes with parameters [[n,1,3]] using measurements performed on graph states. These codes support fault-tolerant syndrome extraction with a single bare ancilla qubit, protecting against hook errors that would otherwise reduce the effective distance. Simulations with a lookup-table decoder under anisotropic and circuit-level depolarizing noise map the performance trade-off against code rate and identify several optimized members of the family. One member of the family improves the code rate relative to the bare-ancilla construction examined by Li et al. while keeping the same distance.

Core claim

A systematic protocol using graph-state measurements produces a family of [[n,1,3]] non-CSS stabilizer codes whose stabilizer structure preserves the fault-tolerant properties of bare-ancilla syndrome extraction against hook errors, as verified by numerical simulation under depolarizing noise models.

What carries the argument

Bare-ancilla syndrome extraction applied to non-CSS codes generated from graph-state measurements; the single-ancilla circuit detects errors while avoiding hook-error propagation that would drop the distance.

If this is right

  • The family admits direct numerical comparison of logical error rates under both anisotropic and circuit-level depolarizing noise.
  • Performance varies with code rate, allowing selection of an optimized member for each noise model.
  • One code in the family has a higher rate than the bare-ancilla example of Li et al. while retaining distance three.
  • Benchmarking against the flag-qubit construction of Chao et al. shows the bare-ancilla approach remains competitive for these codes.

Where Pith is reading between the lines

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

  • The graph-state generation method could be applied to other distance or rate targets to produce additional fault-tolerant non-CSS families.
  • Higher-rate members of the family could lower the physical-qubit overhead needed to store one logical qubit in hardware experiments.
  • Hardware implementation of the extraction circuit on one of the new codes would test whether the simulated fault tolerance holds under realistic device noise.

Load-bearing premise

The stabilizer generators obtained from the graph-state construction allow each bare-ancilla check to catch hook errors without letting any undetected error of weight two or less produce a logical failure.

What would settle it

Circuit-level Monte Carlo simulation of the full extraction and lookup-table decoding on a proposed code that measures whether the logical error rate remains suppressed at the level expected for distance three or falls due to undetected hook errors.

Figures

Figures reproduced from arXiv: 2501.12072 by Ankur Raina, Harsh Gupta, Mainak Bhattacharyya, Ritik Jain.

Figure 2
Figure 2. Figure 2: FIG. 2. The message qubits ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. A Cluster state, which is also depicted as an undi [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Due to [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Comparision of Pseudo-threshold results for [[ [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Pseudo-threshold results for [[ [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Six qubit graph state with one message qubit(orange) [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
read the original abstract

The reliability of quantum computation critically depends on the performance of quantum error-correcting codes (QECCs). Performance of QECCs can be severely degraded by hook errors, which effectively reduce the code distance. In this work, we construct a family of $[[n,1,3]]$ non-CSS QECCs, which are fault-tolerant (FT) against noisy syndrome measurements. We employ the bare-ancilla method of Muyuan Li \emph{et al.} to demonstrate fault tolerance against hook errors during syndrome extraction. We present a systematic protocol for generating these QECCs using graph codes and propose a family of $[[n,1,3]]$ codes that preserve the fault-tolerant properties of the bare ancilla codes. We use a custom lookup-table decoder and simulate the code's performance under both anisotropic and circuit-level depolarizing noise. Our results reveal a trade-off in performance with respect to the code rate and identify optimized codes under these noise models. We benchmark our results against the flag-qubit method of Chao \emph{et al}. Notably, we report a new bare ancilla code with improved code rate while maintaining the same distance compared to the bare code used in the work of Muyuan Li \emph{et al.}

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

2 major / 2 minor

Summary. The manuscript constructs a family of [[n,1,3]] non-CSS quantum error-correcting codes via measurements on graph states. It invokes the bare-ancilla syndrome extraction protocol of Li et al. to claim fault tolerance against hook errors, presents a systematic generation protocol, performs simulations under anisotropic and circuit-level depolarizing noise with a custom lookup-table decoder, benchmarks against the flag-qubit method of Chao et al., and identifies members with improved code rate relative to the Li et al. example while preserving distance 3.

Significance. If the fault-tolerance claim holds, the work supplies a graph-state route to higher-rate non-CSS codes that support bare-ancilla syndrome extraction. The simulations under two distinct noise models and the explicit rate comparison constitute concrete, reproducible data points that could inform code selection for near-term devices.

major comments (2)
  1. [Abstract and graph-state construction section] Abstract and the section describing the graph-state protocol: the central claim that the generated non-CSS stabilizers 'preserve the fault-tolerant properties of the bare ancilla codes' against hook errors is asserted by invoking the Li et al. construction as a black box, but no explicit check (enumeration of weight-2 hook-error patterns on the measurement graph or verification that every such error remains detectable by the new stabilizers) is supplied once the stabilizers cease to be CSS. This verification is load-bearing for the distance-3 guarantee.
  2. [Construction and code family definition] The claim that the family consists of [[n,1,3]] codes (abstract and results) lacks a derivation or explicit check that the graph-state measurement protocol produces stabilizers whose minimum weight is 3; the distance is stated rather than demonstrated for the new non-CSS family.
minor comments (2)
  1. [Abstract and simulation results] Simulations are described but the abstract and results section supply no error bars, trial counts, or confidence intervals, which would strengthen the reported performance trade-offs.
  2. [Protocol description] Notation for the graph-state vertices and the mapping to stabilizer generators could be clarified with an explicit example for the smallest n>7 member of the family.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the two major comments point by point below, agreeing that explicit verifications would strengthen the claims, and we will incorporate the necessary additions in the revised version.

read point-by-point responses

  1. Referee: [Abstract and graph-state construction section] Abstract and the section describing the graph-state protocol: the central claim that the generated non-CSS stabilizers 'preserve the fault-tolerant properties of the bare ancilla codes' against hook errors is asserted by invoking the Li et al. construction as a black box, but no explicit check (enumeration of weight-2 hook-error patterns on the measurement graph or verification that every such error remains detectable by the new stabilizers) is supplied once the stabilizers cease to be CSS. This verification is load-bearing for the distance-3 guarantee.

    Authors: We agree that the manuscript would benefit from an explicit verification rather than relying solely on the invocation of Li et al. Although the graph-state protocol is constructed to extend the bare-ancilla syndrome extraction while maintaining the relevant error-detection properties, we acknowledge that a direct check for the non-CSS stabilizers is warranted. In the revised manuscript we will add an explicit enumeration of weight-2 hook-error patterns on the measurement graph together with verification that each such error remains detectable by the new stabilizers. revision: yes

  2. Referee: [Construction and code family definition] The claim that the family consists of [[n,1,3]] codes (abstract and results) lacks a derivation or explicit check that the graph-state measurement protocol produces stabilizers whose minimum weight is 3; the distance is stated rather than demonstrated for the new non-CSS family.

    Authors: The referee is correct that the distance-3 property is stated on the basis of the construction without an accompanying derivation or explicit check for the non-CSS family. We will revise the manuscript to include a general argument, based on the structure of the graph-state measurements, showing that the resulting stabilizers have minimum weight 3, supplemented by explicit verification for representative codes in the family. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation relies on external citation.

full rationale

The paper constructs a new family of [[n,1,3]] non-CSS codes via graph-state measurements and applies the bare-ancilla syndrome extraction protocol from the external reference Li et al. (distinct authors) to assert preservation of fault tolerance against hook errors. No quoted step reduces a claimed prediction or uniqueness result to a self-definition, fitted input, or self-citation chain; the higher-rate example is presented as an explicit construction rather than forced by input definitions. The central claim therefore remains independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the domain assumption that the bare-ancilla syndrome extraction protocol remains fault-tolerant when applied to the newly generated non-CSS codes; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The bare-ancilla method of Li et al. provides fault tolerance against hook errors for the graph-state-derived non-CSS codes.
    Invoked to demonstrate FT properties without additional derivation in the abstract.

pith-pipeline@v0.9.0 · 5772 in / 1253 out tokens · 31562 ms · 2026-05-23T04:55:01.024993+00:00 · methodology

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