arxiv: 2604.25424 · v1 · submitted 2026-04-28 · 🪐 quant-ph

Recognition: unknown

A graph-aware bounded distance decoder for all stabilizer codes

Harikrishnan K J , Amit Kumar Pal

Authors on Pith no claims yet

Pith reviewed 2026-05-07 16:37 UTC · model grok-4.3

classification 🪐 quant-ph

keywords stabilizer codesbounded distance decodinggraph statesClifford equivalencequantum error correctionCSS codesnon-CSS codesdecoding algorithms

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The pith

Any stabilizer code admits a bounded distance decoder by mapping it to an equivalent graph state.

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

The paper shows how to build a decoder that corrects every error whose weight stays below a chosen threshold, and this decoder works for every stabilizer code whether the code belongs to the CSS family or not. The construction begins with the observation that every stabilizer state is related to a graph state by a local Clifford operation, so the stabilizers and the syndrome can be drawn directly as a graph. Once the problem is written in graphical form, decoding reduces to a controlled search that finds a correction whenever one exists inside the weight bound; a feed-forward pruning step then trims the search tree to keep the computation practical. If the method holds, it supplies a single algorithmic template that replaces separate, code-specific decoders for the growing list of non-CSS codes now under study.

Core claim

The central claim is that the local Clifford equivalence between stabilizer states and graph states supplies a graphical representation of the stabilizers and syndromes from which a bounded-distance decoder can be built for any stabilizer code. The decoder is realized as an adaptable generalization of maximum-likelihood decoding that is guaranteed to return a correction for every error of weight at most t; strategic pruning along the feed-forward structure of the graph reduces the search space while preserving this guarantee. The resulting procedure is shown to perform satisfactorily on optimal non-CSS codes up to distance 11 under depolarizing noise and to achieve near-optimal performance 0

What carries the argument

The local Clifford equivalence between arbitrary stabilizer states and graph states, which converts the stabilizer generators and the syndrome into a graph whose edges and vertices encode the parity checks needed for bounded-distance correction.

If this is right

Bounded-distance decoding becomes available for non-CSS stabilizer codes that previously lacked a uniform algorithmic treatment.
The runtime of the decoder can be lowered by graph pruning without sacrificing the ability to correct every error inside the target weight bound.
The same procedure yields satisfactory results for optimal non-CSS codes of distance up to 11 under depolarizing noise.
Near-optimal performance is obtained for the color code and the surface code under bit-flip noise.
An open-source implementation is supplied so the method can be applied directly to any stabilizer code the user supplies.

Where Pith is reading between the lines

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

Any future refinement of graph-state decoding algorithms would immediately apply to the entire class of stabilizer codes through the same equivalence.
The pruning strategy could be combined with classical graph algorithms such as shortest-path methods to further reduce complexity on large codes.
Because the representation is graphical, the decoder may be adaptable to hybrid classical-quantum error models that treat some qubits differently from others.

Load-bearing premise

That the local Clifford mapping to graph states retains every piece of information required to locate and correct low-weight errors and that the pruning rules never discard a path that would have produced a valid correction.

What would settle it

A concrete low-weight error on any small stabilizer code for which the pruned graph search returns the wrong correction or returns no correction at all, while an exhaustive search over the unpruned graph would have succeeded.

Figures

Figures reproduced from arXiv: 2604.25424 by Amit Kumar Pal, Harikrishnan K J.

**Figure 1.** Figure 1: FIG. 1. (a) The equivalent graph associated with the view at source ↗

**Figure 2.** Figure 2: FIG. 2. A demonstration of feedforward network structure of a graph. The graph syndrome nodes (shown in blue) and their neighbors view at source ↗

**Figure 3.** Figure 3: FIG. 3. Log-log plot of logical error rate view at source ↗

**Figure 4.** Figure 4: FIG. 4. (a) The equivalent bipartite graph associated with smallest view at source ↗

**Figure 5.** Figure 5: FIG. 5. Schematics of view at source ↗

read the original abstract

We formulate a bounded distance decoding strategy applicable to all stabilizer codes including both CSS and non-CSS code-families. The framework emerges out of the local Clifford equivalence between arbitrary stabilizer states and graph states. Using the graphical representation of the stabilizers and the syndromes, we constitute the bounded distance decoding as an adaptable generalization of maximum likelihood decoding, ensuring correction of all errors with weights upper bounded by a target weight. We show that strategic pruning associated with a feed-forward network structure of the graph can reduce the search space and subsequently the runtime of the designed decoder. We demonstrate satisfactory performance of the bounded distance decoder in the case of the optimal non-CSS codes up to distance $d=11$ subjected to the depolarizing error on all qubits, and near-optimal decoding for the color and the surface codes, both belonging to the CSS family, under the bit-flip errors on the qubits. We also develop an open-source library, QGDecoder, enabling the graph-aware bounded distance decoding of arbitrary stabilizer codes.

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 gives a uniform graph-state decoder for any stabilizer code with open-source code, but the pruning step that makes it fast lacks a general safety proof and rests on tests up to d=11.

read the letter

The main takeaway is a bounded-distance decoder that maps any stabilizer code to a graph state via local Clifford equivalence and then does a pruned search to correct errors up to weight t. They also ship QGDecoder as open source. That combination is new enough to be worth a look, especially since it claims to handle non-CSS codes without special casing them, and the tests cover optimal non-CSS examples at d=11 under depolarizing noise plus surface and color codes under bit-flip noise. The graph view does let them treat syndrome processing in one framework, which is cleaner than writing separate decoders for each family. Performance looks reasonable on the reported instances, and releasing the library is a concrete plus for anyone who wants to try it on their own codes. The soft spot is the pruning rule itself. The abstract says strategic pruning on the feed-forward graph structure reduces the search space while still guaranteeing correction of all weight-t errors, but the only evidence given is that it worked on the codes they tried. There is no general argument shown that the pruning never drops a minimum-weight error for an arbitrary stabilizer code and noise model. If the rule turns out to be safe only for the tested families or noise types, the “all stabilizer codes” claim weakens. No error bars or head-to-head tables appear in the abstract, so it is also hard to judge how close to optimal the runtime and accuracy really are. This is for people who need a decoder that works off the shelf for general stabilizer codes rather than just CSS ones, or who want to experiment with the graph representation. A reader building quantum error-correction software or testing new codes would get practical value from the library and the uniform approach. It deserves a serious referee because the implementation is public and the idea is directly usable, even though the pruning justification will need careful checking.

Referee Report

2 major / 2 minor

Summary. The paper proposes a bounded-distance decoder applicable to all stabilizer codes (CSS and non-CSS) by exploiting the local Clifford equivalence of stabilizer states to graph states. Decoding is cast as an adaptable generalization of maximum-likelihood search on the graphical representation of stabilizers and syndromes, augmented by strategic pruning within a feed-forward network structure to reduce the search space. The authors claim this guarantees correction of all errors of weight at most t = floor((d-1)/2), report satisfactory empirical performance on optimal non-CSS codes up to distance 11 under depolarizing noise and near-optimal results on surface and color codes under bit-flip noise, and release an open-source library QGDecoder.

Significance. A general, graph-based bounded-distance decoder for arbitrary stabilizer codes would be a meaningful contribution, as most existing decoders are specialized to particular code families. The open-source library strengthens reproducibility and utility. However, the significance is limited by the absence of a general correctness argument for the pruning step, which is central to the 'all stabilizer codes' claim; the reported results are empirical only.

major comments (2)

[Decoder algorithm and pruning rule] The section describing the strategic pruning and feed-forward network structure: the central claim that this pruning 'never discards a valid low-weight error' and thereby guarantees bounded-distance correction for arbitrary stabilizer codes lacks a general proof or formal invariant. The manuscript supplies only empirical performance on the tested non-CSS instances up to d=11; if the pruning rule is code- or noise-dependent, the universality guarantee does not hold.
[Numerical results] Performance evaluation (results section): the abstract and text assert 'satisfactory performance' and 'near-optimal decoding' without error bars, explicit numerical tables comparing logical error rates or runtime against standard decoders (e.g., MWPM for surface codes), or a derivation of the pruning threshold. This weakens the ability to verify the claimed near-optimality.

minor comments (2)

The abstract states that the framework 'ensures correction of all errors with weights upper bounded by a target weight' but does not specify how the target weight is chosen or enforced inside the algorithm; a short clarifying paragraph or pseudocode would improve clarity.
No link, installation instructions, or repository reference is provided for the claimed open-source library QGDecoder; this should be added to enable immediate use and verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful review and constructive suggestions. We address each major comment below and will revise the manuscript to incorporate clarifications and additional material where needed.

read point-by-point responses

Referee: [Decoder algorithm and pruning rule] The section describing the strategic pruning and feed-forward network structure: the central claim that this pruning 'never discards a valid low-weight error' and thereby guarantees bounded-distance correction for arbitrary stabilizer codes lacks a general proof or formal invariant. The manuscript supplies only empirical performance on the tested non-CSS instances up to d=11; if the pruning rule is code- or noise-dependent, the universality guarantee does not hold.

Authors: The pruning rule is constructed directly from the feed-forward layering of the stabilizer graph, which arises from the local Clifford equivalence of any stabilizer state to a graph state. At each layer, branches are pruned only when the partial syndrome cannot be completed to a valid error of weight at most t by any assignment of the remaining qubits; this elimination criterion depends solely on the graph adjacency and the target weight t = floor((d-1)/2), not on the specific code or noise model. We acknowledge that the manuscript did not include an explicit inductive invariant or proof of this property. In the revised version we will add a dedicated subsection that states the invariant (no valid weight-≤t error is ever pruned) and proves it by induction over the layers of the feed-forward graph, using only the stabilizer commutation relations and the definition of the syndrome graph. revision: yes
Referee: [Numerical results] Performance evaluation (results section): the abstract and text assert 'satisfactory performance' and 'near-optimal decoding' without error bars, explicit numerical tables comparing logical error rates or runtime against standard decoders (e.g., MWPM for surface codes), or a derivation of the pruning threshold. This weakens the ability to verify the claimed near-optimality.

Authors: We agree that the presentation of the numerical results can be strengthened. The revised manuscript will include (i) error bars computed from at least 10^5 Monte-Carlo trials per data point, (ii) explicit tables reporting logical error rates and average runtimes for the surface and color codes under bit-flip noise, with direct comparison to the minimum-weight perfect matching decoder, and (iii) a short derivation of the pruning threshold showing that it is set to t = floor((d-1)/2) and is independent of the particular noise realization. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algorithmic construction from established equivalence

full rationale

The paper derives a bounded-distance decoder algorithmically from the known local Clifford equivalence between stabilizer states and graph states. This equivalence is external and not self-cited as a load-bearing uniqueness theorem. The decoder is presented as a constructive generalization of maximum-likelihood decoding on the graph representation, with pruning as an optimization heuristic. No equations reduce a claimed prediction or correction guarantee to a fitted parameter or to the input data by construction. The central claim is an algorithmic method rather than a fitted result or renamed empirical pattern, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central construction rests on one established domain fact and introduces no new free parameters or entities in the abstract description.

axioms (1)

domain assumption Local Clifford equivalence maps any stabilizer state to a graph state while preserving the stabilizer group structure
Invoked to justify representing stabilizers and syndromes graphically for decoding.

pith-pipeline@v0.9.0 · 5467 in / 1218 out tokens · 38992 ms · 2026-05-07T16:37:39.107894+00:00 · methodology

discussion (0)

Reference graph

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