Information locality of a quantum locally recoverable code

Ryutaroh Matsumoto

REVIEW 2 major objections 5 minor 41 references

Information locality of a quantum locally recoverable code

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper introduces a quantum information locality parameter for stabilizer codes, proves a linear-algebraic routine that finds the smallest repair group and fewest measured observables, and demonstrates on explicit codes that the older…

desk verdict A well-built paper that introduces quantum information locality, proves a useful repair-group-shrinking algorithm, and honestly maps its limits; it deserves refereeing and likely acceptance after minor revisions. read the letter →

arxiv 2608.04403 v1 pith:SM7UN5OF submitted 2026-08-05 quant-ph cs.ITmath.IT

Ryutaroh Matsumoto This is my paper · ORCID

classification quant-phcs.ITmath.IT MSC 81P7394B6594B35

keywords quantumerrorcorrectionlocalrecoverylocallyrecoverablecodesinformationlocalitysymbolerasuresstabilizermeasurementreduction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper gives quantum local recovery a parameter that tells the truth about how many extra qudits are needed to repair several erasures. Classical codes already solved this problem with information locality, which counts the dimension of the code's projection onto a repair group rather than the repair group's raw size. The paper translates that idea to quantum stabilizer codes built from Hermitian dual-containing linear codes, defines quantum information locality, and provides an O(|J|^3) linear-algebra procedure that finds a repair group achieving the bound while minimizing the number of observables one must measure. An explicit [[16,2,3]]_2 code shows that the previously proposed quantum symbol locality overestimates the needed qudits (6 instead of 5). A second example shows why the same definition cannot be carried over to Euclidean CSS codes with different X and Z component codes: the dimension-based bound can fail outright.

What carries the argument

The load-bearing object is the repair-group projection identity, π_I(C) = π_I(σ_J($C^{{⊥_h}}$)), which the paper inherits from earlier work and uses as the exact criterion for whether erasures in I can be corrected using qudits only in J. The algorithm works by decomposing the shortened dual code σ_{J_j}($C^{{⊥_h}}$) into three summands, row-reducing the part supported off the erasure set, and eliminating its pivot columns from the remaining basis vectors; the support of the resulting rows defines the smaller repair group J. The same decomposition yields the direct-sum relation σ_{J_j}($C^{{⊥_h}}$) = σ_{J_j\I}($C^{{⊥_h}}$) ⊕ σ_J($C^{{⊥_h}}$), which is what lets the procedure read off a minimum-size set of measured observables.

What would settle it

For the explicit [[16,2,3]]_2 code of Section 4, enumerate all 7-symbol subsets J ⊂ {1,...,16} and compute w_H(π_J(C) \ σ_J($C^{{⊥_h}}$)); Proposition 10 asserts that every such J has value at most 2, so finding any 7-symbol repair group with quantum local distance at least 3 would refute the claimed separation between information and symbol locality.

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Extended reading notes

Core claim

The paper's central claim is that the classical information-locality idea, using the dimension of the punctured code rather than the repair-group size to measure recovery cost, does have a working quantum analogue within the Hermitian stabilizer construction. Definition 3 sets quantum information locality (r_Q,i, δ_Q) by requiring, for each index j, a repair group J_j with j in J_j, w_H(π_{J_j}(C) \ σ_{J_j}($C^{{⊥_h}}$)) ≥ δ_Q and dim π_{J_j}(C) ≤ r_Q,i. Proposition 4 then says that any I ⊆ J_j with |I| ≤ δ_Q − 1 can be corrected using at most r_Q,i qudits in addition to the erased ones. The supporting algorithm, Proposition 1, starts from any repair group satisfying the correctability identity π_I(C) = π_I(σ_J($C^{{⊥_h}}$)) and shrinks it to a subset J with |J| − |I| ≤ dim π_{J_j}(C) − dim σ_I($C^{{⊥_h}}$), in O(|J_j|^3) arithmetic operations, while also producing observables whose number is minimal by a theorem from the author's earlier work. The paper demonstrates the advantage on an explicit [[16,2,3]]_2 code with information locality (5,3) versus symbol locality (6,3), and it shows the obstacle for Euclidean CSS codes with C_X ≠ C_Z by an explicit length-6 failure of the corresponding size bound.

Load-bearing premise

The entire framework inherits the earlier if-and-only-if criterion, stated as Eq. (3), that erasures in I can be repaired inside J exactly when the projection of the code onto I equals the projection of the shortened dual onto I, and the minimal-observable claim additionally depends on a theorem from the author's prior work that is cited but not restated here; if either gives way, the size bound, the algorithm, and the new locality parameter all collapse.

Editorial extensions

If this is right

For any Hermitian-constructed stabilizer code with quantum information locality (r_Q,i, δ_Q), erasure sets of size at most δ_Q − 1 are repairable using at most r_Q,i qudits in addition to the erased ones.
The Proposition 1 algorithm produces, in O(|J_j|^3) arithmetic operations, a repair group J ⊆ J_j that still satisfies the correctability identity and meets the dimension-based size bound.
The same computation yields a set of measured observables whose size is the minimum possible for the punctured code to correct the erasures, not merely a convenient set.
The explicit [[16,2,3]]_2 stabilizer code has quantum information locality (5,3) and quantum symbol locality (6,3), so the older definition overestimates the number of extra qudits needed for erasure correction.
For pure Hermitian-constructed stabilizer codes, the quantum Singleton-like bound applies to information locality, via the classical bound and the paper's Proposition 6.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: the same 'shrink the repair group by row-reducing the shortened dual' trick could be used as a general measurement-reduction preprocessing step in any stabilizer erasure-correction routine, independent of locality, since the direct-sum relation is a structural property of the stabilizer.
Editorial extension: the paper fixes the erasure set I before shrinking J; a natural next test is whether choosing J with I unknown (worst-case or random erasures) still yields near-minimal repair groups on average.
Editorial extension: the Section 5 obstacle suggests a possible definition for asymmetric CSS codes using max(dim π_{J_j}(C_X), dim π_{J_j}(C_Z)) or a min-type formulation, which the paper does not explore.
Editorial extension: because r_C,i = r_Q,i when C contains its Hermitian dual, known classical information-locality constructions that are dual-containing automatically yield quantum codes with the same information locality; the paper notes the classical-to-quantum direction but does not build new code families from it.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know three things. First, this paper defines quantum information locality for Hermitian stabilizer codes, which is genuinely new: the classical notion had no quantum counterpart before. Second, it gives a concrete O(|J|^3) linear-algebra algorithm that finds a smaller repair group than the given one, with a size bound in terms of punctured-code dimension, and that simultaneously minimizes the number of measured observables. Third, it shows by explicit example that the earlier symbol-locality definition overestimates the needed qudits, and by a second example that a naive translation of information locality to CSS codes with C_X ≠ C_Z fails. The paper is honest about this limitation and does not oversell the scope.

The math holds up. Proposition 1's proof is sound; the pivot-column elimination argument is clear and the size bound follows by dimension counting. The Section 4 example is worked out carefully, and the Section 5 counterexample is explicit and convincing. I checked the construction rather than assuming it, and I did not find a hidden gap. The C program supplied for brute-force verification is a plus, and the author discloses AI assistance in finding examples, which is fine.

Soft spots are minor but worth naming. The whole local-recovery framework rests on the Galindo et al. iff criterion (Eq. 3); the paper's results live or die with that external result. That is a published criterion with no reason for suspicion, but it is load-bearing. The claim that the computed observables are minimum possible depends on the author's prior Theorem 5, cited but not restated; a referee will want that theorem stated or the dependence made explicit. Propositions 5 and 6 have omitted proofs, but they are straightforward analogs of [14]. Finally, Proposition 7 states information locality (5,3) as exact, while the written proof gives an upper bound and proves only the symbol-locality lower bound; exactness relies on the brute-force program. That is a small gap between proof text and statement.

This is a paper for the quantum LRC community, and for anyone doing erasure correction with measurement reduction in mind. It deserves a serious referee and likely acceptance after the minor issues are cleaned up. I would cite it in my own work on quantum local recovery, and I would bring it to a reading group focused on quantum LRCs. My recommendation: send it out, ask for the C program to be inspected or the lower-bound argument expanded, and accept after minor revision.

Referee Report

2 major / 5 minor

Summary. The paper introduces a quantum analogue of classical information locality for stabilizer codes constructed from Hermitian dual-containing codes. The central object is Definition 3, which replaces the repair-group size bound of the quantum symbol locality of Galindo et al. by a bound on dim π_{J_j}(C). Proposition 1 gives an O(|J_j|^3) linear-algebraic algorithm that, given an erasure set I and a repair group J_j, constructs a smaller repair group J satisfying the correction criterion (3) and simultaneously provides a basis of observables for local recovery; the size bound (7) and the direct-sum decomposition (11) are the key technical results. Proposition 4 translates the locality parameter into an operational statement about the number of additional qudits needed. The paper then gives two examples: a [[16,2,3]]_2 Hermitian stabilizer code claimed to have symbol locality (6,3) and information locality (5,3), and a CSS/Euclidean example showing that the averaged-dimension bound (17) can fail when C_X ≠ C_Z. The Section 4 example is supported by a brute-force C program included with the arXiv submission.

Significance. If correct, the paper fills a clear gap in the quantum LRC literature: classical information locality was known to estimate repair cost more accurately than symbol locality, and the quantum analogue proposed here does the same for Hermitian stabilizer codes. The algorithmic content of Proposition 1 is a genuine contribution: it not only finds a smaller repair group but also minimizes the number of measured observables, and the proof is a detailed constructive linear-algebra argument. The paper is also honest about the limits of the approach, showing by an explicit CSS example why a natural Euclidean translation fails. The inclusion of a brute-force verifier for the Section 4 claims is a strength, though the manuscript should make the role of that verifier explicit in the proofs. The main caveats are the reliance on the Galindo et al. criterion (3) and on the author's prior Theorem 5 for observable minimality; both are published results and are used without restatement.

major comments (2)

[Section 4, Proposition 10] Proposition 10 aims to rule out symbol locality (5,3), but its proof only considers repair groups J with |J| = 7. The definition of symbol locality in Section 3.1 requires only |J| ≤ r + δ − 1, so a repair group of size at most 6 with local distance at least 3 would still witness (5,3). The sentence 'there must exist a repair group J ... such that |J| = 7' is therefore unjustified, and the case |J| < 7 is not analyzed in the text. Please complete the case analysis or state explicitly in the proof that all subsets with |J| < 7 are checked by the accompanying brute-force C program and indicate where the verification is reported.
[Section 4, Proposition 7] The proposition asserts the exact value 'information locality (5,3)', but the written proof establishes only the upper bound via the repair groups K and L. It does not prove that no repair group J (of any size) with local distance at least 3 has dim π_J(C) ≤ 4; this lower bound appears to rest entirely on the C program mentioned in the Data Availability statement. Please make this reliance explicit in the proof, or provide an analytic argument, and state precisely which claims are machine-verified.

minor comments (5)

[Section 3, Propositions 5 and 6] The proofs are omitted with the note that they are analogous to arguments in [14]; since Proposition 6 is used to derive the quantum Singleton-like bound (12), please include the proofs or precise lemma references so the derivation is checkable.
[Section 3, Remark 2] The claim that the computed observables attain the minimum possible number depends on Theorem 5 of [30], which is not stated in the manuscript; please include the statement of the theorem or a short self-contained argument.
[Section 2 and Section 3] The notation I ∋ j appears in Proposition 1 and Proposition 4 without definition; please define that j is an element of I and that I is the actual erasure set.
[Section 4, proof of Proposition 7] The sentence 'The absolute minimum weight is bounded solely by the H0 subcodes and we see w_H(C^{⊥h}) = 4' is terse; please expand the case analysis (for example, noting that any vector involving the row (v,v) has weight at least 6) to make the computation self-contained.
[Data Availability] Since the exactness claims in Section 4 are machine-verified, please archive the C program and its output with the final version and cite it in the proof of Proposition 7, so that the reliance on brute-force verification is transparent.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the repair-group bound and the information-locality guarantee are derived from linear algebra plus the cited Galindo et al. criterion, and the minimum-observable claim rests on a stated prior theorem rather than a fitted input.

full rationale

The main derivation chain is self-contained once the external criterion (3) from Galindo et al. [14] is accepted. Definition 3 defines quantum information locality via the puncture dimension dim π_J(C) and the local distance w_H(π_J(C) \ σ_J(C^⊥h)); Proposition 1 proves, by explicit row reduction and dimension counting, that a repair group J exists with |J|-|I| ≤ dim π_J(C) - dim σ_I(C^⊥h). Proposition 4 then uses this bound together with the standard erasure-correction threshold for Q(π_J(C)) to conclude that at most r_Q,i additional qudits suffice when |I| ≤ δ_Q - 1. This is a nontrivial theorem, not a re-statement of the definition. The minimal-observable claim in Remark 2 is imported from the author's prior work [30, Theorem 5], but [30] is a published, stated external theorem with a DOI; the present paper does not assume the conclusion of that theorem as part of its derivation. The Section 4 example is verified by explicit linear algebra and a brute-force C program, and the Section 5 counterexample is an explicit subset enumeration. A residual caveat is that the exact information-locality value (5,3) in Proposition 7 appears to rely on the supplied brute-force program for the lower bound, but that is a completeness concern, not circularity. Overall, no definition is fitted to a target prediction and no load-bearing step reduces to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the standard stabilizer-code correspondence, the [14] iff criterion for local erasure correction, the quantum distance bound for erasures, and the author's own [30] theorem on minimal observables. Those are external results, not derived here. No free parameters are fitted to data; the examples are explicit constructions checked by brute force.

assumptions (4)

domain assumption The Hermitian dual-containing correspondence C ⊋ C^⊥h yields a stabilizer code Q(C) with parameters [[n, 2 dim C − n, w_H(C ∖ C^⊥h)]].
Invoked in Section 2 and used throughout to connect linear codes to quantum codes.
domain assumption Erasures in I can be corrected using qudits only in J if and only if π_I(C) = π_I(σ_J(C^⊥h)) (Eq. (3)).
This is the theorem from ref [14] stated in Section 2; Proposition 1 and Definition 3 both depend on it as the operational definition of a repair group.
domain assumption A punctured stabilizer code Q(π_J(C)) can correct any erasure set of size less than w_H(π_J(C) ∖ σ_J(C^⊥h)).
Used in Proposition 4's proof to convert the local distance into a number of correctable erasures.
domain assumption Theorem 5 of [30] characterizes the minimum number of observables needed to identify erasures in I, and the computed basis attains that minimum.
Remark 2 delegates the minimality claim to the author's own prior paper [30], not re-proved here.

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Cite this review

Pith. "Pith review of Information locality of a quantum locally recoverable code." pith.science (2026). https://pith.science/paper/SM7UN5OF

@misc{pith2026260804403,
  author       = {Pith},
  title        = {Pith review of: Information locality of a quantum locally recoverable code},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SM7UN5OF}},
  note         = {Machine review of arXiv:2608.04403}
}

abstract

A classical linear code $C$ of length $n$ is said to have symbol locality $(r, \delta)$ if for any index $j$ there exists a repair group $J_j \subseteq \{1, \ldots, n\}$ with $j\in J_j$ and $|J_j| \leq r+\delta-1$ such that any $\delta-1$ or fewer erasures in $J_j$ can be corrected by using codeword symbols only in $J_j$. Later it turned out that this way of defining $r$ overestimates the number of necessary codeword symbols for multiple-erasure correction, and information locality was proposed to define $r$ as the dimension of the punctured code of $C$ onto $J_j$. Recently locality $(r,\delta)$ was proposed for quantum error-correcting codes by following the original definition of symbol locality $(r, \delta)$. We propose a quantum counterpart of the information locality for quantum stabilizer codes constructed by Hermitian orthogonality, and a linear algebraic procedure computing a smaller repair group predicted by the proposed information locality and simultaneously reducing the number of measured observables in decoding to its minimum possible value. Then we demonstrate that the previously proposed definition of quantum locality $(r,\delta)$ has the same drawback of overestimating the number of necessary codeword symbols for erasure correction by providing an explicit example of a quantum stabilizer code. Finally, we will give another example of a quantum stabilizer code constructed by Euclidean orthogonality and two different linear codes, with which a natural translation of the classical information locality into the quantum setting underestimates the number of necessary codeword symbols for erasure correction.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [14]

Galindo, F

C. Galindo, F. Hernando, H. Martín-Cruz, R. Matsumoto, Quantum (𝑟,𝛿)-locallyrecoverablecodes, FiniteFieldsandTheirApplications111 (2026)102785.doi:10.1016/j.ffa.2025.102785

work page arXiv 2026

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