arxiv: 2602.20635 · v2 · submitted 2026-02-24 · 💻 cs.IT · math.IT

Recognition: no theorem link

Insertion Correcting Capability for Quantum Deletion-Correcting Codes

Ken Nakamura , Takayuki Nozaki

Authors on Pith no claims yet

Pith reviewed 2026-05-15 20:12 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords quantum error-correcting codesdeletion-correcting codesinsertion errorsindel distanceerror spheresquantum codes

0 comments

The pith

Quantum codes that correct t deletions also correct any combination of insertions and deletions adding up to t errors when their error spheres are disjoint.

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

The paper establishes that any quantum code capable of correcting t deletion errors will also correct mixtures of insertions and deletions whose total count is t, provided the code is defined by disjoint error spheres around each valid state. This matches the classical coding-theory approach of separating spheres rather than using inner-product conditions. The authors introduce a quantum indel distance to express the insertion-deletion correcting power of such codes. A reader should care because the result lets existing deletion-correcting constructions serve for a broader class of errors without new design.

Core claim

Any quantum t-deletion-correcting code also corrects a total of t insertion and deletion errors when the error spheres of its code states remain disjoint. The quantum indel distance is defined to measure the minimum insertions plus deletions needed to map one state to another, and the paper shows that this distance determines the combined correcting capability for both error types.

What carries the argument

The quantum indel distance, which counts the fewest insertions and deletions required to change one quantum state into another.

If this is right

Existing quantum deletion-correcting codes can be applied directly to channels that produce both insertions and deletions.
Code construction efforts can concentrate on deletion correction alone and still obtain indel correction as a byproduct.
The indel distance supplies a single metric that ranks quantum codes by their combined insertion-deletion performance.
Bounds derived for deletion correction translate immediately to bounds on the total number of indel errors that can be corrected.

Where Pith is reading between the lines

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

The result may allow quantum code designers to ignore insertion errors during initial construction and still obtain robustness against mixed indel noise.
It raises the question of whether analogous extensions hold for other asymmetric error models such as deletions plus substitutions.
Practical implementations could test small codes on hardware by injecting controlled numbers of insertions and deletions to verify the predicted capability.

Load-bearing premise

A quantum code is defined exactly by the requirement that the error spheres around its code states do not overlap.

What would settle it

An explicit quantum code that corrects every t-deletion pattern yet fails on at least one pattern consisting of k insertions and t-k deletions for some k between 0 and t.

read the original abstract

This paper proves that any quantum t-deletion-correcting codes also correct a total of t insertion and deletion errors under a certain condition. Here, this condition is that a set of quantum states is defined as a quantum error-correcting code if the error spheres of its states are disjoint, as classical coding theory. In addition, this paper proposes the quantum indel distance and describes insertion and deletion errors correcting capability of quantum codes by this distance.

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 shows t-deletion quantum codes correct t mixed indels under a disjoint-sphere definition and introduces a quantum indel distance, but that definition may not match the Knill-Laflamme condition for length-changing errors.

read the letter

The main point is that any quantum code whose states have disjoint t-deletion spheres will also correct a total of t insertions and deletions combined, provided we accept the classical-style disjoint-sphere definition for quantum states. The paper also defines a quantum indel distance to capture this capability in one number. That is the core result, and it follows directly once the definition is granted. No fitting or post-selection is involved; it is a straightforward proof under an explicit condition. The argument re-uses the classical sphere-packing idea without adding new machinery, which keeps it clean but also limits how far it moves beyond existing classical indel work. The new naming and the explicit statement for mixed errors are the main additions. The paper does a reasonable job stating the condition up front and showing how the distance works for both pure deletions and mixed cases. The logic inside that framework is consistent. The soft spot is the definition itself. Standard quantum error correction requires the Knill-Laflamme condition on the inner products of error operators acting on the code subspace, including operators that change the length of the state. Disjoint spheres alone do not automatically enforce the required orthogonality or scalar structure when superpositions are involved or when insertions and deletions alter the Hilbert space dimension. The result therefore holds only inside the chosen definition, which may not cover the codes people actually use for quantum channels with indel noise. This paper is aimed at researchers already comfortable treating quantum codes via classical sphere packing. It will be useful to anyone trying to bridge deletion-only results to mixed indel channels, but readers wanting explicit constructions or verification against Knill-Laflamme will need to do extra work. I would send it to peer review. The claim is narrow and clearly conditional, so referees can focus on whether the definition is acceptable and whether the distance adds anything beyond re-labeling classical concepts.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that any quantum t-deletion-correcting code, defined via the condition that its states have disjoint t-deletion error spheres (as in classical coding theory), also corrects any combination of insertions and deletions whose total number is at most t. It further introduces the quantum indel distance as a metric to characterize the insertion-deletion correcting capability of such codes.

Significance. If the adopted disjoint-sphere definition is accepted as sufficient, the result directly transfers classical indel-correcting properties to the quantum setting without additional parameters, and the quantum indel distance supplies a new tool for code analysis. This could simplify constructions for quantum channels with length-changing errors, provided the definition aligns with standard quantum requirements.

major comments (1)

[Abstract and definition of quantum codes] The central claim rests on defining a quantum code solely by disjoint t-deletion spheres (condition stated in the abstract). This definition is load-bearing, yet the manuscript does not verify that such codes satisfy the Knill-Laflamme condition <c_i|E_a^† E_b|c_j> = δ_ij λ_ab for the combined set of insertion and deletion operators (including those that change Hilbert-space dimension). A concrete check or counter-example discussion is required in the proof section.

minor comments (2)

[Section introducing quantum indel distance] The quantum indel distance should be formally defined with an equation and compared to existing quantum distances (e.g., quantum Hamming distance) to clarify its contribution.
Add at least one explicit low-dimensional example of a code satisfying the disjoint-sphere condition to illustrate the claimed indel correction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. We address the major comment point by point below, clarifying our definition and strengthening the presentation as needed.

read point-by-point responses

Referee: [Abstract and definition of quantum codes] The central claim rests on defining a quantum code solely by disjoint t-deletion spheres (condition stated in the abstract). This definition is load-bearing, yet the manuscript does not verify that such codes satisfy the Knill-Laflamme condition <c_i|E_a^† E_b|c_j> = δ_ij λ_ab for the combined set of insertion and deletion operators (including those that change Hilbert-space dimension). A concrete check or counter-example discussion is required in the proof section.

Authors: We thank the referee for highlighting this point. Our definition of a quantum t-deletion-correcting code is intentionally based on the disjointness of t-deletion error spheres (as in classical coding theory), which is stated explicitly in the abstract and formalized in Section II. This condition guarantees that distinct code states remain perfectly distinguishable after any sequence of at most t deletions. Theorem 1 then shows that the same disjoint-sphere property extends to any combination of insertions and deletions whose total count is at most t, because each such indel error maps the original codeword into a unique enlarged sphere. For the Knill-Laflamme condition, we note that standard KL applies to fixed-dimension errors; when operators change the Hilbert-space dimension, the condition must be adapted to the relevant subspaces. The disjoint-sphere assumption directly implies that post-error states belonging to different codewords have orthogonal supports, which enforces the off-diagonal vanishing <c_i | E† F | c_j> = 0 (i ≠ j) for any pair of indel operators E, F of total weight ≤ t. We will add a dedicated subsection in the proof of Theorem 1 that explicitly verifies this adapted orthogonality condition for the combined insertion-deletion operators and briefly discusses the dimension-changing case. revision: yes

Circularity Check

0 steps flagged

No circularity; proof follows directly from adopted disjoint-sphere definition

full rationale

The paper adopts the classical definition of error-correcting codes via disjoint error spheres and proves that any code satisfying this for t deletions also handles t total indels. This is a direct logical consequence of the sphere-disjointness property under the stated definition, with no reduction of the central claim to a fitted parameter, self-citation chain, or ansatz smuggled from prior work. The quantum indel distance is introduced as a descriptive measure based on the same framework. The derivation remains self-contained against the external benchmark of the disjoint-spheres condition and does not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The result rests on importing the classical disjoint error-sphere definition into quantum coding without additional quantum-specific justification for why spheres remain disjoint under insertion errors.

axioms (1)

domain assumption A set of quantum states forms a quantum error-correcting code precisely when the error spheres of its codewords are disjoint.
Invoked directly in the abstract as the defining condition for the codes under study.

invented entities (1)

quantum indel distance no independent evidence
purpose: To characterize the insertion-deletion correcting capability of quantum codes.
Newly proposed metric; no independent evidence of its utility beyond the paper's own definitions is given in the abstract.

pith-pipeline@v0.9.0 · 5355 in / 1168 out tokens · 19506 ms · 2026-05-15T20:12:58.098963+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Decoding Algorithm to Composite Errors Consisting of Deletions and Insertions for Quantum Deletion-Correcting Codes Based on Quantum Reed-Solomon Codes
cs.IT 2026-05 unverdicted novelty 6.0

A decoding algorithm is provided for composite deletion-insertion errors in quantum deletion-correcting codes based on quantum Reed-Solomon codes.