arxiv: 2605.11510 · v1 · submitted 2026-05-12 · 💻 cs.IT · math.IT

Recognition: 1 theorem link

· Lean Theorem

Decoding Algorithm to Composite Errors Consisting of Deletions and Insertions for Quantum Deletion-Correcting Codes Based on Quantum Reed-Solomon Codes

Koki Sasaki , Ken Nakamura , Takayuki Nozaki

Authors on Pith no claims yet

Pith reviewed 2026-05-13 01:56 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords quantum error correctiondeletion-correcting codesReed-Solomon codesdecoding algorithminsertionsdeletionsHagiwara codes

0 comments

The pith

Hagiwara quantum codes now have an efficient decoder for mixed deletion and insertion errors.

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

The paper constructs a decoding algorithm that recovers quantum information from composite errors made of both deletions and insertions in Hagiwara codes. These codes are built from quantum Reed-Solomon codes and are already known to possess the theoretical ability to correct such errors. Until this work, no practical algorithm existed to locate and fix the errors in polynomial time. The new procedure turns the abstract correction capability into a concrete recovery method. A general reader should care because quantum channels frequently produce exactly these mixed loss-and-gain errors, and usable codes are required for reliable quantum communication.

Core claim

The authors give an explicit decoding algorithm for Hagiwara codes that handles any combination of deletions and insertions whose total weight lies within the code's designed distance. The algorithm works by computing syndromes from the received quantum state, then solving for error positions and values using the algebraic structure inherited from the underlying quantum Reed-Solomon construction.

What carries the argument

The decoding algorithm that adapts classical Reed-Solomon syndrome and error-locator techniques to locate and correct the positions and types of deletions and insertions in the quantum codeword.

If this is right

Hagiwara codes become implementable in quantum channels that produce insertion-deletion noise.
Quantum Reed-Solomon constructions gain a complete error-correction pipeline.
Design of larger quantum codes for composite error models can now include explicit decoders.

Where Pith is reading between the lines

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

The same algebraic approach may extend to other quantum algebraic-geometric codes that inherit Reed-Solomon-like properties.
Numerical simulation of the decoder on small qubit registers would provide concrete performance data under realistic noise models.
Hybrid quantum-classical protocols could combine this decoder with classical insertion-deletion codes for mixed classical-quantum links.

Load-bearing premise

The received quantum state must contain an error pattern whose total number of deletions plus insertions stays within the code's guaranteed correction capability.

What would settle it

Apply the algorithm to a small Hagiwara code instance with a known correctable deletion-insertion pattern and observe whether it fails to output the original codeword.

Figures

Figures reproduced from arXiv: 2605.11510 by Ken Nakamura, Koki Sasaki, Takayuki Nozaki.

**Figure 1.** Figure 1: βb, γb, and yb B. Definition of Hagiwara Codes [5], [6] Let C1 be an (N, K1)-RS code over F2E and C2 be an (N, K2)-RS code over F2E such that C2 ⊂ C1. Then, the Quantum RS code R [13] is defined by the Calderbank–Shor–Steane (CSS) code [15], [16] of B(C1) over B(C2), i.e., R := nPk i=1 αi [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Example of Algorithm 1 continuous positions (i.e., Si = [[u, v]] for some u, v) and τi is a permutation on Si such that S1, S2, ..., Sk are pairwise disjoint and τ = τ1 ◦ τ2 ◦ · · · ◦ τk. Then, IQ ◦ DP is denoted by a complex linear combination of the Pauli errors for the qubits in P ∪ Sk i=1 Si . Corollary 1: A block-transformation error for Rb is decomposed into the Pauli errors for qubits in Rb. These … view at source ↗

**Figure 3.** Figure 3: Change c[[γb]] (Left) and y[[γb+τi]] (Right) equals (kb + ξb) =: kb. To summarize, y[[γb−jb+kb]] is given by jb-deletions and kb-insertions to c[[γb]], as shown on the left side of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 5.** Figure 5: Example of a Change by IP ◦ DP and τ Algorithm 2 Making Si Input: t ∈ Z +, P = {p1, p2, ..., pt}, Q = {q1, q2, ..., qt} Output: k ∈ {0} ∪ Z +, {Si} k i=1 1: for j = 1, 2, ..., t do 2: if pj ̸= qj then 3: Tj ← [[pj , qj ]] ∪ [[qj , pj ]] 4: else 5: Tj ← ∅ 6: end if 7: end for 8: k ← 0 9: for j = 1, 2, ..., t − 1 do 10: if Tj ∩ Tj+1 ̸= ∅ then 11: Tj+1 ← Tj ∪ Tj+1, Tj ← ∅ 12: else if Tj ̸= ∅ then 13: k ← k + … view at source ↗

read the original abstract

This paper focuses on Hagiwara codes, which are quantum deletion-correcting codes constructed by the quantum Reed-Solomon codes. Although Hagiwara codes can correct composite errors consisting of deletions and insertions, an efficient decoding algorithm to such errors remains an open problem. In this paper, we provide a decoding algorithm to such errors for Hagiwara 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 an explicit decoder for deletion-plus-insertion errors on Hagiwara codes by reducing the task to classical Reed-Solomon decoding plus a quantum syndrome step.

read the letter

The main takeaway is that this work supplies the concrete decoding procedure that the abstract says was missing for Hagiwara codes under composite deletion-insertion errors. It reduces the quantum problem to a classical Reed-Solomon decoding step on the underlying code, then uses syndrome extraction to recover the inserted symbols while respecting the code's minimum distance and the total error weight bound. No extra restrictions on length, field size, or error support are added beyond what Hagiwara already requires. That reduction is the actual new piece; the rest follows from known properties of quantum Reed-Solomon codes. The construction is direct and avoids circularity. It is useful because it turns an existence claim into a workable algorithm that a practitioner could follow. The soft spots are limited. The argument rests on the classical decoder being correct and on the error weight staying inside the correction radius, which is standard but means any weakness in those classical steps carries over. There is no new distance proof or asymptotic analysis here, and the high-level description does not include runtime bounds or simulation results, so the paper reads more as an algorithmic note than a full performance study. Readers already working on quantum deletion channels or on extending classical Reed-Solomon techniques will get the most out of it; others will find it too narrow. The paper shows clear thinking and honest use of prior results, so it is worth sending to a serious referee even if the final version needs added complexity or numerical checks.

Referee Report

0 major / 1 minor

Summary. The manuscript claims to provide a decoding algorithm for composite deletion and insertion errors in Hagiwara codes, which are quantum deletion-correcting codes constructed using quantum Reed-Solomon codes. The algorithm is said to reduce the problem to a classical Reed-Solomon decoding step followed by a quantum syndrome extraction to recover the inserted symbols, relying on the code's minimum distance and a bounded total error weight.

Significance. Should the algorithm prove correct and efficient, this work would address an open problem in quantum coding theory by enabling decoding for more general error patterns beyond pure deletions. The approach leverages existing classical techniques, which is a strength for potential implementability. It builds directly on the Hagiwara construction without introducing new restrictions on code parameters.

minor comments (1)

[Abstract] The abstract is very brief and does not outline the key steps of the proposed algorithm or any performance guarantees, which makes it hard for readers to quickly assess the contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review and recommendation of minor revision. We appreciate the acknowledgment that the work addresses an open problem in quantum coding theory by providing a decoding algorithm for composite deletion-insertion errors in Hagiwara codes based on quantum Reed-Solomon codes, leveraging classical techniques for potential implementability.

Circularity Check

0 steps flagged

No significant circularity in the decoding algorithm construction

full rationale

The paper constructs an explicit decoding procedure for composite deletion-insertion errors on Hagiwara codes by reducing the problem to a classical Reed-Solomon decoding step followed by quantum syndrome extraction. This relies on the established minimum distance of the underlying quantum Reed-Solomon code and a bounded total error weight that does not exceed the code's correction capability. No steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the argument is self-contained against the prior Hagiwara code definition and presents an independent algorithmic contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the background properties of Hagiwara codes as quantum deletion-correcting codes constructed from quantum Reed-Solomon codes.

axioms (1)

domain assumption Hagiwara codes can correct composite errors consisting of deletions and insertions
Stated as given in the abstract.

pith-pipeline@v0.9.0 · 5358 in / 1045 out tokens · 38763 ms · 2026-05-13T01:56:57.725311+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Hagiwara code is constructed by inserting the marker 0^(t)1^(t) to quantum RS codes... Algorithm 1 transforms y to a sequence z with e block-erasures and m block-substitutions where e+2m≤t

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

[1]

Quantum insertion-deletion chan- nels,

J. Leahy, D. Touchette, and P. Yao, “Quantum insertion-deletion chan- nels,”arXiv preprint arXiv:1901.00984, 2019

work page arXiv 1901
[2]

The first quantum error-correcting code for single deletion errors,

A. Nakayama and M. Hagiwara, “The first quantum error-correcting code for single deletion errors,”IEICE Communications Express, vol. 9, no. 4, pp. 100–104, 2020

work page 2020
[3]

A four-qubits code that is a quantum deletion error-correcting code with the optimal length,

M. Hagiwara and A. Nakayama, “A four-qubits code that is a quantum deletion error-correcting code with the optimal length,”2020 IEEE International Symposium on Information Theory (ISIT), pp. 1870–1874, 2020

work page 2020
[4]

Constructions ofl-adict-deletion- correcting quantum codes,

R. Matsumoto and M. Hagiwara, “Constructions ofl-adict-deletion- correcting quantum codes,”IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. 105, no. 3, pp. 571–575, 2022

work page 2022
[5]

Quantum deletion codes derived from quantum reed- solomon codes,

M. Hagiwara, “Quantum deletion codes derived from quantum reed- solomon codes,”arXiv preprint arXiv:2306.13399, 2023

work page arXiv 2023
[6]

Quantum multi deletion codes derived from quantum reed- solomon codes,

——, “Quantum multi deletion codes derived from quantum reed- solomon codes,”2025 IEEE International Symposium on Information Theory (ISIT), pp. 1–6, 2025

work page 2025
[7]

The four qubits deletion code is the first quantum insertion code,

——, “The four qubits deletion code is the first quantum insertion code,” IEICE Communications Express, vol. 10, no. 5, pp. 243–247, 2021

work page 2021
[8]

Multiple-insertion-correcting non-binary quantum codes and decoding algorithm,

K. Nakamura and T. Nozaki, “Multiple-insertion-correcting non-binary quantum codes and decoding algorithm,”IEICE Transactions on Fun- damentals of Electronics, Communications and Computer Sciences, vol. 108, no. 2, pp. 123–128, 2025

work page 2025
[9]

Insertion correcting algorithm for quantum deletion correcting codes based on quantum reed-solomon codes,

K. Sasaki and T. Nozaki, “Insertion correcting algorithm for quantum deletion correcting codes based on quantum reed-solomon codes,”2024 International Symposium on Information Theory and Its Applications (ISITA), pp. 92–97, 2024

work page 2024
[10]

Decoding algorithm correcting single- insertion plus single-deletion for non-binary quantum codes,

K. Nakamura and T. Nozaki, “Decoding algorithm correcting single- insertion plus single-deletion for non-binary quantum codes,”2024 International Symposium on Information Theory and Its Applications (ISITA), pp. 86–91, 2024

work page 2024
[11]

Single-insertion plus single-deletion correct- ing algorithm for quantum deletion correcting codes based on quantum reed-solomon codes,

K. Sasaki and T. Nozaki, “Single-insertion plus single-deletion correct- ing algorithm for quantum deletion correcting codes based on quantum reed-solomon codes,”IEICE Tech. Rep., vol. 125, no. 37, pp. 13–18, 2025

work page 2025
[12]

Insertion Correcting Capability for Quantum Deletion-Correcting Codes

K. Nakamura and T. Nozaki, “Insertion correcting capability for quan- tum deletion-correcting codes,”arXiv preprint arXiv:2602.20635, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[13]

Quantum reed-solomon codes,

M. Grassl, W. Geiselmann, and T. Beth, “Quantum reed-solomon codes,” International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, pp. 231–244, 1999

work page 1999
[14]

Binary codes capable of correcting deletions, in- sertions, and reversals,

V . I. Levenshtein, “Binary codes capable of correcting deletions, in- sertions, and reversals,”Soviet Physics Doklady, vol. 10, pp. 707–710, 1966

work page 1966
[15]

Good quantum error-correcting codes exist,

A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist,”Physical Review A, vol. 54, no. 2, p. 1098, 1996

work page 1996
[16]

Multiple-particle interference and quantum error correction,

A. Steane, “Multiple-particle interference and quantum error correction,” Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, vol. 452, no. 1954, pp. 2551–2577, 1996

work page 1954
[17]

M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information. Cambridge University Press, 2010

work page 2010
[18]

Codes for the quantum erasure channel,

M. Grassl, T. Beth, and T. Pellizzari, “Codes for the quantum erasure channel,”Physical Review A, vol. 56, no. 1, p. 33, 1997

work page 1997
[19]

Quantum error correction and fault-tolerance,

D. Gottesman, “Quantum error correction and fault-tolerance,”Quantum Information Processing: From Theory to Experiment, vol. 199, p. 159, 2006

work page 2006
[20]

Watrous,The Theory of Quantum Information

J. Watrous,The Theory of Quantum Information. Cambridge university press, 2018

work page 2018