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Characterizing the Burst Error Correction Ability of Quantum Cyclic Codes

Jihao Fan, Min-Hsiu Hsieh

TL;DR

This work tackles burst error correction for quantum cyclic codes (QCCs) under memory-affected quantum channels, by developing polynomial-time methods to determine the burst-limit for QCCs and the true burst limit for quantum Reed-Solomon codes. It extends classical cyclic-code techniques to the quantum realm using CSS and Hermitian constructions, and introduces a linear-time quantum error-trapping decoder (QETD) that handles both degenerate and nondegenerate burst errors. The results show many QCCs saturate the quantum Reiger Bound $n-k \ge 4\ell$, and that quantum RS codes can exceed previous lower bounds on burst-correction capability, with practical decoding insights. Collectively, these advances enhance the design and decoding of QBECCs for memory-affected quantum channels and provide efficient, scalable tools for quantum error correction in realistic settings.

Abstract

Quantum burst error correction codes (QBECCs) are of great importance to deal with the memory effect in quantum channels. As the most important family of QBECCs, quantum cyclic codes (QCCs) play a vital role in the correction of burst errors. In this work, we characterize the burst error correction ability of QCCs constructed from the Calderbank-Shor-Steane (CSS) and the Hermitian constructions. We determine the burst error correction limit of QCCs and quantum Reed-Solomon codes with algorithms in polynomial-time complexities. As a result, lots of QBECCs saturating the quantum Reiger bound are obtained. We show that quantum Reed-Solomon codes have better burst error correction abilities than the previous results. At last, we give the quantum error-trapping decoder (QETD) of QCCs for decoding burst errors. The decoder runs in linear time and can decode both degenerate and nondegenerate burst errors. What's more, the numerical results show that QETD can decode much more degenerate burst errors than the nondegenerate ones.

Characterizing the Burst Error Correction Ability of Quantum Cyclic Codes

TL;DR

This work tackles burst error correction for quantum cyclic codes (QCCs) under memory-affected quantum channels, by developing polynomial-time methods to determine the burst-limit for QCCs and the true burst limit for quantum Reed-Solomon codes. It extends classical cyclic-code techniques to the quantum realm using CSS and Hermitian constructions, and introduces a linear-time quantum error-trapping decoder (QETD) that handles both degenerate and nondegenerate burst errors. The results show many QCCs saturate the quantum Reiger Bound , and that quantum RS codes can exceed previous lower bounds on burst-correction capability, with practical decoding insights. Collectively, these advances enhance the design and decoding of QBECCs for memory-affected quantum channels and provide efficient, scalable tools for quantum error correction in realistic settings.

Abstract

Quantum burst error correction codes (QBECCs) are of great importance to deal with the memory effect in quantum channels. As the most important family of QBECCs, quantum cyclic codes (QCCs) play a vital role in the correction of burst errors. In this work, we characterize the burst error correction ability of QCCs constructed from the Calderbank-Shor-Steane (CSS) and the Hermitian constructions. We determine the burst error correction limit of QCCs and quantum Reed-Solomon codes with algorithms in polynomial-time complexities. As a result, lots of QBECCs saturating the quantum Reiger bound are obtained. We show that quantum Reed-Solomon codes have better burst error correction abilities than the previous results. At last, we give the quantum error-trapping decoder (QETD) of QCCs for decoding burst errors. The decoder runs in linear time and can decode both degenerate and nondegenerate burst errors. What's more, the numerical results show that QETD can decode much more degenerate burst errors than the nondegenerate ones.
Paper Structure (9 sections, 15 theorems, 32 equations, 1 figure, 4 tables, 3 algorithms)

This paper contains 9 sections, 15 theorems, 32 equations, 1 figure, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Burst Error Correction Ability of Quantum Cyclic Codes
  4. The True Burst Error Correction Ability of Quantum Reed-Solomon Codes
  5. Error-Trapping Decoder of Quantum Cyclic Codes
  6. Conclusion and Discussion
  7. Proof of the Quantum Reiger Bound
  8. Proof of Theorem \ref{['theoremquantumburst1']}
  9. Proofs of Lemma \ref{['relationshipofRSandExpan']}, Lemma \ref{['RSMBl']}, and Theorem \ref{['theoremQRSburstlimit']}

Key Result

Proposition 1

The quantum code $Q$ can correct any quantum burst error of length $l$ or less if and only if for all $\langle c_i|c_j\rangle = \delta_{ij}$ and for all $\textrm{bl}(E),\textrm{bl}(E')\leq \ell$, where $|c_i\rangle$ and $|c_j\rangle\in Q$, $E$ and $E'\in\mathcal{G}_n$, and $a_{(E,E')}$ is a constant which depends only on $E$ and $E'$. If $\langle c_i|E^\dagger E'|c_j\rangle=0$ for all $|c_i\

Figures (1)

  • Figure 1: Quantum error-trapping decoder for QCCs.

Theorems & Definitions (15)

  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Theorem 1: Quantum Reiger Bound
  • Lemma 3
  • Lemma 4: matt1980determining
  • Lemma 5: lay2011linear
  • Theorem 2
  • Lemma 6
  • Lemma 7
  • ...and 5 more