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Decoding Algorithm Correcting Single-Insertion Plus Single-Deletion for Non-binary Quantum Codes

Ken Nakamura, Takayuki Nozaki

TL;DR

This paper provides a decoding algorithm for non-binary quantum codes constructed by Matsumoto and Hagiwara by assuming an error such that a single insertion occurs and then a single deletion occurs.

Abstract

In this paper, we assume an error such that a single insertion occurs and then a single deletion occurs. Under such an error model, this paper provides a decoding algorithm for non-binary quantum codes constructed by Matsumoto and Hagiwara.

Decoding Algorithm Correcting Single-Insertion Plus Single-Deletion for Non-binary Quantum Codes

TL;DR

This paper provides a decoding algorithm for non-binary quantum codes constructed by Matsumoto and Hagiwara by assuming an error such that a single insertion occurs and then a single deletion occurs.

Abstract

In this paper, we assume an error such that a single insertion occurs and then a single deletion occurs. Under such an error model, this paper provides a decoding algorithm for non-binary quantum codes constructed by Matsumoto and Hagiwara.
Paper Structure (15 sections, 13 theorems, 35 equations, 2 figures, 2 algorithms)

This paper contains 15 sections, 13 theorems, 35 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notations
  4. Classical Deletion and Insertion
  5. Indel Distance levenshtein1966binary
  6. Set of Sequences Detecting Deletion Indices
  7. Unitary error and Deletion/Insertion
  8. Non-binary Deletion-Correcting Quantum Codes matsumoto2022constructions
  9. Decoding Algorithm Correcting Single-Insertion Plus Single-Deletion
  10. Decoding Algorithm
  11. Justification
  12. Proof of Theorem \ref{['theo:r=m_result_class']}
  13. Lemmas to prove Theorem \ref{['theo:r=m_result_class']}
  14. Proof of Theorem \ref{['theo:r=m_result_class']}
  15. Conclusion

Key Result

Theorem 1

Consider a quantum state $\rho = \sum_{\bm{x},\bm{y} \in \mathbb{Z}_l^n} c_{\bm{x},\bm{y}}\ket{\bm{x}}\bra{\bm{y}} \in S \bigl(\mathcal{H}_{l}^{\otimes {n}} \bigr)$ and a permutation $\tau$ on $[\space[1,n]\space]$. By abuse of notation, we define the index permutation$\tau(\rho) \in S \bigl(\mathc In addition, for $t,n \in \mathbb{Z}^+$ and $J = \{j_1,j_2,...,j_t\} \subset [\space[1,n+t]\space]$

Figures (2)

  • Figure 1: Digraph $\mathtt{G}_{(0,1,2), (1,1,2)}$
  • Figure 2: $\mathtt{P}_{\bot}$ and $\mathtt{P}_{\bot}$

Theorems & Definitions (21)

  • Theorem 1: Theorem $3.3$ of sibayama_single_ins_def
  • Remark 1
  • Remark 2
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Theorem 4
  • Corollary 2
  • Definition 1
  • Lemma 1
  • ...and 11 more