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Necessary and sufficient condition for constructing a single qudit insertion/deletion code and its decoding algorithm

Taro Shibayama

TL;DR

This work extends the Knill-Laflamme condition to quantum errors that change the particle count, showing that single deletion and single insertion errors share the same correctability criteria. It develops a recovery framework for non-square Kraus operators and proves the KL condition remains the key criterion in this broader setting. Two parallel, necessary-and-sufficient condition sets are derived: (C^{del}) for single deletion and (C^{ins}) for single insertion errors, and these are shown to be equivalent. The authors present a concrete 6-qudit, l=3 insertion/deletion code with explicit decoding procedures based on a Gram-Schmidt basis and measurement-driven corrections, illustrating both deletion and insertion correction in a non-binary setting and enabling practical construction of non-binary insertion/deletion codes.

Abstract

This paper shows that Knill-Laflamme condition, known as a necessary and sufficient condition for quantum error-correction, can be applied to quantum errors where the number of particles changes before and after the error. This fact shows that correctabilities of single deletion errors and single insertion errors are equivalent. By applying Knill-Laflamme condition, we generalize the previously known correction conditions for single insertion and deletion errors to necessary and sufficient level. By giving an example that satisfies this condition, we construct a new single qudit insertion/deletion code and explain its decoding algorithm.

Necessary and sufficient condition for constructing a single qudit insertion/deletion code and its decoding algorithm

TL;DR

This work extends the Knill-Laflamme condition to quantum errors that change the particle count, showing that single deletion and single insertion errors share the same correctability criteria. It develops a recovery framework for non-square Kraus operators and proves the KL condition remains the key criterion in this broader setting. Two parallel, necessary-and-sufficient condition sets are derived: (C^{del}) for single deletion and (C^{ins}) for single insertion errors, and these are shown to be equivalent. The authors present a concrete 6-qudit, l=3 insertion/deletion code with explicit decoding procedures based on a Gram-Schmidt basis and measurement-driven corrections, illustrating both deletion and insertion correction in a non-binary setting and enabling practical construction of non-binary insertion/deletion codes.

Abstract

This paper shows that Knill-Laflamme condition, known as a necessary and sufficient condition for quantum error-correction, can be applied to quantum errors where the number of particles changes before and after the error. This fact shows that correctabilities of single deletion errors and single insertion errors are equivalent. By applying Knill-Laflamme condition, we generalize the previously known correction conditions for single insertion and deletion errors to necessary and sufficient level. By giving an example that satisfies this condition, we construct a new single qudit insertion/deletion code and explain its decoding algorithm.
Paper Structure (9 sections, 6 theorems, 41 equations, 1 figure, 1 table)

This paper contains 9 sections, 6 theorems, 41 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Correctable condition for quantum errors with non-square Kraus operators
  4. Equivalence of quantum single deletion and single insertion error-correctabilities
  5. A necessary and sufficient condition for single qudit insertion/deletion error-correction
  6. Example of insertion/deletion qudit code and its decoding algorithm
  7. Decoding algorithm for single deletion errors
  8. Decoding algorithm for single insertion errors
  9. Conclusion

Key Result

Theorem 3.2

For a quantum code $C$ with logical codewords $|0_L\rangle,|1_L\rangle,|2_L\rangle,\dots,|l-1_L\rangle\in\mathbb{C}^{l\otimes n}$ and a quantum error $E:S(\mathbb{C}^{l\otimes n})\rightarrow S(\mathbb{C}^{l\otimes n'})$ with the Kraus set $\mathcal{A}$, the following three conditions are equivalent

Figures (1)

  • Figure 1: Error-correction process for the 6-qudit insertion code when $\sigma=\frac{1}{2}|0\rangle\langle0|+\frac{1}{3}|1\rangle\langle1|+\frac{1}{6}|2\rangle\langle2|\in S(\mathbb{C}^3)$ is inserted in the 4th position

Theorems & Definitions (14)

  • Definition 2.1: Quantum error
  • Definition 3.1: Recovery superoperator
  • Theorem 3.2
  • proof
  • Definition 4.1: Single deletion error
  • Definition 4.2: Single insertion error
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 5.1
  • proof
  • ...and 4 more