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IEEE International Symposium on Information Theory (ISIT)

Equivalence of Quantum Single Insertion and Single Deletion Error-Correctabilities, and Construction of Codes and Decoders

Taro Shibayama, Manabu Hagiwara

Published Jun 26, 2022• 8 citations
Publisher

Abstract

This article contributes to an unsolved quantum information problem: the equivalence between insertion error-correctability and deletion error-correctability. The solution in Shibayama and Ouyang’s sense is provided in the single-error under the assumption of the purity of quantum codeword states On the other hand, a class of codes that can correct both single deletion and single insertion is provided.

DOI
10.1109/isit50566.2022.9834635

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This article contributes to an unsolved quantum information problem: the equivalence between insertion error-correctability and deletion error-correctability. The solution in Shibayama and Ouyang’s sense is provided in the single-error under the assumption of the purity of quantum codeword states On the other hand, a class of codes that can correct both single deletion and single insertion is provided.
Quantum synchronisation errors are a class of errors that change the number of qubits in a quantum system. The classical error correction of synchronisation errors has been well-studied, including an insertion-deletion equivalence more than a half-century ago, but little progress has been made towards the quantum counterpart since the birth of quantum error correction. We address the longstanding problem of a quantum insertion-deletion equivalence on permutation-invariant codes, detailing the conditions under which such codes are ttt-insertion error-correctable. We extend these conditions to quantum insertion-deletion errors, formulating a more restrictive set of conditions under which permutation-invariant codes are (t,s)(t,s)(t,s)-insdel error-correctable. For completeness, we extend our approach to qudit errors on qudit permutation-invariant codes. Our work resolves many of the outstanding questions regarding the quantum error correction of synchronisation errors.
In this paper, we prove the equivalence of inserting separable quantum states and deletions. Hence any quantum code that corrects deletions automatically corrects separable insertions. First, we describe the quantum insertion/deletion error using the Kraus operators. Next, we develop an algebra for commuting Kraus operators corresponding to insertions and deletions. Using this algebra, we prove the equivalence between quantum insertion codes and quantum deletion codes using the Knill-Laflamme conditions.
In this paper, we discuss a construction method of quantum deletion error-correcting codes. First of all, we define deletion errors for quantum states, an encoder, a decoder, and two conditions which is expressed by only the combinatorial language. Then, we prove that quantum deletion error-correcting codes can be constructed by two sets that satisfy the conditions. In other words, problems that correct the deletion errors for quantum states are reduced to problems that find the sets satisfying the condition by this paper. Also, we performed experiment of the codes over IBM Quantum Experience.
This manuscript is an extended abstract version of the paper entitled ``Quantum Deletion Codes derived from Classical Deletion Codes.'' The paper contributes to the fundamental theory for quantum deletion error-correcting codes. The paper proposes a code construction condition for a partition of classical deletion error-correcting codes to derive quantum deletion error-correcting codes. The construction methods in this paper give examples of quantum codes that can correct single-quantum deletion errors and have a code rate arbitrarily close to 1, while the previously known quantum deletion code rates are close to 0 for long length. This manuscript omits the proofs of the statements in the paper.