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IEICE Communications Express

The four qubits deletion code is the first quantum insertion code

Manabu Hagiwara

Published Feb 15, 2021• 12 citations
Publisher

Abstract

The classical insertion codes were discovered at the same time as the classical deletion codes. The quantum deletion code was presented at IEICE Communications Express last year, but quantum insertion codes were never constructed. In this article, we provide the first quantum insertion code. Its encoding is the same as the four-qubits deletion code, known as the theoretically shortest deletion code.

DOI
10.1587/comex.2020xbl0191

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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.

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Research on quantum insertion or deletion error correcting codes has become active in recent years. In 2023, Hagiwara constructed a binary quantum code that corrects multiple deletion errors and has a more flexible code rate than conventional ones. The contribution of this study is to provide an insertion correcting algorithm for the code constructed by Hagiwara.

International Symposium on Information Theory and its Applications•Nov 10, 2024•0 citations
The classical insertion codes were discovered at the same time as the classical deletion codes. The quantum deletion code was presented at IEICE Communications Express last year, but quantum insertion codes were never constructed. In this article, we provide the first quantum insertion code. Its encoding is the same as the four-qubits deletion code, known as the theoretically shortest deletion code.
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.
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.