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International Symposium on Information Theory and its Applications

Pure States in Quantum Deletion Error-Correction

Manabu Hagiwara

Published Nov 10, 2024• 0 citations
Publisher

Abstract

This paper discusses harnessing of pure states in the study of quantum deletion. Quantum deletion is one of the recently prominent aspects in the error model of quantum error correction codes, and pure states represent a class of quantum states widely used in the study of other error models. This study reveals that, under specific encoding and decoding operations, the possibility of deletion error-correction for pure states extends to deletion error-correction for mixed states. Through this paper, it is hoped that the understanding of quantum deletion errors deepens, contributing to the advancement of research in quantum deletion error-correction codes.

DOI
10.23919/isita60732.2024.10858246

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International Symposium on Information Theory and its Applications•Nov 10, 2024•0 citations
This paper discusses harnessing of pure states in the study of quantum deletion. Quantum deletion is one of the recently prominent aspects in the error model of quantum error correction codes, and pure states represent a class of quantum states widely used in the study of other error models. This study reveals that, under specific encoding and decoding operations, the possibility of deletion error-correction for pure states extends to deletion error-correction for mixed states. Through this paper, it is hoped that the understanding of quantum deletion errors deepens, contributing to the advancement of research in quantum deletion error-correction codes.
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.
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.
This paper serves as an introductory overview of quantum deletion error-correction codes, a burgeoning field within quantum coding theory. Covering foundational concepts, existing research, and open questions, it aims to be the first accessible resource on the subject. This paper contains basic definitions of terms so that readers can read it regardless of their background. This paper invites readers to explore this primer and take their initial steps into the realm of quantum deletion error-correcting codes research.
In recent years, significant progress has been made in the research on errors in quantum deletion/insertion channels along with the development of error-correcting codes. However, the current channel model may not be applicable in specific scenarios, such as when using coherent-state qubits as funda-mental qubits. Here, we materialize the conventional error model of synchronization errors to accommodate coherent-state qubits and their superposition states. Additionally, we quantitatively assess the discrepancies between the conventional error model and our more realistic approach, underscoring the limitations of the former in practical applications.