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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Introduction to Quantum Deletion Error-Correcting Codes

Manabu Hagiwara

Published May 21, 2024• 3 citations
Publisher

Abstract

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.

DOI
10.1587/transfun.2024tai0001

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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.
Error correcting codes is an area of mathematics that combines many theoretical concepts with a wide variety of applications.In this expository thesis, we will begin by introducing many of the fundamental ideas in classical error correcting codes.The goal of this is to introduce the reader to the field of coding theory and discuss a few of the wide range of classical codes that have been studied.The second section will discuss the recent findings in error correcting codes, namely their use in quantum error correcting codes.This section uses much more theoretical areas of math, when compared to the classical section, in the construction of quantum error correcting codes.Most of the research discussed in this section has been completed within the past six years.All of the research discoveries presented in this thesis have been previously known.
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.
An overview of the quantum computing paradigm with a focus on reliability is provided in a tutorial form intended for practitioners and researchers in the test and reliability community. It is assumed that readers have little prior knowledge of quantum informatics. An introductory description of the mathematical models of a qubit and quantum information processing operations and projective measurement is presented as background. Discussions of quantum error sources and associated fault models are included using the concepts and notation explained in the background section. Topics related to the decoherence problem are also included. The concept of a logical qubit and how quantum error detection and correction can be applied to enhance reliability of quantum computations is formulated using the notions of classical fault models and error detection and correction techniques. The general concept of quantum error detection and correction as applied to enhancing quantum computational reliability is discussed including an example of one of the first such methods originally introduced in 1995.
This is an expository article aiming to introduce the reader to the underlying mathematics and geometry of quantum error correction. Information stored on quantum particles is subject to noise and interference from the environment. Quantum error-correcting codes allow the negation of these effects in order to successfully restore the original quantum information. We briefly describe the necessary quantum-mechanical background to be able to understand how quantum error correction works. We go on to construct quantum codes: firstly qubit stabilizer codes, then qubit non-stabilizer codes, and finally codes with a higher local dimension. We will delve into the geometry of these codes. This allows one to deduce the parameters of the code efficiently, deduce the inequivalence between codes that have the same parameters, and presents a useful tool in deducing the feasibility of certain parameters. We also include sections on quantum maximum distance separable codes and the quantum MacWilliams identities.
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.