Algebraic quantum codes: linking quantum mechanics and discrete mathematics

We discuss the connection between quantum error-correcting codes (QECCS) and algebraic coding theory. We start with an introduction to the relevant concepts of quantum mechanics, including the general error model. A quantum error-correcting code is a subspace of a complex Hilbert space, and its erro...

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Bibliographic Details
Published in:International journal of computer mathematics. Computer systems theory Vol. 6; no. 4; pp. 243 - 259
Main Author: Grassl, Markus
Format: Journal Article
Language:English
Published: Taylor & Francis 02.10.2021
Subjects:
algebraic coding theory
puncture code
quantum construction X
Quantum error-correction
stabilizer codes
ISSN:2379-9927, 2379-9935
Online Access:Get full text
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Summary:We discuss the connection between quantum error-correcting codes (QECCS) and algebraic coding theory. We start with an introduction to the relevant concepts of quantum mechanics, including the general error model. A quantum error-correcting code is a subspace of a complex Hilbert space, and its error-correcting properties are characterized by the Knill-Laflamme conditions. Using the stabilizer formalism, we illustrate how QECCs for can be constructed using techniques from algebraic coding theory. We also sketch how the information obtained via a quantum measurement can be interpreted as syndrome of the related classical code. Additionally, we present secondary constructions for QECCs, leading to propagation rules for the parameters of QECCs. This includes the puncture code by Rains and construction X for quantum codes.
ISSN:2379-9927
2379-9935
DOI:10.1080/23799927.2020.1850530