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World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering

Algebraic Quantum Error Correction Codes

Ming-Chung Tsai, Kuan-Peng Chen, Zheng-Yao

Published Mar 20, 2012• 0 citations
Publisher

Abstract

A systematic and exhaustive method based on the group structure of a unitary Lie algebra is proposed to generate an enormous number of quantum codes. With respect to the algebraic structure, the orthogonality condition, which is the central rule of generating quantum codes, is proved to be fully equivalent to the distinguishability of the elements in this structure. In addition, four types of quantum codes are classified according to the relation of the codeword operators and some initial quantum state. By linking the unitary Lie algebra with the additive group, the classical correspondences of some of these quantum codes can be rendered.

DOI
10.5281/zenodo.1060485

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A systematic and exhaustive method based on the group structure of a unitary Lie algebra is proposed to generate an enormous number of quantum codes. With respect to the algebraic structure, the orthogonality condition, which is the central rule of generating quantum codes, is proved to be fully equivalent to the distinguishability of the elements in this structure. In addition, four types of quantum codes are classified according to the relation of the codeword operators and some initial quantum state. By linking the unitary Lie algebra with the additive group, the classical correspondences of some of these quantum codes can be rendered.
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.
The group of transversal gates and the group of local symmetries are important features of quantum error correcting codes and pure quantum states, respectively; the former provides fault-tolerant operations on a code while the latter tells us about a state's reachability via stochastic local operations with classical communication. We prove that there exists a bijection between local unitary (LU) orbits of absolutely maximally entangled (AME) states in (CD)⊗n(\mathbb{C}^D)^{\otimes n}(CD)⊗n where nnn is even, also known as perfect tensors, and LU orbits of ((n−1,D,n/2))D((n-1,D,n/2))_D((n−1,D,n/2))D​ quantum error correcting codes. Furthermore, there is a close connection between the local symmetries of an AME state and the transversal gates of its corresponding quantum error correcting code. We explore in detail the 4-qutrit AME state ∣Φ⟩|Φ\rangle∣Φ⟩ and its corresponding ((3,3,2))3((3,3,2))_3((3,3,2))3​ qutrit code C\mathcal{C}C. We show that ∣Φ⟩|Φ\rangle∣Φ⟩ and C\mathcal{C}C are both unique up to the action of the LU group. We find generators of the local symmetry group of ∣Φ⟩|Φ\rangle∣Φ⟩ and the group of transversal gates on C\mathcal{C}C. Our proofs rely on prior results by Huber and Grassl (2020), Hebenstreit et al. (2016), and Rather et al. (2023). We use Vinberg's theory of Lie algebras to study the special case of ∣Φ⟩|Φ\rangle∣Φ⟩ and C\mathcal{C}C.
Codeword-stabilized quantum codes provide a unified approach to constructing quantum error-correcting codes, including both additive and nonadditive quantum codes. Standard codeword-stabilized quantum codes encode quantum information into subspaces. The more general notion of encoding quantum information into a subsystem is known as an operator (or subsystem) quantum error-correcting code. Most operator codes studied to date are based in the usual stabilizer formalism. We introduce operator quantum codes based on the codeword-stabilized quantum code framework. Based on the necessary and sufficient conditions for operator quantum error correction, we derive an error-correction condition for operator codeword-stabilized quantum codes. Based on this condition, the word operators of a operator codeword-stabilized quantum code are constructed from a set of classical binary errors induced by generators of the gauge group. We use this scheme to construct examples of both additive and nonadditive codes that encode quantum information into a subsystem.
Abstract Certain classical codes can be viewed isomorphically as ideals of group algebras, while studying their algebraic structures help extracting the code properties. Research has shown that this was remarkably efficient in the case when the code generators are idempotents. In quantum error correction, the theory of stabilizer formalism requires classical self-orthogonal additive codes over the finite field GF(4), which, via the lens of group algebras, are essentially F2F_2F2​ F 2 -submodules over GF(4). Therefore, this paper provides a classification on idempotents in commutative group algebra GF(4)G, followed by a criterion that allows idempotents to generate stabilizer subgroups. Later, the construction of quantum stabilizer codes is done in the case when G is a cyclic group CnC_nCn​ C n , for n=2m−1n=2^m-1n=2m−1 n = 2 m - 1 and n=2m+1n=2^m+1n=2m+1 n = 2 m + 1 . Quantum bounds on their burst error minimum distance are subsequently determined.
We present a universal framework for quantum error-correcting codes, i.e., a framework that applies to the most general quantum error-correcting codes. This framework is based on the group algebra, an algebraic notation associated with nice error bases of quantum systems. The nicest thing about this framework is that we can characterize the properties of quantum codes by the properties of the group algebra. We show how it characterizes the properties of quantum codes as well as generates some new results about quantum codes.