Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0901.0376 · Entropy

Universal Framework for Quantum Error-Correcting Codes

Zhuo Li, Lijuan Xing

Published Jul 23, 2021• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.3390/e23080937

Related papers

Error bases, group algebra and quantum codes

Zhuo Li, Lijuan Xing

We find a new approach to study nice error bases and quantum error-correcting codes, namely the group algebra which gives us an algebraic notation for nice error bases and quantum codes. From this algebraic notation we establish a series of linear programming bounds on the most general quantum error-correcting codes.

Acta Physica Sinica•Jan 1, 2013•1 citation

Algebraic Quantum Error-Correction Codes

Ming-Chung Tsai, Pochung Chen, Kuan-Peng Chen +1 more

Based on the group structure of a unitary Lie algebra, a scheme is provided to systematically and exhaustively generate quantum error correction codes, including the additive and nonadditive codes. The syndromes in the process of error-correction distinguished by different orthogonal vector subspaces, the coset subspaces. Moreover, the generated codes can be classified into four types with respect to the spinors in the unitary Lie algebra and a chosen initial quantum state.

arXiv:1310.7514•Oct 28, 2013•0 citations

Error Correcting Quantum Codes and Algebraic Curves

Artur Elezi

The goal of this paper is twofold. First, to provide a self contained, detailed and rigorous mathematical introduction to some aspects of the quantum error-correcting codes and especially quantum stabilizer codes and their connection to self-orthogonal linear codes. This has been done without venturing that much, if at all, into the world of physics. While most of the results presented are not new, it is not easy to extract a precise mathematical formulation of results and to provide their rigorous proofs by reading the vast number of papers in the field, quite a few of which are written by computer scientists or physicists. It is this formulation and proofs, some of which are new, that we present here. Techniques from algebra of finite fields as well as representations of finite abelian groups have been employed. The second goal is the construction of some stabilizer codes via self-orthogonal linear codes associated to algebraic curves.

NATO science for peace and security series. D, Information and communication security•Jan 1, 2015•0 citations

Quantum Error Correction and Orthogonal Geometry

A.R. Calderbank, Eric M. Rains, Peter W. Shor +1 more

A group theoretic framework is introduced that simplifies the description of known quantum error-correcting codes and greatly facilitates the construction of new examples. Codes are given which map 3 qubits to 8 qubits correcting 1 error, 4 to 10 qubits correcting 1 error, 1 to 13 qubits correcting 2 errors, and 1 to 29 qubits correcting 5 errors.

arXiv:quant-ph/9605005•Physical Review Letters•Jan 20, 1997•801 citations

Quantum Anticodes

ChunJun Cao, Giuseppe Cotardo, Brad Lackey

This work introduces a symplectic framework for quantum error correcting codes in which local structure is analyzed through an anticode perspective. In this setting, a code is treated as a symplectic space, and anticodes arise as maximal symplectic subspaces whose elements vanish on a prescribed set of components, providing a natural quantum analogue of their classical counterparts. This framework encompasses several families of quantum codes, including stabilizer and subsystem codes, provides a natural extension of generalized distances in quantum codes, and yields new invariants that capture local algebraic and combinatorial features. The notion of anticodes also naturally leads to operations such as puncturing and shortening for symplectic codes, which in turn provide algebraic interpretations of key phenomena in quantum error correction, such as the cleaning lemma and complementary recovery and yield new descriptions of weight enumerators.

quant-pharXiv:2512.13891•Dec 17, 2025•0 citations

Stabilizing quantum information

Paolo Zanardi A

The dynamical-algebraic structure underlying all the schemes for quantum information stabilization is argued to be fully contained in the reducibility of the operator algebra describing the interaction with the environment of the coding quantum system. This property amounts to the existence of a nontrivial group of symmetries for the global dynamics. We provide a unified framework that allows us to build systematically additional classes of error correcting codes and noiseless subsystems. It is shown that by using symmetrization strategies one can artificially produce noiseless subsystems supporting universal quantum computation.

arXiv:quant-ph/9910016•Physical Review A•Dec 5, 2000•207 citations
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.
The goal of this paper is twofold. First, to provide a self contained, detailed and rigorous mathematical introduction to some aspects of the quantum error-correcting codes and especially quantum stabilizer codes and their connection to self-orthogonal linear codes. This has been done without venturing that much, if at all, into the world of physics. While most of the results presented are not new, it is not easy to extract a precise mathematical formulation of results and to provide their rigorous proofs by reading the vast number of papers in the field, quite a few of which are written by computer scientists or physicists. It is this formulation and proofs, some of which are new, that we present here. Techniques from algebra of finite fields as well as representations of finite abelian groups have been employed. The second goal is the construction of some stabilizer codes via self-orthogonal linear codes associated to algebraic curves.
This work introduces a symplectic framework for quantum error correcting codes in which local structure is analyzed through an anticode perspective. In this setting, a code is treated as a symplectic space, and anticodes arise as maximal symplectic subspaces whose elements vanish on a prescribed set of components, providing a natural quantum analogue of their classical counterparts. This framework encompasses several families of quantum codes, including stabilizer and subsystem codes, provides a natural extension of generalized distances in quantum codes, and yields new invariants that capture local algebraic and combinatorial features. The notion of anticodes also naturally leads to operations such as puncturing and shortening for symplectic codes, which in turn provide algebraic interpretations of key phenomena in quantum error correction, such as the cleaning lemma and complementary recovery and yield new descriptions of weight enumerators.
The dynamical-algebraic structure underlying all the schemes for quantum information stabilization is argued to be fully contained in the reducibility of the operator algebra describing the interaction with the environment of the coding quantum system. This property amounts to the existence of a nontrivial group of symmetries for the global dynamics. We provide a unified framework that allows us to build systematically additional classes of error correcting codes and noiseless subsystems. It is shown that by using symmetrization strategies one can artificially produce noiseless subsystems supporting universal quantum computation.