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:0909.5103

Nested Quantum Error Correction Codes

Zhuo Wang, Kai Sun, Hen Fan, Vlatko Vedral

Published Sep 28, 2009• 2 citations
arXivHTMLPDF

Abstract

The theory of quantum error correction was established more than a decade ago as the primary tool for fighting decoherence in quantum information processing. Although great progress has already been made in this field, limited methods are available in constructing new quantum error correction codes from old codes. Here we exhibit a simple and general method to construct new quantum error correction codes by nesting certain quantum codes together. The problem of finding long quantum error correction codes is reduced to that of searching several short length quantum codes with certain properties. Our method works for all length and all distance codes, and is quite efficient to construct optimal or near optimal codes. Two main known methods in constructing new codes from old codes in quantum error-correction theory, the concatenating and pasting, can be understood in the framework of nested quantum error correction codes.

DOI
10.48550/arxiv.0909.5103

Related papers

Good additive cyclic quantum error-correcting codes

Ai Luo

This thesis is about the quantum error-correcting codes. Although there are many methods to construct them, finding a good quantum code is still a complicated and difficult task, for we do not know which methods to use. To solve this problem, we did a thorough research on a special class of quantum codes--additive cyclic quantum codes. A new search algorithm has been designed and a lot of good additive cyclic quantum codes found by this algorithm are presented in this thesis. By showing the success of this algorithm and great value of additive cyclic quantum codes, we have greatly reduced the complexity of finding good quantum codes. Quantum error-correction theory is key part of quantum information theory. As quantum information theory is a quite new field which has been developing fast during the last decade, we spent much time explaining the primary concepts of it. We introduced linear algebra, quantum mechanics, quantum operations, quantum error-correction theory and quantum error-correcting codes. In a word, this thesis serves two functions: (1) an exploration of quantum error-correcting codes and (2) an introduction to quantum information theory.

•Jan 1, 2004•1 citation

Codes and designs for quantum error correction : some recent developments through combinatorics (Designs, Codes, Graphs and Related Areas)

Yuichiro Fujiwara

Suppressing the effect of decoherence plays a vital role in the theory of quantum information processing.Despite many skepticisms about its feasibility, the existence of error correction schemes in the quantum do- main was proved in the last century, giving the birth to the field of quantum error correction.Since then, the new field has seen remarkably rapid progress in various directions including experimental realizations of quantum error-correcting codes.While quantum information science draws on many branches of physics, computer science, and mathematics, combinatorics is rapidly becoming an indispensable mathematical tool to the study of quantum error correction.In this note, we review some of the latest developments in quantum error correction where combinatorial design theory and coding theory play the central role.

数理解析研究所講究録•Apr 1, 2014•0 citations

Some Binary Quantum Codes with Good Burst-Error-Correcting Capabilities

欣一朗 常盤, 和敬 木山, 高弘 山崎

Quantum error-correcting codes have been developed as one of the promising tools for protecting quantum information against quantum errors. A great deal of effort has been made mainly to construct efficient quantum random-error-correcting codes. In this article, we investigate a class of quantum codes capable of correcting quantum burst errors, and present a list of some new good quantum burst-error-correcting codes of length less than or equal to 51.

•Feb 28, 2005•2 citations

Entanglement-assisted quantum error-correcting codes and asymmetric entanglement-assisted quantum error-correcting codes of length 5pt over 𝔽pm

Hai Quang Dinh, Bac Trong Nguyen, Van‐Nui Nguyen +1 more

Classical cyclic and constacyclic codes play a fundamental role in coding theory, thanks to their rich algebraic structures and efficient encoding/decoding algorithms. These codes are especially important in the construction of quantum error-correcting codes, since their inherent algebraic properties facilitate the derivation of self-orthogonality or dual-containing conditions that are crucial for ensuring quantum error correction capability. In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of length [Formula: see text] over [Formula: see text]. We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. Moreover, our EAQEC codes have shorter lengths than the known codes, yet exhibit significantly larger quantum distances. This is highly meaningful in the context of quantum error correction, as it enables the construction of codes with small lengths while maintaining large quantum distances. We also construct asymmetric EAQEC (AEAQEC) codes of length [Formula: see text] over [Formula: see text]. We evaluate our AEAQEC codes against all existing ones and find that they are both new and better.

International Journal of Modern Physics C•Feb 27, 2026•0 citations

A Novel Construction of Quantum Stabilizer Codes

Shengmei Zhao

Quantum error correcting code is an efficient tool to correct and prevent quantum error.It is the basic for quantum computation and quantum communications.Taking advantage of the cyclic difference set,we present a novel approach to construct quantum stabilizer codes in this paper.It's not difficult to construct the well-known quantum code by our approach.By analyzing the property of the quantum code constructed by this method,such as ,as well as ,we find all the constructed codes are the quantum Single-bound codes,and we can use this method to encode different length information.Besides,the code rate is high.

Journal of Nanjing University of Posts and Telecommunications•Jan 1, 2011•0 citations

Nonbinary Stabilizer Codes

Pradeep Kiran Sarvepalli, Andreas Klappenecker, Salah A. Aly

Recently, the field of quantum error-correcting codes has rapidly emerged as an important discipline. As quantum information is extremely sensitive to noise, it seems unlikely that any large scale quantum computation is feasible without quantum error-correction. In this paper we give a brief exposition of the theory of quantum stabilizer codes. We review the stabilizer formalism of quantum codes, establish the connection between classical codes and stabilizer codes and the main methods for constructing quantum codes from classical codes. In addition to the expository part, we include new results that cannot be found elsewhere. Specifically, after reviewing some important bounds for quantum codes, we prove the nonexistence of pure perfect quantum stabilizer codes with minimum distance greater than 3. Finally, we illustrate the general methods of constructing quantum codes from classical codes by explicitly constructing two new families of quantum codes and conclude by showing how to construct new quantum codes by shortening.

Chapman & Hall/CRC applied mathematics and nonlinear science series•Sep 19, 2007•7 citations
The theory of quantum error correction was established more than a decade ago as the primary tool for fighting decoherence in quantum information processing. Although great progress has already been made in this field, limited methods are available in constructing new quantum error correction codes from old codes. Here we exhibit a simple and general method to construct new quantum error correction codes by nesting certain quantum codes together. The problem of finding long quantum error correction codes is reduced to that of searching several short length quantum codes with certain properties. Our method works for all length and all distance codes, and is quite efficient to construct optimal or near optimal codes. Two main known methods in constructing new codes from old codes in quantum error-correction theory, the concatenating and pasting, can be understood in the framework of nested quantum error correction codes.
This thesis is about the quantum error-correcting codes. Although there are many methods to construct them, finding a good quantum code is still a complicated and difficult task, for we do not know which methods to use. To solve this problem, we did a thorough research on a special class of quantum codes--additive cyclic quantum codes. A new search algorithm has been designed and a lot of good additive cyclic quantum codes found by this algorithm are presented in this thesis. By showing the success of this algorithm and great value of additive cyclic quantum codes, we have greatly reduced the complexity of finding good quantum codes. Quantum error-correction theory is key part of quantum information theory. As quantum information theory is a quite new field which has been developing fast during the last decade, we spent much time explaining the primary concepts of it. We introduced linear algebra, quantum mechanics, quantum operations, quantum error-correction theory and quantum error-correcting codes. In a word, this thesis serves two functions: (1) an exploration of quantum error-correcting codes and (2) an introduction to quantum information theory.
Suppressing the effect of decoherence plays a vital role in the theory of quantum information processing.Despite many skepticisms about its feasibility, the existence of error correction schemes in the quantum do- main was proved in the last century, giving the birth to the field of quantum error correction.Since then, the new field has seen remarkably rapid progress in various directions including experimental realizations of quantum error-correcting codes.While quantum information science draws on many branches of physics, computer science, and mathematics, combinatorics is rapidly becoming an indispensable mathematical tool to the study of quantum error correction.In this note, we review some of the latest developments in quantum error correction where combinatorial design theory and coding theory play the central role.
Classical cyclic and constacyclic codes play a fundamental role in coding theory, thanks to their rich algebraic structures and efficient encoding/decoding algorithms. These codes are especially important in the construction of quantum error-correcting codes, since their inherent algebraic properties facilitate the derivation of self-orthogonality or dual-containing conditions that are crucial for ensuring quantum error correction capability. In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of length [Formula: see text] over [Formula: see text]. We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. Moreover, our EAQEC codes have shorter lengths than the known codes, yet exhibit significantly larger quantum distances. This is highly meaningful in the context of quantum error correction, as it enables the construction of codes with small lengths while maintaining large quantum distances. We also construct asymmetric EAQEC (AEAQEC) codes of length [Formula: see text] over [Formula: see text]. We evaluate our AEAQEC codes against all existing ones and find that they are both new and better.
Quantum error correcting code is an efficient tool to correct and prevent quantum error.It is the basic for quantum computation and quantum communications.Taking advantage of the cyclic difference set,we present a novel approach to construct quantum stabilizer codes in this paper.It's not difficult to construct the well-known quantum code by our approach.By analyzing the property of the quantum code constructed by this method,such as ,as well as ,we find all the constructed codes are the quantum Single-bound codes,and we can use this method to encode different length information.Besides,the code rate is high.
Recently, the field of quantum error-correcting codes has rapidly emerged as an important discipline. As quantum information is extremely sensitive to noise, it seems unlikely that any large scale quantum computation is feasible without quantum error-correction. In this paper we give a brief exposition of the theory of quantum stabilizer codes. We review the stabilizer formalism of quantum codes, establish the connection between classical codes and stabilizer codes and the main methods for constructing quantum codes from classical codes. In addition to the expository part, we include new results that cannot be found elsewhere. Specifically, after reviewing some important bounds for quantum codes, we prove the nonexistence of pure perfect quantum stabilizer codes with minimum distance greater than 3. Finally, we illustrate the general methods of constructing quantum codes from classical codes by explicitly constructing two new families of quantum codes and conclude by showing how to construct new quantum codes by shortening.