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

Chapman & Hall/CRC applied mathematics and nonlinear science series

Nonbinary Stabilizer Codes

Pradeep Kiran Sarvepalli, Andreas Klappenecker, Salah A. Aly

Published Sep 19, 2007• 7 citations
Publisher

Abstract

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.

DOI
10.1201/9781584889007.ch10

Related papers

Nonbinary Stabilizer Codes

Pradeep Kiran Sarvepalli, Salah A. Aly

Abstract 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 eBooks•Sep 19, 2007•0 citations

Nonbinary Stabilizer Codes Over Finite Fields

Avanti Ulhas Ketkar, Andreas Klappenecker, Santhosh Kumar +1 more

One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum error-correcting codes have been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over$bf F_q$in terms of classical codes over$ bf F_q^2$is provided that generalizes the well-known notion of additive codes over$ bf F_4$of the binary case. This paper also derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum Bose–Chaudhuri–Hocquenghem (BCH) codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.

arXiv:quant-ph/0508070•IEEE Transactions on Information Theory•Oct 30, 2006•627 citations

Code constructions and code families for nonbinary quantum stabilizer code

Avanti Ulhas Ketkar

Stabilizer codes form a special class of quantum error correcting codes. Nonbinary quantum stabilizer codes are studied in this thesis. A lot of work on binary quantum stabilizer codes has been done. Nonbinary stabilizer codes have received much less attention. Various results on binary stabilizer codes such as various code families and general code constructions are generalized to the nonbinary case in this thesis. The lower bound on the minimum distance of a code is nothing but the minimum distance of the currently best known code. The focus of this research is to improve the lower bounds on this minimum distance. To achieve this goal, various existing quantum codes are studied that have good minimum distance. Some new families of nonbinary stabilizer codes such as quantum BCH codes are constructed. Different ways of constructing new codes from the existing ones are also found. All these constructions together help improve the lower bounds.

•Nov 1, 2005•0 citations

Quantum stabilizer codes and beyond

Sarvepalli, Pradeep Kiran

The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. This dissertation makes a threefold contribution to the mathematical theory of quantum error-correcting codes. Firstly, it extends the framework of an important class of quantum codes -- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work establishes a close link between subsystem codes and classical codes showing that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels.

arXiv:0810.2574•Oct 14, 2008•8 citations

Upper bounds on minimum distance of nonbinary quantum stabilizer codes

Santosh Kumar

The most popular class of quantum error correcting codes is stabilizer codes. Binary quantum stabilizer codes have been well studied, and Calderbank, Rains, Shor and Sloane (July 1998) have constructed a table of upper bounds on the minimum distance of these codes using linear programming methods. However, not much is known in the case of nonbinary stabilizer codes. In this thesis, we establish a bridge between selforthogonal classical codes over the finite field containing q2 elements and quantum codes, extending and unifying previous work by Matsumoto and Uyematsu (2000), Ashikhmin and Knill (November 2001), Kim and Walker (2004). We construct a table of upper bounds on the minimum distance of the stabilizer codes using linear programming methods that are tighter than currently known bounds. Finally, we derive code construction techniques that will help us find new codes from existing ones. All these results help us to gain a better understanding of the theory of nonbinary stabilizer codes.

•Nov 1, 2005•0 citations

Quantum error-correcting codes and their geometries

Simeon Ball, Aina Centelles, Felix Huber

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.

arXiv:2007.05992•Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions•Feb 9, 2023•9 citations
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.
Abstract 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.
One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum error-correcting codes have been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes overbfFqbf F_qbfFq​in terms of classical codes over$ bf F_q^2isprovidedthatgeneralizesthewell−knownnotionofadditivecodesoveris provided that generalizes the well-known notion of additive codes overisprovidedthatgeneralizesthewell−knownnotionofadditivecodesover bf F_4$of the binary case. This paper also derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum Bose–Chaudhuri–Hocquenghem (BCH) codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.
Stabilizer codes form a special class of quantum error correcting codes. Nonbinary quantum stabilizer codes are studied in this thesis. A lot of work on binary quantum stabilizer codes has been done. Nonbinary stabilizer codes have received much less attention. Various results on binary stabilizer codes such as various code families and general code constructions are generalized to the nonbinary case in this thesis. The lower bound on the minimum distance of a code is nothing but the minimum distance of the currently best known code. The focus of this research is to improve the lower bounds on this minimum distance. To achieve this goal, various existing quantum codes are studied that have good minimum distance. Some new families of nonbinary stabilizer codes such as quantum BCH codes are constructed. Different ways of constructing new codes from the existing ones are also found. All these constructions together help improve the lower bounds.
The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. This dissertation makes a threefold contribution to the mathematical theory of quantum error-correcting codes. Firstly, it extends the framework of an important class of quantum codes -- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work establishes a close link between subsystem codes and classical codes showing that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels.
The most popular class of quantum error correcting codes is stabilizer codes. Binary quantum stabilizer codes have been well studied, and Calderbank, Rains, Shor and Sloane (July 1998) have constructed a table of upper bounds on the minimum distance of these codes using linear programming methods. However, not much is known in the case of nonbinary stabilizer codes. In this thesis, we establish a bridge between selforthogonal classical codes over the finite field containing q2 elements and quantum codes, extending and unifying previous work by Matsumoto and Uyematsu (2000), Ashikhmin and Knill (November 2001), Kim and Walker (2004). We construct a table of upper bounds on the minimum distance of the stabilizer codes using linear programming methods that are tighter than currently known bounds. Finally, we derive code construction techniques that will help us find new codes from existing ones. All these results help us to gain a better understanding of the theory of nonbinary stabilizer codes.
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.