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

Journal of Physics Conference Series

Establishing the number of distinct stabilizer bases for a quantum qudit error-correcting code

Colin Wilmott

Published Jan 21, 2015• 1 citation
Publisher

Abstract

The class of quantum codes called stabilizer codes is increasingly well-understood. The premise of the stabilizer formalism is that a quantum code can be efficiently described by a subgroup of its error group, and, interestingly, the stabilizer formalism permits correspondences with classical linear codes. In this paper, we examine one such correspondence, and we shall use this to establish the number of distinct stabilizer codes that exist for a fixed parametrisation.

DOI
10.1088/1742-6596/574/1/012096

Related papers

Stabilizer Codes and Quantum Error Correction

Daniel Gottesman

Controlling operational errors and decoherence is one of the major challenges facing the field of quantum computation and other attempts to create specified many-particle entangled states. The field of quantum error correction has developed to meet this challenge. A group-theoretical structure and associated subclass of quantum codes, the stabilizer codes, has proved particularly fruitful in producing codes and in understanding the structure of both specific codes and classes of codes. I will give an overview of the field of quantum error correction and the formalism of stabilizer codes. In the context of stabilizer codes, I will discuss a number of known codes, the capacity of a quantum channel, bounds on quantum codes, and fault-tolerant quantum computation.

•May 28, 1997•1,237 citations

Stabilizer Codes and Quantum Error Correction

Daniel Gottesman

Controlling operational errors and decoherence is one of the major challenges facing the field of quantum computation and other attempts to create specified many-particle entangled states. The field of quantum error correction has developed to meet this challenge. A group-theoretical structure and associated subclass of quantum codes, the stabilizer codes, has proved particularly fruitful in producing codes and in understanding the structure of both specific codes and classes of codes. I will give an overview of the field of quantum error correction and the formalism of stabilizer codes. In the context of stabilizer codes, I will discuss a number of known codes, the capacity of a quantum channel, bounds on quantum codes, and fault-tolerant quantum computation.

arXiv:quant-ph/9705052•May 28, 1997•753 citations

Quantum error correcting codes: From stabilizer codes to induced codes

Edward A. Loeb

The first large class of quantum error-correcting codes that was constructed is the class of stabilizer codes. The construction method used for these codes is based on finding abelian subgroups of general Pauli groups. Stabilizer codes are a special case of a more general class of quantum error-correcting codes---Clifford codes. The construction technique for Clifford codes generalizes the stabilizer code construction by introducing abstract error groups. Abstract error groups generalize the Pauli groups so that codes can be constructed for quantum systems with any dimension. Clifford codes are thus constructed by finding normal subgroups of abstract error groups. In this dissertation we show that Clifford codes are a special case of a larger class of codes, which we refer to as induced codes. In addition to this new construction method, we also present some new results pertaining to a subclass of stabilizer codes called degenerate stabilizer codes.

•Jan 1, 2006•0 citations

An Introduction to Quantum Error Correction

Daniel Gottesman

Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. The theory of quantum error-correcting codes has some close ties to and some striking differences from the theory of classical error-correcting codes. Many quantum codes can be described in terms of the stabilizer of the codewords. The stabilizer is a finite Abelian group, and allows a straightforward characterization of the error-correcting properties of the code. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over GF(4), the finite field with four elements.

arXiv:quant-ph/0004072•Apr 18, 2000•77 citations

Some Results on Qudit Quantum Error-Correction

Lane G. Gunderman

Quantum computing’s seemingly perpetual promise of nearness still has a few hurdles \nto surmount before it can become a reality. Among these hurdles is that of protection of \ninformation from random errors. A potential solution for this challenge is stabilizer codes, \nwhich are the analog of classical linear error-correcting codes, however, with an additional \naxis of error possibility. By and large, quantum computing is discussed in the standard \nqubit, or two-level, language, however, it is worth considering the case of qudits, or more \nthan two-level systems. Often times results follow simply from some form of algebraic \nextension: typically group theory or linear algebra. In this work we consider some features \nthat are not immediately apparent from that approach, more often appealing to physical \nintuition to guide our mathematical ideas, then proving these ideas using the language of \nqudit operators. \n \nHere we consider two particular previously unexplored ideas. The first idea is that of \nembedding and inscribing of codes into spaces of different sizes than the stabilizer code \nwas originally designed for. Here, we show that all codes can be embedded, and that for \ninfinitely many primes we can in fact guarantee that the distance is at least preserved–a \nsomewhat surprising result. The second idea is, in a way an application of those presented \nin the first idea, that of turning many stabilizer codes into hybrid codes by taking advantage \nof relabelling of syndromes and the rapid increase in syndrome space upon using codes in \nlarge spaces. Both of these are somewhat useful in their current forms, however, with some \nadditional mathematical work could be turned into potentially very powerful tools for the \nprotection of quantum information. We finish off by, and along the way, mentioning various \nfuture directions to carry work on this.

•Jan 30, 2020•0 citations

Constructing Quantum Error-Correcting Codes for pm-State System from Classical Error-Correcting Codes

Ryutaroh Matsumoto, Tomohiko Uyematsu

Quantum error-correcting codes have attracted much attention. Among many research articles, the most general and systematic construction is the so called stabilizer code construction [6] or additive code construction [2], which constructs a quantum error-correcting code as an eigenspace of an Abelian subgroup S of the error group. Thereafter Calderbank et al. [3] proposed a construction of S from an additive code over the finite field F4 with 4 elements. These constructions work for tensor products of 2-state quantum systems. However Knill [8], [9] and Rains [13] observed that the construction [2], [6] can be generalized to n-state systems by an appropriate choice of the error basis. Rains [13] also generalized the construction [3] using additive codes over F4 to p-state quantum systems, but his generalization does not relate the problem of quantum code construction to classical error-correcting codes. We propose a construction of quantum error-correcting codes for p-state systems from classical error-correcting codes which is a generalization of [3]. Throughout this paper, p denotes a prime number andm a positive integer. This paper is organized as follows. In Sect. 2, we review the construction of quantum codes for nonbinary systems. In Sect. 3, we propose a construction of quantum codes for p-state systems from classical codes over Fp2 . In Sect. 4, we propose a construction of quantum codes for p-state systems from classical linear codes over Fp2m . In Sect. 5, we discuss a

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences•Oct 25, 2000•26 citations
The class of quantum codes called stabilizer codes is increasingly well-understood. The premise of the stabilizer formalism is that a quantum code can be efficiently described by a subgroup of its error group, and, interestingly, the stabilizer formalism permits correspondences with classical linear codes. In this paper, we examine one such correspondence, and we shall use this to establish the number of distinct stabilizer codes that exist for a fixed parametrisation.
Controlling operational errors and decoherence is one of the major challenges facing the field of quantum computation and other attempts to create specified many-particle entangled states. The field of quantum error correction has developed to meet this challenge. A group-theoretical structure and associated subclass of quantum codes, the stabilizer codes, has proved particularly fruitful in producing codes and in understanding the structure of both specific codes and classes of codes. I will give an overview of the field of quantum error correction and the formalism of stabilizer codes. In the context of stabilizer codes, I will discuss a number of known codes, the capacity of a quantum channel, bounds on quantum codes, and fault-tolerant quantum computation.
Controlling operational errors and decoherence is one of the major challenges facing the field of quantum computation and other attempts to create specified many-particle entangled states. The field of quantum error correction has developed to meet this challenge. A group-theoretical structure and associated subclass of quantum codes, the stabilizer codes, has proved particularly fruitful in producing codes and in understanding the structure of both specific codes and classes of codes. I will give an overview of the field of quantum error correction and the formalism of stabilizer codes. In the context of stabilizer codes, I will discuss a number of known codes, the capacity of a quantum channel, bounds on quantum codes, and fault-tolerant quantum computation.
The first large class of quantum error-correcting codes that was constructed is the class of stabilizer codes. The construction method used for these codes is based on finding abelian subgroups of general Pauli groups. Stabilizer codes are a special case of a more general class of quantum error-correcting codes---Clifford codes. The construction technique for Clifford codes generalizes the stabilizer code construction by introducing abstract error groups. Abstract error groups generalize the Pauli groups so that codes can be constructed for quantum systems with any dimension. Clifford codes are thus constructed by finding normal subgroups of abstract error groups. In this dissertation we show that Clifford codes are a special case of a larger class of codes, which we refer to as induced codes. In addition to this new construction method, we also present some new results pertaining to a subclass of stabilizer codes called degenerate stabilizer codes.
Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. The theory of quantum error-correcting codes has some close ties to and some striking differences from the theory of classical error-correcting codes. Many quantum codes can be described in terms of the stabilizer of the codewords. The stabilizer is a finite Abelian group, and allows a straightforward characterization of the error-correcting properties of the code. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over GF(4), the finite field with four elements.
Quantum computing’s seemingly perpetual promise of nearness still has a few hurdles \nto surmount before it can become a reality. Among these hurdles is that of protection of \ninformation from random errors. A potential solution for this challenge is stabilizer codes, \nwhich are the analog of classical linear error-correcting codes, however, with an additional \naxis of error possibility. By and large, quantum computing is discussed in the standard \nqubit, or two-level, language, however, it is worth considering the case of qudits, or more \nthan two-level systems. Often times results follow simply from some form of algebraic \nextension: typically group theory or linear algebra. In this work we consider some features \nthat are not immediately apparent from that approach, more often appealing to physical \nintuition to guide our mathematical ideas, then proving these ideas using the language of \nqudit operators. \n \nHere we consider two particular previously unexplored ideas. The first idea is that of \nembedding and inscribing of codes into spaces of different sizes than the stabilizer code \nwas originally designed for. Here, we show that all codes can be embedded, and that for \ninfinitely many primes we can in fact guarantee that the distance is at least preserved–a \nsomewhat surprising result. The second idea is, in a way an application of those presented \nin the first idea, that of turning many stabilizer codes into hybrid codes by taking advantage \nof relabelling of syndromes and the rapid increase in syndrome space upon using codes in \nlarge spaces. Both of these are somewhat useful in their current forms, however, with some \nadditional mathematical work could be turned into potentially very powerful tools for the \nprotection of quantum information. We finish off by, and along the way, mentioning various \nfuture directions to carry work on this.
Quantum error-correcting codes have attracted much attention. Among many research articles, the most general and systematic construction is the so called stabilizer code construction [6] or additive code construction [2], which constructs a quantum error-correcting code as an eigenspace of an Abelian subgroup S of the error group. Thereafter Calderbank et al. [3] proposed a construction of S from an additive code over the finite field F4 with 4 elements. These constructions work for tensor products of 2-state quantum systems. However Knill [8], [9] and Rains [13] observed that the construction [2], [6] can be generalized to n-state systems by an appropriate choice of the error basis. Rains [13] also generalized the construction [3] using additive codes over F4 to p-state quantum systems, but his generalization does not relate the problem of quantum code construction to classical error-correcting codes. We propose a construction of quantum error-correcting codes for p-state systems from classical error-correcting codes which is a generalization of [3]. Throughout this paper, p denotes a prime number andm a positive integer. This paper is organized as follows. In Sect. 2, we review the construction of quantum codes for nonbinary systems. In Sect. 3, we propose a construction of quantum codes for p-state systems from classical codes over Fp2 . In Sect. 4, we propose a construction of quantum codes for p-state systems from classical linear codes over Fp2m . In Sect. 5, we discuss a