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Computational matematics series/Computational mathematics series

Clifford codes

Andreas Klappenecker, Martin Rötteler

Published Feb 14, 2002• 13 citations
Publisher

Abstract

Quantum error control codes allow to detect and correct errors that are due to decoherence effects. We review some basic properties of these codes and give some constructions. Our main focus will be on a construction of quantum error control codes that have been introduced by Knill in 1996 with the intention to generalize stabilizer codes. These so-called Clifford codes can be constructed and analyzed with tools from representation theory of finite groups. We show that a large class of Clifford codes are actually stabilizer codes. And we construct the smallest example of a Clifford code that is not a stabilizer code.

DOI
10.1201/9781420035377.ch10

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Abstract Quantum error control codes allow to detect and correct errors that are due to decoherence effects. We review some basic properties of these codes and give some constructions. Our main focus will be on a construction of quantum error control codes that have been introduced by Knill in 1996 with the intention to generalize stabilizer codes. These so-called Clifford codes can be constructed and analyzed with tools from representation theory of finite groups. We show that a large class of Clifford codes are actually stabilizer codes. And we construct the smallest example of a Clifford code that is not a stabilizer code.

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Clifford codes are a class of quantum error control codes that form a natural generalization of stabilizer codes. These codes were introduced in 1996 by Knill, but only a single Clifford code was known, which is not already a stabilizer code. We derive a necessary and sufficient condition that allows to decide when a Clifford code is a stabilizer code, and compile a table of all true Clifford codes for error groups of small order.

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Quantum error control codes allow to detect and correct errors that are due to decoherence effects. We review some basic properties of these codes and give some constructions. Our main focus will be on a construction of quantum error control codes that have been introduced by Knill in 1996 with the intention to generalize stabilizer codes. These so-called Clifford codes can be constructed and analyzed with tools from representation theory of finite groups. We show that a large class of Clifford codes are actually stabilizer codes. And we construct the smallest example of a Clifford code that is not a stabilizer code.
Abstract Quantum error control codes allow to detect and correct errors that are due to decoherence effects. We review some basic properties of these codes and give some constructions. Our main focus will be on a construction of quantum error control codes that have been introduced by Knill in 1996 with the intention to generalize stabilizer codes. These so-called Clifford codes can be constructed and analyzed with tools from representation theory of finite groups. We show that a large class of Clifford codes are actually stabilizer codes. And we construct the smallest example of a Clifford code that is not a stabilizer code.
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.
Recently, operator quantum error-correcting codes have been proposed to unify and generalize decoherence free subspaces, noiseless subsystems, and quantum error-correcting codes. This correspondence introduces a natural construction of such codes in terms of Clifford codes, an elegant generalization of stabilizer codes due to Knill. Character-theoretic methods are used to derive a simple method to construct operator quantum error-correcting codes from any classical additive code over a finite field, which obviates the need for self-orthogonal codes.
Clifford codes are a natural generalization of quantum stabilizer codes based primarily on representation theory. This class of codes has previously been extended to the setting of quantum subsystem codes. We formulate a two-fold generalization of Clifford codes, for both the hybrid classical and quantum information and projective representation theory settings. This leads to new classes of hybrid subspace and subsystem Clifford codes. We extend the fundamental representation theoretic quantum error correction theorem to include these codes, based on the operator algebra quantum error correction framework. We also discuss several examples throughout the presentation, of both stabilizer and non-stabilizer type.
Quantum codes are subspaces of the state space of a quantum system that are used to protect quantum information. Some common classes of quantum codes are stabilizer (or additive) codes, non-stabilizer (or non-additive) codes obtained from stabilizer codes, and Clifford codes. These are analyzed in a framework using the Fourier transform on finite groups, the finite group in question being a subgroup of the quantum error group considered. All the classes of codes that can be obtained in this framework are explored, including codes more general than Clifford codes. The error detection properties of one of these more general classes ("direct sums of translates of Clifford codes") are characterized. Examples codes are constructed, and computer code search results presented and analysed.