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IEEE International Symposium on Information Theory (ISIT)

Clifford subsystem codes

Andreas Klappenecker

Published Jun 1, 2010• 0 citations
Publisher

Abstract

Subsystem codes are a generalization of decoherence free subspaces, noiseless subsystems, and quantum error-correcting codes. The known constructions of subsystem codes from classical codes are limited to quantum systems that all have the same dimension, and this dimension must be a power of a prime. It is shown that one can remove these restrictions and obtain subsystem codes in quantum systems of arbitrary finite dimension from classical codes that are subgroups of an abelian group. The constructions are derived from Clifford codes over abstract error groups with abelian index groups.

DOI
10.1109/isit.2010.5513672

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Subsystem codes are a generalization of decoherence free subspaces, noiseless subsystems, and quantum error-correcting codes. The known constructions of subsystem codes from classical codes are limited to quantum systems that all have the same dimension, and this dimension must be a power of a prime. It is shown that one can remove these restrictions and obtain subsystem codes in quantum systems of arbitrary finite dimension from classical codes that are subgroups of an abelian group. The constructions are derived from Clifford codes over abstract error groups with abelian index groups.
Subsystem codes are quantum error correcting schemes unifying stabilizer codes, decoherence free subspaces and noiseless subsystems. Subsystem codes were most commonly based on the generalized Pauli basis with dimensions of a power of prime. Recently, a class of nice error bases indexed by a nearring were introduced by the second author. We give a construction of subsystem codes over nice nearrings. Furthermore, we show that free subsystem codes over a finite chain ring cannot perform better than those over a finite field.
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.
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.
Subsystem codes are the most versatile class of quantum error-correcting codes known to date that combine the best features of all known passive and active error-control schemes. The subsystem code is a subspace of the quantum state space that is decomposed into a tensor product of two vector spaces: the subsystem and the co-subsystem. A generic method to derive subsystem codes from existing subsystem codes is given that allows one to trade the dimensions of subsystem and co-subsystem while maintaining or improving the minimum distance. As a consequence, it is shown that all pure MDS subsystem codes are derived from MDS stabilizer codes. The existence of numerous families of MDS subsystem codes is established. Propagation rules are derived that allow one to obtain longer and shorter subsystem codes from given subsystem codes. Furthermore, propagation rules are derived that allow one to construct a new subsystem code by combining two given subsystem 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.