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IEEE Transactions on Information Theory

Beyond stabilizer codes II: Clifford codes

Andreas Klappenecker, Martin Rötteler

Published Aug 1, 2002• 34 citations
Publisher

Abstract

For pt. I see ibid., vol.48, no.8, p.2392-95 (2002). Knill (1996) introduced a generalization of stabilizer codes, called Clifford codes. It remained unclear whether or not Clifford codes can be superior to stabilizer codes. We show that Clifford codes are stabilizer codes provided that the abstract error group has an Abelian index group. In particular, if the errors are modeled by tensor products of Pauli matrices, then the associated Clifford codes are necessarily stabilizer codes.

DOI
10.1109/tit/2002.800473

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For pt. I see ibid., vol.48, no.8, p.2392-95 (2002). Knill (1996) introduced a generalization of stabilizer codes, called Clifford codes. It remained unclear whether or not Clifford codes can be superior to stabilizer codes. We show that Clifford codes are stabilizer codes provided that the abstract error group has an Abelian index group. In particular, if the errors are modeled by tensor products of Pauli matrices, then the associated Clifford codes are necessarily stabilizer 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.
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.