Operator Algebraic Formulation of the Stabilizer Formalism for Quantum Error Correction
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Operator Algebraic Formulation of the Stabilizer Formalism for Quantum Error Correction N. Johnston · D.W. Kribs · C.-W. Teng
Received: 10 September 2008 / Accepted: 29 December 2008 / Published online: 10 January 2009 © Springer Science+Business Media B.V. 2009
Abstract We give an operator algebraic formulation of the stabilizer formalism for error correction in quantum computing. The approach relies on an analysis of commutant structures, and gives a natural extension of the classic stabilizer formalism to the general case of arbitrary (not necessarily abelian) Pauli subgroups and subsystem codes. We show how to identify the largest stabilizer subsystem for every Pauli subgroup and discuss examples. Keywords Stabilizer formalism · Pauli group · Quantum error correction · Operator algebra
1 Introduction The classic stabilizer formalism of Gottesman [1, 2] provides the simplest technique to generate codes for the standard model of error correction (QEC) in quantum computing [3–6]. Recently the formalism was extended by Poulin [7] to the case of subsystem codes in “operator quantum error correction” (OQEC) [8, 9]. An important subsystem refinement of Shor’s 9-qubit stabilizer code [3] was recently discovered by Bacon [10], and also elucidated by Poulin [7]. The so-called “Bacon-Shor code” has now been used by Aliferis-Cross [11] to improve the crucial threshold theorem for fault-tolerant quantum computation. The mathematical starting point for the classic stabilizer formalism is an abelian subgroup of the n-qubit Pauli group. There are a number of naïve and natural questions that can be asked, such as: Does there exist an extension of the stabilizer formalism that begins with an arbitrary (not necessarily abelian) subgroup of the Pauli group? If so, how does it relate to Poulin’s extension? Moreover, is there an operator algebraic formulation of the stabilizer formalism? In this paper we present a natural subsystem extension of the stabilizer N. Johnston · D.W. Kribs () · C.-W. Teng Department of Mathematics and Statistics, University of Guelph, Guelph, ON, N1G 2W1, Canada e-mail: [email protected] D.W. Kribs Institute for Quantum Computing, University of Waterloo, Waterloo, ON, N2L 3G1, Canada
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formalism that starts with an arbitrary Pauli subgroup. The operator algebraic approach we take may be viewed as complementary to Poulin’s, in that the same subsystem codes are obtained from a different perspective. In particular, we argue that this affirms the subsystem generalization of [7] is indeed the “right” generalization of the stabilizer formalism to arbitrary Pauli subgroups. Intuitively, the approach allows the “gauge freedom” of stabilizer subsystem codes to be injected at an earlier stage in the process. Formally, stabilizer subsystems are obtained via the operator commutant structure of Pauli subgroups. The operator algebra approach has the advantage that it yields a simple way to generate stabilizer subsystem codes. However, it has the drawback that more complicated Pauli groups mu
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