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arXiv:quant-ph/0507213 · Physical Review Letters

Method to Find Quantum Noiseless Subsystems

Man-Duen Choi, David W. Kribs

Published Feb 6, 2006• 60 citations
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Abstract

We develop a structure theory for decoherence-free subspaces and noiseless subsystems that applies to arbitrary (not necessarily unital) quantum operations. The theory can be alternatively phrased in terms of the superoperator perspective, or the algebraic noise commutant formalism. As an application, we propose a method for finding all such subspaces and subsystems for arbitrary quantum operations. We suggest that this work brings the fundamental passive technique for error correction in quantum computing an important step closer to practical realization.

DOI
10.1103/physrevlett.96.050501

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We begin with an examination of the assumptions made in the theory of passive quantum error correction. Physical motivation is discussed for the use of completely positive trace-preserving maps as a description of the evolution of quantum systems. The Lindblad master equation is presented as an example of the dynamics of a system thusly described, in the case of a Markovian system. We review noiseless subsystems and decoherence-free subspaces. Building on the generalization of the noiseless subsystem approach, we discuss recent results that provide a robust procedure for finding noiseless subsystems, given a specific, not necessarily unital, channel. Unitarily correctable subsystems are introduced, which form a type of active quantum error correction that does not require a measurement. The key result for these systems is that we can apply our procedure for finding noiseless subsystems to find unitarily correctable subsystems, in the case of a unital noise model.