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Completely positive maps and passive quantum error correction

Aron Pasieka

Published Jan 1, 2007• 0 citations

Abstract

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.

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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.

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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.

This paper is an expanded and more detailed version of the work [1] in which the Operator Quantum Error Correction protocol was introduced.This is a new scheme for the error correction of quantum operations that incorporates the known techniques -i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem methodas special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations.We also discuss a number of examples and introduce the notion of "unitarily noiseless subsystems".

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Abstract. This paper is an expanded and more detailed version of the work [1] in which the Operator Quantum Error Correction protocol was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques — i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method — as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of “unitarily noiseless subsystems”. A unified and generalized approach to quantum error correction, called Operator Quantum Error Correction (OQEC), was recently introduced in [1]. This model unifies all of the known techniques for the error correction of quantum operations – i.e. the standard model [2, 3, 4, 5], the method of decoherence-free subspaces [6, 7, 8, 9]

The prevailing description for dissipative quantum dynamics is given by the Lindblad form of a Markovian master equation, used under the assumption that memory effects are negligible. However, in certain physical situations, the master equation is essentially of a non-Markovian nature. In this paper we examine master equations that possess a memory kernel, leading to a replacement of white noise by colored noise. The conditions under which this leads to a completely positive, trace-preserving map are discussed for an exponential memory kernel.