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arXiv:1412.5086 · Quantum Information Processing

Central limit theorem for reducible and irreducible open quantum walks

Przemysław Sadowski, Łukasz Pawela

Published Apr 13, 2016• 20 citations
arXivHTMLPDF

Abstract

In this work we aim at proving central limit theorems for open quantum walks on $${\mathbb {Z}}^d$$ . We study the case when there are various classes of vertices in the network. In particular, we investigate two ways of distributing the vertex classes in the network. First, we assign the classes in a regular pattern. Secondly, we assign each vertex a random class with a transition invariant distribution. For each way of distributing vertex classes, we obtain an appropriate central limit theorem, illustrated by numerical examples. These theorems may have application in the study of complex systems in quantum biology and dissipative quantum computation.

DOI
10.1007/s11128-016-1314-z

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In this work we aim at proving central limit theorems for open quantum walks on Zd{\mathbb {Z}}^dZd . We study the case when there are various classes of vertices in the network. In particular, we investigate two ways of distributing the vertex classes in the network. First, we assign the classes in a regular pattern. Secondly, we assign each vertex a random class with a transition invariant distribution. For each way of distributing vertex classes, we obtain an appropriate central limit theorem, illustrated by numerical examples. These theorems may have application in the study of complex systems in quantum biology and dissipative quantum computation.
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Discrete quantum walk is one of the de facto models of quantum computation and as an efficient tool to develop quantum search algorithms. Although the theoretical model of quantum walks is straightforward, there are many complex scenarios such as coherence decay and/or decoherence in the implementations. It is hard to test experimentally if quantum walk works, or it just decays into a version of classic random walk. We propose a quantum central limit theorem (QCLT) for discrete quantum walks and conduct the statistical hypothesis testing for the standard or decayed walker probability distribution for imperfect quantum walks based on the QCLT. A reliable statistical analysis result is obtained for the imperfect distribution by the experimental quantum walk study.
Open quantum walks (OQWs) describe a quantum walker on an underlying graph whose dynamics is purely driven by dissipation and decoherence. Mathematically, they are formulated as completely positive trace preserving (CPTP) maps on the space of density matrices for the walker on the graph. Any microscopically derived OQW must include the possibility of remaining on the same site on the graph when the map is applied. We extend the CPTP map to describe a lazy OQW. We derive a central limit theorem for lazy OQWs on a ddd-dimensional lattice, where the distribution converges to a Gaussian. We show that the properties of this Gaussian computed using conventional methods agree with the general formulas derived from our central limit theorem.
Abstract. We study open quantum random walks (OQRW) for which the underlying graph is a lattice, and the generators of the walk are translation-invariant. Using the results obtained recently in [5], we study the quantum trajectory associated with the OQRW, which is described by a position process and a state process. We obtain a central limit theorem and a large deviation principle for the position process. We study in detail the case of homogeneous OQRWs on a lattice, with internal space h = C2. 1
Abstract We consider homogeneous open quantum walks on a lattice with finite dimensional local Hilbert space and we study in particular the position process of the quantum trajectories of the walk. We prove that the properly rescaled position process asymptotically approaches a mixture of Gaussian measures. We can generalize the existing central limit type results and give more explicit expressions for the involved asymptotic quantities, dropping any additional condition on the walk. We use deformation and spectral techniques, together with reducibility properties of the local channel associated with the open quantum walk. Further, we can provide a large deviation principle in the case of a fast recurrent local channel and at least lower and upper bounds in the general case.