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arXiv:0906.3692 · Physical Review A

Inhomogeneous quantum walks

Noah Linden, James Sharam

Published Nov 20, 2009• 46 citations
arXivHTMLPDF

Abstract

We study a natural construction of a general class of inhomogeneous quantum walks (namely, walks whose transition probabilities depend on position). Within the class we analyze walks that are periodic in position and show that, depending on the period, such walks can be bounded or unbounded in time; in the latter case we analyze the asymptotic speed. We compare the construction to others in the existing literature. As an example we give a quantum version of a nonirreducible classical walk: the P\'olya Urn.

DOI
10.1103/physreva.80.052327

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We study a natural construction of a general class of inhomogeneous quantum walks (namely, walks whose transition probabilities depend on position). Within the class we analyze walks that are periodic in position and show that, depending on the period, such walks can be bounded or unbounded in time; in the latter case we analyze the asymptotic speed. We compare the construction to others in the existing literature. As an example we give a quantum version of a nonirreducible classical walk: the P'olya Urn.
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We define and analyze quantum computational variants of \nrandom walks on one-dimensional lattices. In particular, we analyze a quantum analog of the symmetric random \nwalk, which we call the Hadamard walk. Several striking \ndifferences between the quantum and classical cases are ob- served. For example, when unrestricted in either direction, \nthe Hadamard walk has position that is nearly uniformly \ndistributed in the range [-t/√2, t/√2] after t steps, which \nis in sharp contrast to the classical random walk, which has \ndistance O(√t) from the origin with high probability. With \nan absorbing boundary immediately to the left of the starting position, the probability that the walk exits to the left is 2/π, and with an additional absorbing boundary at location n, the probability that the walk exits to the left actually increases, approaching 1/√2 in the limit. In the classical case both values are 1.
We present an alternative definition of discrete-time coined quantum walks convenient for capturing a rather broad spectrum of a walker's behavior on arbitrary graphs. It includes and covers both the geometry of possible walker's positions with interconnecting links and the prescribed rule in which directions the walker will move at each vertex. While the former allows for the analysis of inhomogeneous quantum walks on graphs with vertices of varying degree, the latter offers us to choose, investigate, and compare quantum walks with different shift operators. The synthesis of both key ingredients constitutes a well-suited playground for analyzing percolated quantum walks on a quite general class of graphs. Analytical treatment of the asymptotic behavior of percolated quantum walks is presented and worked out in details for the Grover walk on graphs with maximal degree 3. We find that for these walks with cyclic shift operators, the existence of an edge-3 coloring of the graph allows for nonstationary asymptotic behavior of the walk. For different shift operators, the general structure of localized attractors is investigated, which determines the overall efficiency of a source-to-sink quantum transport across a dynamically changing medium. As a simple nontrivial example of the theory, we treat a single-excitation transport on a percolated cube.
The discrete-time quantum walk (QW) is a quantum version of the random walk (RW) and has been widely investigated for the last two decades. Some remarkable properties of QW are well known. For example, QW has a ballistic spreading, i.e., QW is quadratically faster than RW. For some cases, localization occurs: a walker stays at the starting position forever. In this paper, we consider stationary measures of two-state QWs on the line. It was shown that for any space-homogeneous model, the uniform measure becomes the stationary measure. However, the corresponding result for space-inhomogeneous model is not known. Here, we present a class of space-inhomogeneous QWs on the line and cycles in which the uniform measure is stationary. Furthermore, we briefly discuss a difference between QWs and RWs.