Disordered quantum walks in one lattice dimension

被引:78
作者
Ahlbrecht, Andre [1 ]
Scholz, Volkher B. [1 ]
Werner, Albert H. [1 ]
机构
[1] Leibniz Univ Hannover, Inst Theoret Phys, D-30167 Hannover, Germany
关键词
Hadamard matrices; Hilbert spaces; quantum computing; random processes; MULTISCALE ANALYSIS; LOCALIZATION; STATES; DIFFUSION; BERNOULLI; DENSITY; ABSENCE; PROOF;
D O I
10.1063/1.3643768
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
We study a spin-1/2-particle moving on a one-dimensional lattice subject to disorder induced by a random, space-dependent quantum coin. The discrete time evolution is given by a family of random unitary quantum walk operators, where the shift operation is assumed to be deterministic. Each coin is an independent identically distributed random variable with values in the group of two-dimensional unitary matrices. We derive sufficient conditions on the probability distribution of the coins such that the system exhibits dynamical localization. Put differently, the tunneling probability between two lattice sites decays rapidly for almost all choices of random coins and after arbitrary many time steps with increasing distance. Our findings imply that this effect takes place if the coin is chosen at random from the Haar measure, or some measure continuous with respect to it, but also for a class of discrete probability measures which support consists of two coins, one of them being the Hadamard coin. (C) 2011 American Institute of Physics. [doi: 10.1063/1.3643768]
引用
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页数:48
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