Quantum Walks and Reversible Cellular Automata

We investigate a connection between a property of the distribution and a conserved quantity for the reversible cellular automaton derived from a discrete-time quantum walk in one dimension. As a corollary, we give a detailed information of the quantum walk.Comment: 15 pages, minor corrections, some references adde

Crossover from Diffusive to Ballistic Transport in Periodic Quantum Maps

We derive an expression for the mean square displacement of a particle whose motion is governed by a uniform, periodic, quantum multi-baker map. The expression is a function of both time, $t$, and Planck's constant, $\hbar$, and allows a study of both the long time, $t\to\infty$, and semi-classical, $\hbar\to 0$, limits taken in either order. We evaluate the expression using random matrix theory as well as numerically, and observe good agreement between both sets of results. The long time limit shows that particle transport is generically ballistic, for any fixed value of Planck's constant. However, for fixed times, the semi-classical limit leads to diffusion. The mean square displacement for non-zero Planck's constant, and finite time, exhibits a crossover from diffusive to ballistic motion, with crossover time on the order of the inverse of Planck's constant. We argue, that these results are generic for a large class of 1D quantum random walks, similar to the quantum multi-baker, and that a sufficient condition for diffusion in the semi-classical limit is classically chaotic dynamics in each cell. Some connections between our work and the other literature on quantum random walks are discussed. These walks are of some interest in the theory of quantum computation.Comment: Final version to appear in Physica D, Proceedings of the International Workshop and Seminar on Microscopic Chaos and Transport in Many-Particle Systems, Dresden, 2002; corrected a minor error in section 3.1, new section 4.

Quantum walks: a comprehensive review

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing Journa