Quantization and Asymptotic Behaviour of $\epsilon_{V^{k}}$ Quantum Random Walk on Integers

Abstract

Quantization and asymptotic behaviour of a variant of discrete random walk on integers are investigated. This variant, the $\epsilon_{V^{k}}$ walk, has the novel feature that it uses many identical quantum coins keeping at the same time characteristic quantum features like the quadratically faster than the classical spreading rate, and unexpected distribution cutoffs. A weak limit of the position probability distribution (pd) is obtained, and universal properties of this arch sine asymptotic distribution function are examined. Questions of driving the walk are investigated by means of a quantum optical interaction model that reveals robustness of quantum features of walker's asymptotic pd, against stimulated and spontaneous quantum noise on the coin system.