Abstract
This work is a study on quantum computational formulations of Parrondo walks, that is, positively trending random walks formed as combinations of negatively trending random walks. We reanalyse the position-dependent walk proposed by Košík et al (2007 J. Mod. Opt. 54 2275), correcting the parameter choices in that paper to achieve the Parrondo effect. We also devise a quantum analogue of the cooperative Parrondo walk of Toral (2002 Fluct. Noise Lett. 2 L305), in which it is the interaction between multiple participants, rather than position-dependence, that allows the Parrondo effect to occur. We give a general formulation of a quantum analogue of the classical walk of Toral (2002 Fluct. Noise Lett. 2 L305), and demonstrate the Parrondo effect numerically. Lastly, we highlight a qualitative difference in asymptotic behaviour between quantum Parrondo walks and their classical counterparts. In particular, we draw attention to an intuitive but unreliable assumption, based on classical random walks, which may pose extra challenges for applications of the Parrondo effect in the quantum setting seeking to separate or classify data or particles.
Highlights
This work is a study on quantum computational formulations of Parrondo walks, that is, positively trending random walks formed as combinations of negatively trending random walks
We highlight a qualitative difference in asymptotic behaviour between quantum Parrondo walks and their classical counterparts
We have reanalysed the position-dependent walk proposed by [4], correcting the parameter choices in that paper to achieve the Parrondo effect, that is, a coherent combination of quantum walks such that the combination shows an increasing trend in expected position, while each of its constituent walks shows a decreasing trend in expected position
Summary
The original, non-quantum Parrondo walk has a simple structure. It was presented in [1]–[3] in some generality, but the parameter values described here were given as an example. Perhaps the simplest way (and the only one considered here) is simple stochastic mixing: at each time step, one of the two subwalks is chosen at random to govern the particle’s motion This reversal, a positive drift in the combined walk with negative drifts in the individual walks, is the Parrondo effect. I sin cos where θ is a parameter depending on the particle’s position modulo m and on the state of the subwalk selection register. The space HW is the subwalk selection space, determining which position-dependent subwalk is used on each iteration: the state of this register is rotated by U. at the beginning of each iteration, and together with the particle’s position, modulo m, determines an entry of the parameter matrix , parameterizing the rotation of the HX register. Subwalk 0 is described in [4] (where it is called B) as a position-dependent combination of two further subwalks: one used when the position is divisible by three, and represented by the upper-left entry π/5 − 1/50 of ; and the other used when the position is not divisible by three, and represented by ’s remaining entries, both equal to 3π/2 − 1/50
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