Scouring Parrondo's paradox in discrete-time quantum walks

We propose a quantum game based on coin-based quantum walks. Given a quantum walk and a Hermitian operator on the coin-position composite space, winning this game involves choosing an initial coin state such that the given quantum walk leads to a composite state in which the expectation value of the given Hermitian operator is greater than a certain value. Parrondo's paradox is a phenomenon where a combination of losing strategies becomes a winning strategy. We give a deterministic scheme for identifying Parrondo's paradox in our game, in the sense that, given a collection of distinct quantum steps, we identify initial coin states which happen to be losing states for all quantum walks comprised solely of these steps individually, but turn out to be winning states for a quantum walk comprised of all the given steps taken in a sequence. Unlike traditional quantum steps that allow for equal magnitude forward and backward strides based on the outcome of the coin toss, the steps of the quantum walks employed here, though still contingent upon coin toss, permit the strides to be of unequal magnitude, and not necessarily in opposite directions. We believe the results presented here will contribute to a deeper understanding of evolution of expectation values of observables in quantum walks, and facilitate the development of novel quantum algorithms.