Some limit laws for quantum walks with applications to a version of the Parrondo paradox

A quantum walker moves on the integers with four extra degrees of freedom, performing a coin-shift operation to alter its internal state and position at discrete units of time. The time evolution is described by a unitary process. We focus on finding the limit probability law for the position of the walker and study it by means of Fourier analysis. The quantum walker exhibits both localization and a ballistic behavior. Our two results are given as limit theorems for a 2-period time-dependent walk, and they describe the location of the walker after it has repeated the unitary process a large number of times. The theorems give an analytical tool to study some of the Parrondo-type behavior in a quantum game which was studied by Rajendran and Benjamin (R Soc Open Sci 5:171599, 2018) by means of very nice numerical simulations. With our analytical tools at hand we can easily explore the “phase space” of parameters of one of the games, similar to the winning game in their papers. We include numerical evidence that our two games, similar to theirs, exhibit a Parrondo-type paradox.