Quantum walk with deterministic dynamical systems

Presenting Author: Sivaprasad Omanakuttan, University of New Mexico
Contributing Author(s): Prof. Arul Lakshminarayan

We studied the discrete-time quantum walks using simple deterministic dynamical systems like coins. The dynamical system under consideration, in the classical limit, shows a range of behavior from the integrable to chaotic, or deterministically random. For the integrable case, the Fourier coin that generalizes the Hadamard coin was studied and was found that the overall nature of the walk was similar to the well understood two-dimensional coin. It shows the famous ballistic growth of standard deviation, and hence the coin dimensionality has very little significance on the quantum walk. Using kicked Harper map as the deterministic model of random walk, we studied the effect of coin chaos on the quantum walk. We found that the walk becomes classical in the chaotic regime of the kicked Harper and hence the ballistic growth paves way to the diffusive one. The probability distribution in the chaotic regime was studied using coins from the Gaussian unitary ensemble (GUE) and we obtained an approximate expression that resembles the Gaussian distribution for the classical random walk. For a finite-dimensional walker with chaotic coin we obtained a classical to quantum transition in the behavior of the walker. The diffusive growth becomes ballistic after some time of unitary evolution and the time at which the transition occurs depends on the coin dimension but not on the walker dimension.

(Session 5 : Saturday from 5:00pm - 7:00pm)


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