Parameter estimation with reluctant quantum walks

The parametric maximum likelihood estimation problem is addressed in the context of quantum walk theory for quantum walks on the line, or on a finite ring. Two different coin reshuffling actions are presented, with the real parameter θ to be estimated being identified either with the angular argument of an orthogonal reshuffling matrix, or the phase of a unitary reshuffling matrix, acting in a 2 state coin space, respectively. We provide analytic results for the probability distribution for a quantum walker to be displaced by d units from its initial position after k steps. For k large, we show that the likelihood is sharply peaked at a displacement determined by the ratio d/k, which is correlated with the reshuffling parameter θ. We suggest that this ‘reluctant walker’ behaviour provides the framework for maximum likelihood estimation analysis, allowing for robust parameter estimation of θ via measurement of the walker ‘reluctance index’ r = d/k.