Parameter estimation with reluctant quantum walks: a maximum likelihood approach

The parametric maximum likelihood estimation problem is addressed in the context of quantum walk theory for quantum walks on the lattice of integers. A coin action is presented, with the real parameter theta to be estimated identified with the angular argument of an orthogonal reshuffling matrix. 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 theta. We suggest that this 'reluctant walker' behaviour provides the framework for maximum likelihood estimation analysis, allowing for robust parameter estimation of theta via return probabilities of closed evolution loops and quantum measurements of the position of quantum walker with 'reluctance index' r = d/k.