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' $r=d/k$.