Parameter Estimation with Reluctant Quantum Walks: a Maximum Likelihood approach

TL;DR

This paper develops a maximum likelihood estimation method for a parameter in quantum walks, analyzing the probability distribution of displacements and introducing a 'reluctant walker' behavior to estimate the parameter robustly.

Contribution

It provides an analytic framework for parameter estimation in quantum walks using likelihood peakedness and return probabilities, a novel approach in quantum walk theory.

Findings

Likelihood sharply peaked at displacement ratio d/k for large k

Reluctant walker behavior enables robust parameter estimation

Analytic probability distribution for quantum walk displacements

Abstract

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' $r=d/k$.