Optimizing topology for quantum probing with discrete-time quantum walks

TL;DR

This paper investigates how the topology of the space in discrete-time quantum walks influences the precision of quantum parameter estimation, identifying optimal configurations that maximize Fisher information.

Contribution

It introduces a method to optimize the topology of quantum walks to enhance quantum metrology performance, specifically maximizing Fisher information for parameter estimation.

Findings

Optimal initial states for maximum Fisher information identified.

Topology optimization improves quantum Fisher information.

Position Fisher information can match quantum Fisher information.

Abstract

Discrete-time quantum walk (DTQW) represents a convenient mathematical framework for describing the motion of a particle on a discrete set of positions when this motion is conditioned by the values of certain internal degrees of freedom, which are usually referred to as the {\em coin} of the particle. As such, and owing to the inherent dependence of the position distribution on the coin degrees of freedom, DTQWs naturally emerge as promising candidates for quantum metrology. In this paper, we explore the use of DTQWs as quantum probes in scenarios where the parameter of interest is encoded in the internal degree of freedom of the walker, and investigate the role of the topology of the walker's space on the attainable precision. In particular, we start considering the encoding of the parameter by rotations for a walker on the line, and evaluate the quantum Fisher information (QFI) and the position Fisher information (FI), explicitly determining the optimal initial state in position space that maximizes the QFI across all encoding schemes. This allows us to understand the role of interference in the position space and to introduce an optimal topology, which maximizes the QFI of the coin parameter and makes the position FI equal to the QFI.