Deciding factor for detecting a particle within a subspace via dark and bright states

TL;DR

This paper develops a method using dark and bright states to detect a particle in a subspace during a quantum walk, with conditions ensuring high detection probability and implications for quantum computing.

Contribution

It introduces a novel approach based on bright and dark states for subspace detection in quantum walks, utilizing the rank-nullity theorem and analyzing detection probabilities.

Findings

Conditions for unit detection probability in subspace detection.

Dependence of detection probability on the number of dark states.

High rank projectors reduce divergence in the average number of measurements.

Abstract

In a measurement-induced continuous-time quantum walk, we address the problem of detecting a particle in a subspace, instead of a fixed position. In this configuration, we develop an approach of bright and dark states based on the unit and vanishing detection probability respectively for a particle-detection in the subspace. Specifically, by employing the rank-nullity theorem, we determine several properties of dark and bright states in terms of energy spectrum of the Hamiltonian used for a quantum walk and the projectors applied to detect the subspace. We provide certain conditions on the position and the rank of the subspace to be detected, resulting in the unit total detection probability, which has broad implications for quantum computing. Further, we illustrate the forms of dark as well as bright states and the dependence of detection probability on the number of dark states by considering a cyclic graph with nearest-neighbor and next nearest-neighbor hopping. Moreover, we observe that the divergence in the average number of measurements for detecting a particle successfully in a subspace can be reduced by performing high rank projectors.