The walker speaks its graph: global and nearly-local probing of the tunnelling amplitude in continuous-time quantum walks

TL;DR

This paper investigates how quantum measurements on continuous-time quantum walks can optimally estimate tunnelling amplitudes, revealing the impact of graph topology and initial states on measurement precision.

Contribution

It computes the quantum Fisher information for various graphs and identifies optimal measurements and initial states for maximum estimation precision.

Findings

Quantum Fisher information varies with graph topology.

Optimal measurements depend on the graph's structure.

Connectivity influences the achievable measurement precision.

Abstract

We address continuous-time quantum walks on graphs, and discuss whether and how quantum-limited measurements on the walker may extract information on the tunnelling amplitude between the nodes of the graphs. For a few remarkable families of graphs, we evaluate the ultimate quantum bound to precision, i.e. we compute the quantum Fisher information (QFI), and assess the performances of incomplete measurements, i.e. measurements performed on a subset of the graph's nodes. We also optimize the QFI over the initial preparation of the walker and find the optimal measurement achieving the ultimate precision in each case. As the topology of the graph is changed, a non-trivial interplay between the connectivity and the achievable precision is uncovered.