Quantum-classical dynamical distance and quantumness of quantum walks

TL;DR

This paper introduces a fidelity-based measure to compare classical and quantum walks on graphs, revealing how quantum features influence their dynamical differences over time.

Contribution

It provides universal, analytic expressions for the quantum-classical dynamical distance, linking short-time quantum coherence to the measure and analyzing long-time behavior.

Findings

At short times, the distance is proportional to quantum coherence.

At long times, the distance depends only on graph size.

Intermediate times show dependence on graph topology.

Abstract

We introduce a fidelity-based measure $\text{D}_{\text{CQ}}(t)$ to quantify the differences between the dynamics of classical (CW) and quantum (QW) walks over a graph. We provide universal, graph-independent, analytic expressions of this quantum-classical dynamical distance, showing that at short times $\text{D}_{\text{CQ}}(t)$ is proportional to the coherence of the walker, i.e. a genuine quantum feature, whereas for long times it depends only on the size of the graph. At intermediate times, $\text{D}_{\text{CQ}}(t)$ does depend on the graph topology through its algebraic connectivity. Our results show that the difference in the dynamical behaviour of classical and quantum walks is entirely due to the emergence of quantum features at short times. In the long time limit, quantumness and the different nature of the generators of the dynamics, e.g. the open system nature of CW and the unitary nature of QW, are instead contributing equally.