Quantum-classical distance as a tool to design optimal chiral quantum walk

TL;DR

This paper introduces the quantum-classical distance as a metric to optimize Hamiltonian parameters in continuous-time quantum walks, enhancing quantum transport and search efficiency on specific graph topologies.

Contribution

It demonstrates how quantum-classical distance guides the design of Hamiltonians for improved quantum transport and search on cycle and complete graphs, linking it to coherence and participation ratio.

Findings

Quantum-classical distance guides Hamiltonian optimization.

Enhanced quantum transport on cycle graphs.

Achieved quantum speed limit without an oracle.

Abstract

Continuous-time quantum walks (CTQWs) provide a valuable model for quantum transport, universal quantum computation and quantum spatial search, among others. Recently, the empowering role of new degrees of freedom in the Hamiltonian generator of CTQWs, which are the complex phases along the loops of the underlying graph, was acknowledged for its interest in optimizing or suppressing transport on specific topologies. We argue that the quantum-classical distance, a figure of merit which was introduced to capture the difference in dynamics between a CTQW and its classical, stochastic counterpart, guides the optimization of parameters of the Hamiltonian to achieve better quantum transport on cycle graphs and spatial search to the quantum speed limit without an oracle on complete graphs, the latter also implying fast uniform mixing. We compare the variations of this quantity with the 1-norm of coherence and the Inverse Participation Ratio, showing that the quantum-classical distance is linked to both, but in a topology-dependent relation, which is key to spot the most interesting quantum evolution in each case.