Quantum-classical dynamical distance and quantumness of quantum walks

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.

arXiv comment6 pages, 3 figures, largely rewritten

References in corpus (21)

• Quantifying Coherence
• Quantum Computation and Decision Trees
• Universal computation by quantum walk
• Quantum walks: a comprehensive review
• Spatial search by quantum walk
• Distance measures to compare real and ideal quantum processes
• An example of the difference between quantum and classical random walks
• Continuous-Time Quantum Walks: Models for Coherent Transport on Complex Networks
• Limit Theorem for Continuous-Time Quantum Walk on the Line
• A fidelity measure for quantum channels
• Spacetime structures of continuous time quantum walks
• Simulating Open Quantum Systems with Hamiltonian Ensembles and the Nonclassicality of the Dynamics
• Laplacian versus Adjacency Matrix in Quantum Walk Search
• Quantumness of noisy quantum walks: a comparison between measurement-induced disturbance and quantum discord
• Degree Distribution in Quantum Walks on Complex Networks
• Non-Markovian continuous-time quantum walks on lattices with dynamical noise
• A quantum-walk-inspired adiabatic algorithm for graph isomorphism
• "Partial" Fidelities
• Continuous-time quantum walks on dynamical percolation graphs
• Simultaneous decomposition of two states
• Short time behavior of continuous time quantum walks on graphs

Cited by in corpus (11)

• Complex Quantum Networks: a Topical Review
• Quantum advantage in simulating stochastic processes
• Quantum spatial search in two-dimensional waveguide arrays
• Quantum-classical distance as a tool to design optimal chiral quantum walk
• Experimental Control of the Degree of Non-Classicality via Quantum Coherence
• Scaling and crossover behaviour in a truncated long range quantum walk
• Identifying network topologies via quantum walk distributions
• Enhanced quantum transport in chiral quantum walks
• Decoherence and classicalization of continuous-time quantum walks on graphs
• Universality of the fully connected vertex in Laplacian continuous-time quantum walk problems
• Temporal nonclassicality in continuous-time quantum walks