Degree Distribution in Quantum Walks on Complex Networks

TL;DR

This paper provides an analytical relationship between quantum and classical walk distributions on complex networks, revealing how quantum effects influence node connectivity measures like degree distribution.

Contribution

It introduces a theoretical framework linking quantum walk distributions to classical degree distributions, highlighting the role of energy and quantumness in this relationship.

Findings

Quantum walk distribution equals classical degree distribution at zero energy.

Higher energy states show quantum effects bounded by initial state energy.

Quantumness relates to Renyi entropy of node degrees.

Abstract

In this theoretical study, we analyze quantum walks on complex networks, which model network-based processes ranging from quantum computing to biology and even sociology. Specifically, we analytically relate the average long time probability distribution for the location of a unitary quantum walker to that of a corresponding classical walker. The distribution of the classical walker is proportional to the distribution of degrees, which measures the connectivity of the network nodes and underlies many methods for analyzing classical networks including website ranking. The quantum distribution becomes exactly equal to the classical distribution when the walk has zero energy and at higher energies the difference, the so-called quantumness, is bounded by the energy of the initial state. We give an example for which the quantumness equals a Renyi entropy of the normalized weighted degrees, guiding us to regimes for which the classical degree-dependent result is recovered and others for which quantum effects dominate.