On the equivalence between quantum and random walks on finite graphs

TL;DR

This paper establishes a formal equivalence between quantum walks and a class of time-heterogeneous random walks on finite graphs, enabling new simulation methods and insights into their relationship.

Contribution

It introduces a method to construct random walk matrices that exactly replicate quantum walk probability distributions on finite graphs.

Findings

Equivalence between quantum and random walks established for arbitrary finite graphs.

Random walks can be made time-heterogeneous to match quantum walk distributions.

New polynomial-time sampling approach for quantum walks based on local random walk matrices.

Abstract

Quantum walks on graphs are ubiquitous in quantum computing finding a myriad of applications. Likewise, random walks on graphs are a fundamental building block for a large number of algorithms with diverse applications. While the relationship between quantum and random walks has been recently discussed in specific scenarios, this work establishes a formal equivalence between the processes on arbitrary finite graphs and general conditions for shift and coin operators. It requires empowering random walks with time heterogeneity, where the transition probability of the walker is non-uniform and time dependent. The equivalence is obtained by equating the probability of measuring the quantum walk on a given node of the graph and the probability that the random walk is at that same node, for all nodes and time steps. The result is given by the construction procedure of a matrix sequence for the random walk that yields the exact same vertex probability distribution sequence of any given quantum walk, including the scenario with multiple interfering walkers. Interestingly, these matrices allows for a different simulation approach for quantum walks where node samples respect neighbor locality and convergence is guaranteed by the law of large numbers, enabling efficient (polynomial) sampling of quantum graph trajectories (paths). Furthermore, the complexity of constructing this sequence of matrices is discussed in the general case.