On the equivalence between quantum and random walks on finite graphs

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 vertex of the graph and the probability that the random walk is at that same vertex , for all vertices 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 allow for a different simulation approach for quantum walks where vertex 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.