Efficient Quantum Walk on a Quantum Processor

TL;DR

This paper develops efficient quantum circuits for continuous-time quantum walks on circulant graphs, demonstrating potential quantum advantage and implementing a proof-of-principle experiment on a photonic quantum processor.

Contribution

It introduces explicit efficient quantum circuits for continuous-time quantum walks on circulant graphs and links this to quantum supremacy potential.

Findings

Quantum circuits for circulant graphs are efficient and implementable.

Classical intractability of sampling from these quantum walks is supported by complexity theory.

Experimental demonstration on a two-qubit photonic processor confirms feasibility.

Abstract

The random walk formalism is used across a wide range of applications, from modelling share prices to predicting population genetics. Likewise quantum walks have shown much potential as a frame- work for developing new quantum algorithms. In this paper, we present explicit efficient quantum circuits for implementing continuous-time quantum walks on the circulant class of graphs. These circuits allow us to sample from the output probability distributions of quantum walks on circulant graphs efficiently. We also show that solving the same sampling problem for arbitrary circulant quantum circuits is intractable for a classical computer, assuming conjectures from computational complexity theory. This is a new link between continuous-time quantum walks and computational complexity theory and it indicates a family of tasks which could ultimately demonstrate quantum supremacy over classical computers. As a proof of principle we have experimentally implemented the proposed quantum circuit on an example circulant graph using a two-qubit photonics quantum processor.