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 framework for developing new quantum algorithms. Here 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 that could ultimately demonstrate quantum supremacy over classical computers. As a proof of principle, we experimentally implement the proposed quantum circuit on an example circulant graph using a two-qubit photonics quantum processor.
Highlights
The random walk formalism is used across a wide range of applications, from modelling share prices to predicting population genetics
The exact evolution of the continuous-time quantum walks (CTQWs) is governed by connections between the vertices of G : jcðtÞi 1⁄4 expð À itHÞjcð0Þi where the Hamiltonian is given by H 1⁄4 gA for hopping rate per edge per unit time g and where A is the N-by-N symmetric adjacency matrix, whose entries are Ajk 1⁄4 1, if vertices j and k are connected by an edge in G, and Ajk 1⁄4 0 otherwise[1]
The dynamics of a CTQW on a graph with N vertices can be evaluated in time poly(N) on a classical computer
Summary
The random walk formalism is used across a wide range of applications, from modelling share prices to predicting population genetics. Sampling of this form is sufficient to solve various search and characterization problems[4,9], and can be used to deduce critical parameters of the quantum walk, such as mixing time[2]. We adapt the methodology of refs 36–38 to show that if there did exist a classical sampler for a somewhat more general class of circuits, this would have the following unlikely complexity-theoretic implication: the infinite tower of complexity classes known as the polynomial hierarchy would collapse This evidence of hardness exists despite the classical efficiency with which properties of the CTQW, such as the eigenvalues of circulant graphs, can be computed on a classical machine
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