Efficient quantum circuits for Szegedy quantum walks

TL;DR

This paper introduces a general method for constructing efficient quantum circuits for Szegedy quantum walks on certain classes of Markov chains, facilitating their implementation in quantum algorithms like quantum Pagerank.

Contribution

It presents a novel scheme to build efficient quantum circuits for Szegedy walks on Markov chains with columnar symmetry, applicable to non-sparse and tensor product chains.

Findings

Efficient quantum circuits are constructed for cyclic and bipartite graphs.

The scheme applies to non-sparse and tensor product Markov chains.

Facilitates experimental realization of quantum Pagerank algorithm.

Abstract

A major advantage in using Szegedy's formalism over discrete-time and continuous-time quantum walks lies in its ability to define a unitary quantum walk on directed and weighted graphs. In this paper, we present a general scheme to construct efficient quantum circuits for Szegedy quantum walks that correspond to classical Markov chains possessing transformational symmetry in the columns of the transition matrix. In particular, the transformational symmetry criteria do not necessarily depend on the sparsity of the transition matrix, so this scheme can be applied to non-sparse Markov chains. Two classes of Markov chains that are amenable to this construction are cyclic permutations and complete bipartite graphs, for which we provide explicit efficient quantum circuit implementations. We also prove that our scheme can be applied to Markov chains formed by a tensor product. We also briefly discuss the implementation of Markov chains based on weighted interdependent networks. In addition, we apply this scheme to construct efficient quantum circuits simulating the Szegedy walks used in the quantum Pagerank algorithm for some classes of non-trivial graphs, providing a necessary tool for experimental demonstration of the quantum Pagerank algorithm.