Circuit implementation of discrete-time quantum walks via the shunt decomposition method

Several models have been proposed to build evolution operators to perform quantum walks in a theoretical way, although when wanting to map the resulting evolution operators into quantum circuits to run them in quantum computers, it is often the case that the mapping process is in fact complicated. Nevertheless, when the adjacency matrix of a graph can be decomposed into a sum of permutation matrices, we can always build a shift operator for a quantum walk that has a block diagonal matrix representation. In this paper, we analyze the mapping process of block diagonal operators into quantum circuit form and apply this method to obtain quantum circuits that generate quantum walks on the most common topologies found in the literature: the straight line, the cyclic graph, the hypercube and the complete graph. The obtained circuits are then executed on quantum processors of the type Falcon r5.11 L and Falcon r4T (two of each type) through IBM Quantum Composer platform and on the Qiskit Aer simulator, performing three steps for each topology. The resulting distributions were compared against analytical distributions, using the statistical distance ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1$$\end{document} as a performance metric. Regarding experimental executions, we obtained short ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1$$\end{document} distances in the cases of quantum circuits with a low amount of multi-control gates, being the quantum processors of the type Falcon r4T the ones that provided more accurate results.