Efficient quantum circuits for continuous-time quantum walks on composite graphs

TL;DR

This paper introduces efficient quantum circuits for simulating continuous-time quantum walks on certain composite graphs, enabling constant-time simulation and exact results with polylogarithmic efficiency, thus advancing quantum algorithm implementation.

Contribution

It presents a method to fast-forward quantum walks on specific composite graphs, expanding the class of efficiently simulatable graphs in quantum computing.

Findings

Fast-forwarding quantum walks on commuting graphs

Efficient simulation of Cartesian product graphs

Polylogarithmic complexity in quantum circuit simulation

Abstract

In this paper, we investigate the simulation of continuous-time quantum walks on specific classes of graphs, for which it is possible to fast-forward the time-evolution operator to achieve constant-time simulation complexity and to perform the simulation exactly, while maintaining $\mbox{poly}(\mbox{log}(n))$ efficiency. In particular, we discuss two classes of composite graphs, commuting graphs and Cartesian product of graphs, that contain classes of graphs which can be simulated in this fashion. This allows us to identify new families of graphs that we can efficiently simulate in a quantum circuit framework, providing practical and explicit means to explore quantum-walk based algorithms in laboratories.