Implementation of hitting times of discrete time quantum random walks on Cubelike graphs

TL;DR

This paper implements and verifies the hitting times of discrete quantum walks on cubelike graphs using IBM's Qiskit, and explores their relationship with graph degree through numerical analysis.

Contribution

It provides an efficient circuit implementation for quantum walks on cubelike graphs and extends the analysis to augmented cubes, proposing a conjecture on hitting time and graph degree.

Findings

Verified one-shot hitting time on hypercubes.

Extended study to augmented cubes.

Conjecture of linear relationship between degree and hitting time.

Abstract

We demonstrate an implementation of the hitting time of a discrete time quantum random walk on cubelike graphs using IBM's Qiskit platform. Our implementation is based on efficient circuits for the Grover and Shift operators. We verify the results about the one-shot hitting time of quantum walks on a hypercube as proved in [https://link.springer.com/article/10.1007/s00440-004-0423-2]. We extend the study to another family of cubelike graphs called the augmented cubes [https://onlinelibrary.wiley.com/doi/abs/10.1002/net.10033]. Based on our numerical study, we conjecture that for all families of cubelike graphs there is a linear relationship between the degree of a cubelike graph and its hitting time which holds asymptotically. That is, for any cubelike graph of degree $\Delta$, the probability of finding the quantum random walk at the target node at time $\frac{\pi \Delta}{2}$ approaches 1 as the degree $\Delta$ of the cubelike graph approaches infinity.