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

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.