One-Shot and Concurrent Hitting Times for Grover-Coined Quantum Walks on Cubelike Graphs

Abstract

We study the one-shot and concurrent hitting for the discrete-time Grover-coined quantum walk on cubelike graphs $G=\text{Cay}(\mathbb Z_2^d,\Omega)$ of degree $\Delta=|\Omega|$. Starting from the vertex labeled $0$, we identify $\sigma=\bigoplus_{\omega\in\Omega}\omega$ as a natural target vertex; for the hypercube, $\sigma$ is precisely the antipodal vertex. For families with $\Delta\to\infty$, let $T$ be an integer having the same parity as $\Delta$ and satisfying $ \left|T-\frac{\pi\Delta}{2}\right|\leq 1. $ We show that the probability $p_T(\sigma)$ of finding the walker at $\sigma$ when it is measured at time $T$ satisfies $$ p_T(\sigma)=1-O(\Delta^{-1/5}). $$ Thus the target is found with probability tending to one after $\Theta(\Delta)$ steps. For the concurrently measured walk, let $H_T^{\mathrm{Conc}}(\sigma)$ denote the probability that the target is detected at or before time $T$ when it is tested after every step. We prove $$ p_T(\sigma)\leq T H_T^{\mathrm{Conc}}(\sigma), $$ which implies $H_T^{\mathrm{Conc}}(\sigma)=\Omega(\Delta^{-1})$ over the same time scale. The proof uses the Walsh-Fourier decomposition, an exact two-dimensional reduction of each Fourier mode, and a universal second-moment identity for the associated character sums. Our results extend Kempe's hypercube hitting phenomenon (J. Kempe, Probab. Theory Relat. Fields 133, 215-235, 2005) to arbitrary cubelike generating sets and establish the conjectured asymptotic hitting behavior for cublelike and augmented cubes in Mulherkar, Rajdeepak and Sunitha (Int. J. Quantum Inf. 20,2250020, 2022)