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 $G=\text{Cay}(\mathbb Z_2^d,Ω)$ of degree $Δ=|Ω|$. Starting from the vertex labeled $0$, we identify $σ=\bigoplus_{ω\inΩ}ω$ as a natural target vertex; for the hypercube, $σ$ is precisely the antipodal vertex. For families with $Δ\to\infty$, let $T$ be an integer having the same parity as $Δ$ and satisfying $ \left|T-\frac{πΔ}{2}\right|\leq 1. $ We show that the probability $p_T(σ)$ of finding the walker at $σ$ when it is measured at time $T$ satisfies $$ p_T(σ)=1-O(Δ^{-1/5}). $$ Thus the target is found with probability tending to one after $Θ(Δ)$ steps. For the concurrently measured walk, let $H_T^{\mathrm{Conc}}(σ)$ denote the probability that the target is detected at or before time $T$ when it is tested after every step. We prove $$ p_T(σ)\leq T H_T^{\mathrm{Conc}}(σ), $$ which implies $H_T^{\mathrm{Conc}}(σ)=Ω(Δ^{-1})$ 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)