AI 中文总结
本文研究立方图上带Grover硬币的量子行走的单次与并发击中时间，通过Walsh-Fourier分解等方法，将超立方体的击中现象扩展到任意立方生成集，并证实了相关渐近击中行为猜想。
AI 中文摘要
我们研究立方图$G=\text{Cay}(\mathbb Z_2^d,\Omega)$（度数为$\Delta=|\Omega|$）上离散时间带Grover硬币的量子行走的单次与并发击中问题。从标记为0的顶点出发，我们将$\sigma=\bigoplus_{\omega\in\Omega}\omega$识别为自然目标顶点；对于超立方体，$\sigma$恰好是对跖顶点。对于$\Delta\to\infty$的族，设$T$为与$\Delta$奇偶性相同且满足$\left|T-\frac{\pi\Delta}{2}\right|\leq1$的整数。我们证明，在时刻$T$测量行走者时，其出现在$\sigma$处的概率$p_T(\sigma)$满足$p_T(\sigma)=1-O(\Delta^{-1/5})$，即经过$\Theta(\Delta)$步后，以趋近于1的概率找到目标。对于并发测量的行走，设$H_T^{\mathrm{Conc}}(\sigma)$为每一步后都进行测试时，在时刻$T$或之前检测到目标的概率，我们证明$p_T(\sigma)\leq T H_T^{\mathrm{Conc}}(\sigma)$，这意味着在相同时间尺度上$H_T^{\mathrm{Conc}}(\sigma)=\Omega(\Delta^{-1})$。证明使用了Walsh-Fourier分解、每个Fourier模式的精确二维约化以及相关特征和的通用二阶矩恒等式。我们的结果将Kempe的超立方体击中现象（J. Kempe, Probab. Theory Relat. Fields 133, 215-235, 2005）扩展到任意立方生成集，并证实了Mulherkar、Rajdeepak和Sunitha（Int. J. Quantum Inf. 20,2250020, 2022）中关于立方图和增广立方图的渐近击中行为猜想。
英文摘要
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)
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