AI 中文总结
本文研究具有Rosenzweig-Porter相的无序图上的标记顶点搜索，推导有限尺寸局域边界解析估计，发现搜索性能与量子相直接关联，表明无序可作为量子搜索协议的可调参数。
AI 中文摘要
量子标记顶点搜索算法已知优于经典对应算法，但其在存在无序时的行为仍大多未被探索。本文通过研究无序随机图上的标记顶点搜索来解决这一空白。为引入无序，我们在Erdős-Rényi（ER）图上实现Rosenzweig-Porter（RP）模型，该模型是具有可调遍历、非遍历扩展和局域相的随机矩阵系综，由此产生双随机系统：ER图的连通性随机化相互作用是否存在，而RP无序控制相互作用强度和“ onsite”势，提供研究无序网络量子动力学的双参数框架。首先，我们证明RP系综的特征Wigner-Dyson到Poisson的谱交叉在图约束下于稀疏到稠密范围内依然存在，并推导了有限尺寸局域边界的解析估计，其随图边概率p系统移动，与共振杂化论一致。随后，利用该无序图系综，我们研究标记顶点搜索问题，发现搜索性能直接跟踪潜在量子相。反直觉的是，遍历相虽支持快速输运，却产生比局域相更低的成功概率，局域相以显著更长的搜索时间为代价实现高成功概率。这些结果建立了图上随机矩阵无序与连续时间量子行走搜索性能之间直接且定量的联系，表明无序并非仅为障碍，还可作为量子搜索协议中的可调参数被利用。
英文摘要
Quantum marked vertex search algorithms are known to outperform their classical counterparts, yet their behavior in the presence of disorder remains largely unexplored. Here, we address this gap by studying marked vertex search on disordered random graphs. To introduce disorder, we implement the Rosenzweig-Porter (RP) model, a random matrix ensemble with tunable ergodic, non-ergodic extended, and localized phases, on Erdős-Rényi (ER) graphs. This produces a doubly random system where ER graph connectivity randomizes which interactions exist, while RP disorder controls their strength and `on-site' potentials, providing a two-parameter framework to study quantum dynamics on disordered networks. First, we show that the characteristic Wigner-Dyson-to-Poisson spectral crossover of the RP ensemble survives under graph constraints across the sparse-to-dense range, and we derive an analytical estimate for the finite-size localization boundary that shifts systematically with the graph edge probability $p$, consistent with a resonant-hybridization argument. Thereafter, using this disordered graph ensemble, we study the marked vertex search problem and find that search performance tracks the underlying quantum phase directly. Counterintuitively, the ergodic phase, despite supporting fast transport, yields lower success probability than the localized phase, which achieves high success probability at the cost of significantly longer search times. These results establish a direct and quantitative link between random matrix disorder on graphs and the performance of continuous-time quantum walk search, and suggest that disorder, rather than being merely an obstacle, can be exploited as a tunable parameter in quantum search protocols.
Comments13 pages, 10 figures
