AI 中文总结
本文首次解析证明完全图单纯形上的最优量子搜索，确定桥权重为M时最优，权重≥√M可达最优运行时间，权重在√M至M间算法具确定性，还给出非确定性权重下找标记顶点的方法。
AI 中文摘要
完全图单纯形，又称一阶截断单纯形格，是由M+1个相同完全图构成的网络，每个完全图含M个顶点，且每个团与其他所有团之间包含一条边或桥，总顶点数N=M(M+1)。此前利用连续时间量子游走在该图上搜索单个标记顶点的渐近结果，要么数值证明了O(√N)的最优运行时间，要么解析证明了1的确定性成功概率，但即使对桥进行加权处理，也未能同时满足两者。本文首次解析证明了该图上的最优量子搜索，确定当桥的权重等于M时，最优量子搜索发生；此外数值显示，当权重至少为√M时，均可达到最优运行时间，且权重在√M到M之间时，算法具有确定性，这是完全图单纯形上量子搜索首次同时具备渐近最优性和确定性的实例。对于算法非确定性的权重范围，本文提供了通过检查相邻顶点找到标记顶点的方法。最后，尽管已知在不同图族间比较时，连通性并非快速量子搜索的可靠指标，但本文证明，在加权完全图单纯形这一图族内部，连通性同样不可靠。
英文摘要
The simplex of complete graphs, also known as the first-order truncated simplex lattice, is a network of $M+1$ identical complete graphs, each with $M$ vertices, such that each clique contains an edge or bridge to every other clique. It contains $N = M(M+1)$ vertices, and previous asymptotic results using a continuous-time quantum walk to search this graph for a single marked vertex have either numerically demonstrated an optimal runtime of $O(\sqrt{N})$, or analytically proved a deterministic success probability of 1, but not both, even when the bridges are weighted. In this paper, we give the first analytical proof of optimal quantum search on this graph, proving that it occurs when the weight of the bridges equals $M$. In addition, we numerically show that the optimal runtime is achieved more broadly whenever the weight is at least $\sqrt{M}$. Furthermore, the algorithm is also deterministic when the weight scales between $\sqrt{M}$ and $M$, and this is the first example of quantum search on the simplex of complete graphs that is both asymptotically optimal and deterministic. In addition, for weights where the algorithm is nondeterministic, we give a way to find the marked vertex by inspecting neighboring vertices. Finally, while it is known that connectivity is not a reliable indicator of fast quantum search when comparing different graph families, we show that it is also unreliable within the graph family of weighted simplex of complete graphs.
Comments37 pages, 22 figures
