图的量子嵌入用于子图计数
Quantum embedding of graphs for subgraph counting
AI总结：
本文提出一种统一的量子框架用于图中的子图计数，通过量子态编码图结构，实现子图计数的量子算法，具有O(N²)的门复杂度，展示了量子log空间算法在子图计数中的优势。
摘要：
我们开发了一个统一的量子框架用于图中的子图计数。我们将一个有N个顶点的图编码到2⌈log₂N⌉个工作量子比特和2个辅助量子比特的量子态中，使用其邻接表，最坏情况下的门复杂度为O(N²)，称为图邻接态。我们设计了量子测量算子，捕捉目标子图的边结构，通过测量邻接态的m重张量积（m为子图边数）来估计其计数。我们通过三角形、环和完全子图展示了该框架。这种方法为动机计数提供了量子log空间算法，目前没有已知的经典对应方法。
英文摘要：
We develop a unified quantum framework for subgraph counting in graphs. We encode a graph on $N$ vertices into a quantum state on $2\lceil \log_2 N \rceil$ working qubits and $2$ ancilla qubits using its adjacency list, with worst-case gate complexity $O(N^2)$, which we refer to as the graph adjacency state. We design quantum measurement operators that capture the edge structure of a target subgraph, enabling estimation of its count via measurements on the $m$-fold tensor product of the adjacency state, where $m$ is the number of edges in the subgraph. We illustrate the framework for triangles, cycles, and cliques. This approach yields quantum logspace algorithms for motif counting, with no known classical counterpart.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
