超图单纯形查找的量子算法
Quantum algorithms for hypergraph simplex finding
AI总结：
本文研究超图单纯形查找的量子查询算法，利用秩约化性质、嵌套 Johnson 图量子游走与自适应学习图的转换，提出 α-对称学习图框架，并将秩-4 超图中 4-单纯形查找的量子查询复杂度从 O(n^2.5) 改进到 O(n^2.46)。
摘要：
我们研究单纯形查找的量子查询算法，这是三角形查找在超图上的推广。该问题满足秩约化性质：在秩-$r$ 超图中查找单纯形的量子查询算法可以转化为在秩-$(r-1)$ 超图中查找单纯形的更快算法。随后我们证明，每一个嵌套 Johnson 图量子游走（具有任意常数个嵌套层数）都可以转换为自适应学习图。接着，我们引入 $α$-对称学习图的概念，这是设计和分析复杂量子搜索算法的有用框架。受 Le Gall、Nishimura 和 Tani（2016）关于 $3$-单纯形查找工作的启发，我们利用新技术得到一个在秩-$4$ 超图中查找 $4$-单纯形的算法，其量子查询代价为 $O(n^{2.46})$，改进了平凡的 $O(n^{2.5})$ 算法。
英文摘要：
We study the quantum query algorithms for simplex finding, a generalization of triangle finding to hypergraphs. This problem satisfies a rank-reduction property: a quantum query algorithm for finding simplices in rank-$r$ hypergraphs can be turned into a faster algorithm for finding simplices in rank-$(r-1)$ hypergraphs. We then show that every nested Johnson graph quantum walk (with any constant number of nested levels) can be converted into an adaptive learning graph. Then, we introduce the concept of $α$-symmetric learning graphs, which is a useful framework for designing and analyzing complex quantum search algorithms. Inspired by the work of Le Gall, Nishimura, and Tani (2016) on $3$-simplex finding, we use our new technique to obtain an algorithm for $4$-simplex finding in rank-$4$ hypergraphs with $O(n^{2.46})$ quantum query cost, improving the trivial $O(n^{2.5})$ algorithm.
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