It has recently been shown that starting with a classical query algorithm (decision tree) and a guessing algorithm that tries to predict the query answers, we can design a quantum algorithm with query complexity $O(\sqrt{GT})$ where $T$ is the query complexity of the classical algorithm (depth of the decision tree) and $G$ is the maximum number of wrong answers by the guessing algorithm [arXiv:1410.0932, arXiv:1905.13095]. In this paper we show that, given some constraints on the classical algorithms, this quantum algorithm can be implemented in time $\tilde O(\sqrt{GT})$. Our algorithm is based on non-binary span programs and their efficient implementation. We conclude that various graph theoretic problems including bipartiteness, cycle detection and topological sort can be solved in time $O(n^{3/2}\log n)$ and with $O(n^{3/2})$ quantum queries. Moreover, finding a maximal matching can be solved with $O(n^{3/2})$ quantum queries in time $O(n^{3/2}\log n)$, and maximum bipartite matching can be solved in time $O(n^2\log n)$.
翻译:最近已经表明,从古典查询算法(决定树)和试图预测查询答案的猜测算法开始,我们可以设计一个量子算法,其质询复杂度为$O(sqrt{GT})美元,其中美元是古典算法(决定树深度)的质询复杂性,美元是G$G$,这是猜测算法[arXiv:141.0932,arXiv:1905.130955]的最大错误回答数。在本文中,我们显示,鉴于古典算法的一些限制,这种量子算法可以及时执行$\telde O(sqrt{GT})美元。我们的算法以非双线跨线程序及其高效实施为基础。我们的结论是,各种图表学问题,包括双线性、周期检测和表面学类型的问题,可以用时间($(n{%3/2 ⁇ )和$O(n_____}量子查询解决。此外,在时间(n__________}中,可以找到一个最高匹配值(n__________xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx