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 $O(n^2\log n)$.

中文翻译：

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基于决策树的时间和查询最优量子算法

最近显示，从经典查询算法（决策树）和试图预测查询答案的猜测算法开始，我们可以设计一种查询复杂度为$ O（\ sqrt {GT}）$的量子算法，其中$ T $是经典算法的查询复杂度（决策树的深度），而$ G $是猜测算法的最大错误答案数[arXiv：1410.0932，arXiv：1905.13095]。在本文中，我们证明，在经典算法受某些约束的情况下，该量子算法可以在$ \ tilde O（\ sqrt {GT}）$的时间内实现。我们的算法基于非二进制跨度程序及其有效实现。我们得出结论，可以在$ O（n ^ {3/2} \ log n）$的时间和$ O（n ^ {3/2}）$量子的情况下解决各种图形理论问题，包括二部性，周期检测和拓扑排序查询。