Time and Query Optimal Quantum Algorithms Based on Decision Trees

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