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(√ 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 [3, 14]. In this article, we show that, given some constraints on the classical algorithms, this quantum algorithm can be implemented in time Õ(√ 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(n3/2log2n) and with O(n3/2) quantum queries. Moreover, finding a maximal matching can be solved with O(n3/2) quantum queries in time O(n3/2log2n), and maximum bipartite matching can be solved in time O(n2log2n).