Span-program-based quantum algorithm for tree detection

Span program is a linear-algebraic model of computation originally proposed for studying the complexity theory. Recently, it has become a useful tool for designing quantum algorithms. In this paper, we present a time-efficient span-program-based quantum algorithm for the following problem. Let T be an arbitrary tree. Given query access to the adjacency matrix of a graph G with n vertices, we need to determine whether G contains T as a subgraph, or G does not contain T as a minor, under the promise that one of these cases holds. We call this problem the subgraph/not-a-minor problem for T . We show that this problem can be solved by a bounded-error quantum algorithm with O(n) query complexity and Õ(n) time complexity. The query complexity is optimal, and the time complexity is tight up to polylog factors.