Quantum Lower Bound for Graph Collision Implies Lower Bound for Triangle Detection

We show that an improvement to the best known quantum lower bound for Graph-Collision problem implies an improvement to the best known lower bound for Triangle problem in the quantum query complexity model. In Graph-Collision we are given free access to a graph (V,E) and access to a function f : V → {0, 1} as a black box. We are asked to determine if there exist (u, v) ∈ E, such that f(u) = f(v) = 1. In Triangle we have a black box access to an adjacency matrix of a graph and we have to determine if the graph contains a triangle. For both of these problems the known lower bounds are trivial (Ω( √ n) and Ω(n), respectively) and there is no known matching upper bound.