The Bayesian network structure learning (BNSL) problem asks for a directed acyclic graph that maximizes a given score function. For networks with $n$ nodes, the fastest known algorithms run in time $O(2^n n^2)$ in the worst case, with no improvement in the asymptotic bound for two decades. Inspired by recent advances in quantum computing, we ask whether BNSL admits a polynomial quantum speedup, that is, whether the problem can be solved by a quantum algorithm in time $O(c^n)$ for some constant $c$ less than $2$. We answer the question in the affirmative by giving two algorithms achieving $c \le 1.817$ and $c \le 1.982$ assuming the number of potential parent sets is, respectively, subexponential and $O(1.453^n)$. Both algorithms assume the availability of a quantum random access memory. We also prove that one presumably cannot lower the base $2$ for any classical algorithm, as that would refute the strong exponential time hypothesis.