# Quantum Speedups for Bayesian Network Structure Learning

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 ≤ 1.817 and c ≤ 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.

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