Quantum inference on Bayesian networks

Performing exact inference on Bayesian networks is known to be $#P$-hard. Typically approximate inference techniques are used instead to sample from the distribution on query variables given the values e of evidence variables. Classically, a single unbiased sample is obtained from a Bayesian network on n variables with at most m parents per node in time O(nmP(e)^-1), depending critically on $P(e)$, the probability that the evidence might occur in the first place. By implementing a quantum version of rejection sampling, we obtain a square-root speedup, taking O(n2ᵐP(e)^-1/2) time per sample. We exploit the Bayesian network's graph structure to efficiently construct a quantum state, a q-sample, representing the intended classical distribution, and also to efficiently apply amplitude amplification, the source of our speedup. Thus, our speedup is notable as it is unrelativized---we count primitive operations and require no blackbox oracle queries.