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 $O(n{2}^{m}P{(e)}^{$-${}\frac{1}{2}})$ 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\char22{}we count primitive operations and require no blackbox oracle queries.