Quantum Weighted Model Counting

Abstract: In Weighted Model Counting (WMC) we assign weights to Boolean literals and we want to compute the sum of the weights of the models of a Boolean function where the weight of a model is the product of the weights of its literals. WMC was shown to be particularly effective for performing inference in graphical models, with a complexity of $O(n2^w)$ where $n$ is the number of variables and $w$ is the treewidth. In this paper, we propose a quantum algorithm for performing WMC, Quantum WMC (QWMC), that modifies the quantum model counting algorithm to take into account the weights. In turn, the model counting algorithm uses the algorithms of quantum search, phase estimation and Fourier transform. In the black box model of computation, where we can only query an oracle for evaluating the Boolean function given an assignment, QWMC solves the problem approximately with a complexity of $\Theta(2^{\frac{n}{2}})$ oracle calls while classically the best complexity is $\Theta(2^n)$, thus achieving a quadratic speedup.