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.

中文翻译：

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量子加权模型计数

在加权模型计数 (WMC) 中，我们将权重分配给布尔文字，我们想要计算布尔函数模型的权重之和，其中模型的权重是其文字权重的乘积。WMC 被证明对于在图形模型中执行推理特别有效，复杂度为 $O(n2^w)$，其中 $n$ 是变量的数量，$w$ 是树的宽度。在本文中，我们提出了一种用于执行 WMC 的量子算法，即 Quantum WMC (QWMC)，它修改了量子模型计数算法以考虑权重。反过来，模型计数算法使用量子搜索、相位估计和傅立叶变换的算法。在计算的黑盒模型中，我们只能查询一个预言机来评估给定赋值的布尔函数，