Strongly simulating a quantum circuit, that is, computing an output amplitude, amounts to summing the circuit's Feynman paths, a weighted count over assignments to the Boolean ``path'' variables. The circuit's gates induce correlations among these variables, forming a graph whose structure determines the hardness of the simulation task. This sum-of-powers viewpoint underlies recent simulators built on knowledge-representation tools from artificial intelligence, namely binary decision diagrams and weighted model counting. We show that the structural quantity most accurately governing the difficulty is the rank-width of the path-variable graph, and we give an algorithm that evaluates the amplitude in time that is exponential only in this rank-width and polynomial in the circuit size. Rank-width can be far smaller than the widths that control competing methods: as corollaries, our algorithm reproduces a recent decision-diagram simulation breakthrough as a special case and matches the Markov--Shi tensor-network contraction bound. To complement this, we exhibit circuit families on which our algorithm provably beats both competing methods. The new method applies to every circuit built from Hadamard and diagonal gates, in particular to circuits over Clifford+T. In practical terms, general-purpose decision-diagram and model-counting tools can serve as the workhorse, with our specialized algorithm dispatched to exploit a small rank-width of the associated graph when it is present.

翻译：强模拟一个量子电路，即计算其输出振幅，等价于对电路中费曼路径求和，即对布尔型“路径”变量赋值进行加权计数。电路中的门在这些变量间诱导出关联，形成一张图，其结构决定了模拟任务的难度。这种“幂和”观点为近年来基于人工智能知识表示工具（即二叉决策图和加权模型计数）构建的模拟器提供了理论基础。我们发现，最准确刻画模拟难度的结构量是路径变量图的秩宽，并给出一种算法，该算法计算振幅的时间复杂度与秩宽呈指数关系，而与电路规模呈多项式关系。秩宽可能远小于控制竞争对手方法的宽度：作为推论，我们的算法将近期决策图模拟的突破性成果作为特例复现，并匹配了马尔可夫-施张量网络收缩的界限。此外，我们展示了算法在特定电路族上能严格超越两类竞争对手。新方法适用于所有由哈达玛门和对角门构成的电路，特别是克利福德+T电路。在实践中，通用决策图和模型计数工具可作为主引擎，而我们的专用算法可在关联图呈现小秩宽时被调用以发挥优势。