We study the optimal scheduling of graph states in measurement-based quantum computation, establishing an equivalence between measurement schedules and path decompositions of graphs. We define the spatial cost of a measurement schedule based on the number of simultaneously active qubits and prove that an optimal measurement schedule corresponds to a path decomposition of minimal width. Our analysis shows that approximating the spatial cost of a graph is \textsf{NP}-hard, while for graphs with bounded spatial cost, we establish an efficient algorithm for computing an optimal measurement schedule.

翻译：我们研究了测量基量子计算中图态的最优调度问题，建立了测量调度与图路径分解之间的等价关系。根据同时激活量子比特的数量，我们定义了测量调度的空间成本，并证明最优测量调度对应于最小宽度的路径分解。分析表明，近似计算图的空间成本是\textsf{NP}-难的，而对于具有有界空间成本的图，我们建立了计算最优测量调度的高效算法。