Optimal Generation of Loss-tolerant Photonic Tree Cluster States via Time-Delayed Feedback

Photonic cluster states with tree-type encoding are an important resource for resisting photon loss. Here we develop an optimization framework for their efficient generation using time-delayed feedback. By analyzing the relation between feedback loops and entanglement structure through the adjacency matrix, we significantly reduce the resource overhead of state preparation by mapping the nonperiodic structure of the tree cluster state onto a periodic lattice. Combined with reinforcement learning, this framework identifies, under fixed resource constraints, the near-optimal tree structure for maximizing loss tolerance. Furthermore, we determine the practical loss-tolerant threshold set by the size of the near-optimal tree, the number of feedback loops, and realistic imperfections. These results provide a practical design strategy for loss-tolerant photonic cluster states in state-of-the-art quantum platforms.