Efficient optimization and conceptual barriers in variational finite projected entangled pair states

The authors investigate here a sampling-based optimization scheme for finite projected entangled pair states (PEPS), uncovering an entanglement phase transition in random PEPS that corresponds to a computational complexity shift in tensor network contractions, and extend the complexity analysis to physically relevant states. The method is applied to challenging systems, including a chiral spin liquid and Rydberg atom arrays with long-range interactions, demonstrating its ability to model state-of-the-art cold atom experiments.