Hard limits on the postselectability of optical graph states

Abstract

Coherent control of large entangled graph states enables a wide variety of\nquantum information processing tasks, including error-corrected quantum\ncomputation. The linear optical approach offers excellent control and\ncoherence, but today most photon sources and entangling gates---required for\nthe construction of large graph states---are probabilistic and rely on\npostselection. In this work, we provide proofs and heuristics to aid\nexperimental design using postselection. We derive a fundamental limitation on\nthe generation of photonic qubit states using postselected entangling gates:\nexperiments which contain a cycle of postselected gates cannot be postselected.\nFurther, we analyse experiments that use photons from postselected photon pair\nsources, and lower bound the number of classes of graph state entanglement that\nare accessible in the non-degenerate case---graph state entanglement classes\nthat contain a tree are are always accessible. Numerical investigation up to\n9-qubits shows that the proportion of graph states that are accessible using\npostselection diminishes rapidly. We provide tables showing which classes are\naccessible for a variety of up to nine qubit resource states and sources. We\nalso use our methods to evaluate near-term multi-photon experiments, and\nprovide our algorithms for doing so.\n