Hard limits on the postselectability of optical graph states

TL;DR

This paper establishes fundamental limitations on creating large photonic graph states with postselected gates, showing that certain configurations are impossible and that the accessible entanglement classes decrease rapidly with size.

Contribution

It provides theoretical proofs and heuristics that define the limits of postselected optical graph state generation, including cycle restrictions and class accessibility bounds.

Findings

Experiments with cycles of postselected gates cannot be postselected.

Graph states containing a tree are always accessible.

The proportion of accessible graph states decreases rapidly with qubit number.

Abstract

Coherent control of large entangled graph states enables a wide variety of quantum information processing tasks, including error-corrected quantum computation. The linear optical approach offers excellent control and coherence, but today most photon sources and entangling gates---required for the construction of large graph states---are probabilistic and rely on postselection. In this work, we provide proofs and heuristics to aid experimental design using postselection. We derive a fundamental limitation on the generation of photonic qubit states using postselected entangling gates: experiments which contain a cycle of postselected gates cannot be postselected. Further, we analyse experiments that use photons from postselected photon pair sources, and lower bound the number of classes of graph state entanglement that are accessible in the non-degenerate case---graph state entanglement classes that contain a tree are are always accessible. Numerical investigation up to 9-qubits shows that the proportion of graph states that are accessible using postselection diminishes rapidly. We provide tables showing which classes are accessible for a variety of up to nine qubit resource states and sources. We also use our methods to evaluate near-term multi-photon experiments, and provide our algorithms for doing so.