  A central challenge for many quantum technologies concerns the generation of
large entangled states of individually addressable quantum memories. Here, we
show that percolation theory allows the rapid production of arbitrarily large
graph states by heralded photonic entanglement in a lattice of atomic memories.
This approach can greatly reduce the time required to produce large cluster
resource states for quantum information processing, including states required
for universal one-way quantum computing. This reduction puts our architecture
in an operational regime where demonstrated collection, coupling and detection
efficiencies are sufficient for generating resource states for universal
quantum computing within an experimentally demonstrated coherence time. The
approach also dispenses the need for time consuming feed-forward,
high-cooperativity interfaces and ancilla single photons, and can also tolerate
a high rate of site imperfections. We also derive the minimum coherence time
for the atomic memory to scalably create large-scale photonic-entanglement
without feed-forward as a function of collection efficiency, setting a critical
benchmark for future experimental demonstrations. We also propose a variant of
the architecture with long-range connections that makes our architecture even
more resilient to low site yields. We analyze our architecture for
nitrogen-vacancy (NV) centers in diamond, though the approach applies to any
atomic or atom-like system.
