  We propose fault-tolerant architectures based on performing projective
measurements in the Greenberger-Horne-Zeilinger (GHZ) basis on constant-sized,
entangled resource states. We present linear-optical constructions of the
architectures, where the GHZ-state measurements are encoded to suppress the
errors induced by photon loss and the probabilistic nature of linear optics.
Simulations of our constructions demonstrate high single-photon loss thresholds
compared to the state-of-the-art linear-optical architecture realized with
encoded two-qubit fusion measurements performed on constant-sized resource
states. We believe this result shows a resource-efficient path to achieving
photonic fault-tolerant quantum computing.
