Resource-efficient photonic quantum computation with high-dimensional cluster states

Quantum computers can revolutionize science and technology, but their realization remains challenging across all platforms. A promising route to scalability is photonic-measurement-based quantum computation, where single-qubit measurements on large cluster states, together with feedforward steps, enable fault-tolerant quantum computation; however, generating large cluster states at high rates is notoriously difficult as detection probabilities drop exponentially with the number of photons comprising the state. We tackle this challenge by encoding multiple qubits on each photon through high-dimensional spatial encoding, generating cluster states with over nine qubits at a rate of 100 Hz. We also demonstrate that high-dimensional encoding substantially reduces the computation duration by enabling instantaneous feedforward between qubits encoded in the same photon. Our findings pave the way for resource-efficient measurement-based quantum computation using high-dimensional entanglement.