Tailoring fusion-based error correction for high thresholds to biased fusion failures

Abstract: We introduce fault-tolerant (FT) architectures for error correction with the XZZX cluster state based on performing measurements of two-qubit Pauli operators $Z\otimes Z$ and $X\otimes X$, or fusions, on a collection of few-body entangled resource states. Our construction is tailored to be effective against noise that predominantly causes faulty $X\otimes X$ measurements during fusions. This feature offers practical advantage in linear optical quantum computing with dual-rail photonic qubits, where failed fusions only erase $X\otimes X$ measurement outcomes. By applying our construction to this platform, we find a record high FT threshold to fusion failures exceeding $25\%$ in the experimentally relevant regime of non-zero loss rate per photon, considerably simplifying hardware requirements.