High-threshold fault-tolerance in measurement-based error correction with tailored fusion circuits

A practical approach to implement measurement-based error correction (MBEC) is by stitching together many copies of few-body resource states into a larger entangled state, called the cluster state, using fusions or Bell measurements. Fusions measure Z⊗Z and X⊗X correlations. We propose new four- and six- body resource states that can be input to fusion circuits to build the RHG cluster state and the recently discovered XZZX cluster state. Our construction offers a practical advantage in realistic platforms in which the outcome of an X⊗X measurement in a fusion is more likely to be unreliable than the outcome of a Z⊗Z measurement. This is because the resource states are tailored so that errors introduced in these cluster states due to faulty X⊗X measurements obey a parity conservation law, which gives rise to a two-dimensional system symmetry. This considerably simplifies the decoding problem and can lead to substantial improvement in thresholds. We study the applicability of fusion-based MBEC using these resource states in (i) two specific qubit platforms: dual-rail photonic qubits in linear optics and Yb atoms and (ii) a broad class of qubits in which dephasing errors are dominant. In all cases, we find a simplification in the requirements for fault-tolerant error correction.