A fault-tolerant pairwise measurement-based code on eight qubits

We construct a pairwise measurement-based code on eight qubits that is error correcting for circuit noise, with fault distance 3. The code can be implemented on a subset of a rectangular array of qubits with nearest neighbor connectivity of pairwise Pauli measurements, with a syndrome extraction circuit of depth 28. We describe fault-tolerant logical operations on patches of this eight-qubit code that generate the full Clifford group. We estimate the performance under circuit noise both during logical idle and during a logical two-qubit measurement. We estimate the pseudo-threshold to be between $10^{-5}$ and $2\times 10^{-4}$, depending on the amount of noise on idle physical qubits. The use of post-selection in addition to error correction (correcting all degree one faults and rejecting a subset of the higher degree faults) can improve the pseudo-threshold by up to an order of magnitude.