Noise thresholds for the [4,2,2]-concatenated toric code

We analyze the properties of a 2D topological code derived by concatenating the J4, 2, 2K code with the toric/surface code, or alternatively by removing check operators from the 2D square-octagon or 4.8.8 color code. We show that the resulting code has a circuit-based noise threshold of ~ 0.41% (compared to ~ 0.6% for the toric code in a similar scenario), which is higher than any known 2D color code. We believe that the construction may be of interest for hardware in which one wants to use both long-range two-qubit gates as well as short-range gates between small clusters of qubits.