Noise Thresholds for the [[4, 2, 2]]-concatenated Toric Code

We analyze the properties of a 2D topological code derived by concatenating the [[4, 2, 2]] 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 $\sim 0.41\mathrm{\%}$ (compared to $\sim 0.6\mathrm{\%}$ 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.