Low-distance surface codes under realistic quantum noise

Experimental implementation of the surface code will be a significant milestone for quantum computing. We develop a circuit and a decoder targeted for near-term implementation of a distance-3 surface code. We simulate the code under amplitude and phase damping and compare the threshold to a Pauli-twirl approximation. We find that the approximation yields a pessimistic threshold estimate. From numerical Monte Carlo simulations, we identify the gate and measurement speeds required to achieve reliable error correction. For superconductor devices, a qubit encoded in a 17-qubit surface code demonstrates a lower error rate than an unencoded qubit assuming gate times of 5--40 ns and T₁ times of at least 1--2 μs. If T₁≥10 ns, the difference is significant and can be experimentally measured, allowing near-term implementation and verification of a small surface code. For ion trap devices, gates times of 1 μs and T₁≥40 ms admit measurable differences in error rate.