Error-corrected Hadamard gate simulated at the circuit level

We simulate the logical Hadamard gate in the surface code under a circuit-level noise model, compiling it to a physical circuit on square-grid connectivity hardware. Our paper is the first to do this for a logical unitary gate on a quantum error-correction code. We consider two proposals, both via patch-deformation: one that applies a transversal Hadamard gate (i.e. a domain wall through time) to interchange the logical $X$ and $Z$ strings, and another that applies a domain wall through space to achieve this interchange. We explain in detail why they perform the logical Hadamard gate by tracking how the stabilisers and the logical operators are transformed in each quantum error-correction round. We optimise the physical circuits and evaluate their logical failure probabilities, which we find to be comparable to those of a quantum memory experiment for the same number of quantum error-correction rounds. We present syndrome-extraction circuits that maintain the same effective distance under circuit-level noise as under phenomenological noise. We also explain how a $SWAP$-quantum error-correction round (required to return the patch to its initial position) can be compiled to only four two-qubit gate layers. This can be applied to more general scenarios and, as a byproduct, explains from first principles how the "stepping" circuits of the recent Google paper [McEwen, Bacon, and Gidney, Quantum 7, 1172 (2023)] can be constructed.