Non-Pauli errors in the three-dimensional surface code

A powerful feature of stabiliser error correcting codes is the fact that measurement projects arbitrary errors to Pauli errors, greatly simplifying physical correction process as well classical simulations code performance. However, logical non-Clifford operations can map non-Pauli (Clifford) and while subsequent measurements will project Clifford back resulting distributions possess additional correlations depend on both nature operation structure code. Previous work has studied these effects when applying a transversal $T$ gate three-dimensional colour shown existence non-local "linking charge" phenomenon between membranes intersecting errors. In this we generalise results case $CCZ$ in surface find many aspects problem are much more easily understood setting. particular, emergence linking charge local effect rather than one. We use relative simplicity setting simulate their performance single-shot magic state preparation (the first such simulation account for full errors) threshold largely determined by probability $X$ occurring immediately prior application gate, after most recent measurement.