Non-Pauli Errors in the Three-Dimensional Surface Code

TL;DR

This paper investigates how non-Pauli errors, induced by logical gates like CCZ, affect the performance of the 3D surface code, revealing local linking charge phenomena and their impact on error thresholds.

Contribution

It generalizes previous results to the 3D surface code with CCZ gates, showing that linking charge is a local effect and simulating its impact on magic state preparation.

Findings

Linking charge is a local phenomenon in the 3D surface code.

Non-Pauli errors influence the error threshold mainly through pre-gate X error probabilities.

Simulation of full error effects on magic state prep is performed for the first time.

Abstract

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