Scruby, TR; Vasmer, M; Browne, DE; (2022) Non-Pauli errors in the three-dimensional surface code. Physical Review Research , 4 (4) , Article 043052. 10.1103/PhysRevResearch.4.043052.

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Abstract

A powerful feature of stabilizer error correcting codes is the fact that stabilizer 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 stabilizer 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 color code and shown the existence of a nonlocal "linking charge"phenomenon between membranes of intersecting errors. In this paper 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 nonlocal 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 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 stabilizer measurement.

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