The disjointness of stabilizer codes and limitations on fault-tolerant logical gates

TL;DR

This paper introduces the disjointness metric for stabilizer codes, revealing fundamental limitations on fault-tolerant logical gates, especially constraining non-Clifford operations in certain topological codes.

Contribution

The paper defines disjointness for stabilizer codes and uses it to prove bounds on the levels of the Clifford hierarchy achievable by transversal and constant-depth circuits.

Findings

Transversal gates on stabilizer codes are limited to a finite Clifford hierarchy level.

Symmetric 2D surface codes cannot implement non-Clifford gates with non-local constant depth circuits.

Disjointness provides a new tool to understand fault-tolerance limitations in quantum codes.

Abstract

Stabilizer codes are a simple and successful class of quantum error-correcting codes. Yet this success comes in spite of some harsh limitations on the ability of these codes to fault-tolerantly compute. Here we introduce a new metric for these codes, the disjointness, which, roughly speaking, is the number of mostly non-overlapping representatives of any given non-trivial logical Pauli operator. We use the disjointness to prove that transversal gates on error-detecting stabilizer codes are necessarily in a finite level of the Clifford hierarchy. We also apply our techniques to topological code families to find similar bounds on the level of the hierarchy attainable by constant depth circuits, regardless of their geometric locality. For instance, we can show that symmetric 2D surface codes cannot have non-local constant depth circuits for non-Clifford gates.