Classification of topologically protected gates for local stabilizer codes

TL;DR

This paper classifies which encoded quantum gates can be implemented by constant-depth circuits in topological stabilizer codes, revealing limitations in fault-tolerant quantum computation and identifying some non-Clifford gates in 3D systems.

Contribution

It provides a comprehensive classification of topologically protected gates achievable with constant-depth circuits in 2D and 3D stabilizer codes, highlighting fundamental constraints.

Findings

2D codes only allow Clifford gates with constant-depth circuits.

3D codes permit certain non-Clifford gates like /8 rotations.

Topological protection cannot support universal gates without breaking some protection.

Abstract

Given a quantum error correcting code, an important task is to find encoded operations that can be implemented efficiently and fault-tolerantly. In this Letter we focus on topological stabilizer codes and encoded unitary gates that can be implemented by a constant-depth quantum circuit. Such gates have a certain degree of protection since propagation of errors in a constant-depth circuit is limited by a constant size light cone. For the 2D geometry we show that constant-depth circuits can only implement a finite group of encoded gates known as the Clifford group. This implies that topological protection must be "turned off" for at least some steps in the computation in order to achieve universality. For the 3D geometry we show that an encoded gate U is implementable by a constant-depth circuit only if the image of any Pauli operator under conjugation by U belongs to the Clifford group. This class of gates includes some non-Clifford gates such as the \pi/8 rotation. Our classification applies to any stabilizer code with geometrically local stabilizers and sufficiently large code distance.