Protected gates for topological quantum field theories

TL;DR

This paper investigates the limitations of locality-preserving logical gates in two-dimensional topological quantum codes, showing they are severely restricted for non-abelian anyons and must be Pauli gates for Ising anyons, impacting fault-tolerant quantum computation.

Contribution

It establishes fundamental constraints on fault-tolerant logical gates in topological quantum codes using topological field theory and algebraic methods, especially for non-abelian anyons.

Findings

No locality-preserving gates for codes with universal braiding.

Local gates for Ising anyons are limited to the Pauli group.

Constraints derived from automorphisms of the Verlinde algebra.

Abstract

We study restrictions on locality-preserving unitary logical gates for topological quantum codes in two spatial dimensions. A locality-preserving operation is one which maps local operators to local operators --- for example, a constant-depth quantum circuit of geometrically local gates, or evolution for a constant time governed by a geometrically-local bounded-strength Hamiltonian. Locality-preserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized, and hence sufficiently dilute errors remain correctable. By invoking general properties of two-dimensional topological field theories, we find that the locality-preserving logical gates are severely limited for codes which admit non-abelian anyons; in particular, there are no locality-preserving logical gates on the torus or the sphere with M punctures if the braiding of anyons is computationally universal. Furthermore, for Ising anyons on the M-punctured sphere, locality-preserving gates must be elements of the logical Pauli group. We derive these results by relating logical gates of a topological code to automorphisms of the Verlinde algebra of the corresponding anyon model, and by requiring the logical gates to be compatible with basis changes in the logical Hilbert space arising from local F-moves and the mapping class group.