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.
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In the language of this paper, braiding/mapping class group elements belong to locality-preserving unitaries if the model is abelian. However, for a general non-abelian model, braiding is not locality-preserving according to our definition.
In principle, we could consider unitaries/isometries (or sequences thereof) of the form which map between different systems and . By a slight modification of the arguments here, we could then obtain restrictions on locality-preserving isomorphisms (instead of automorphisms, cf. Section III). Such a scenario was discussed in Ref. 10 in the context of stabilizer codes. Here, we restrict to the case where the systems (and associated ground spaces) are identical for simplicity, since the main conclusions are identical.
As a side remark, we mention that our terminology is chosen with spin lattices in mind. However, the notion of locality-preservation can be relaxed. As will become obvious below, our results apply more generally to the set of homology-preserving automorphisms U. The latter can be defined as follows: if the support of an operator X is contained in a region which deformation retracts to a closed curve C, then the support of UXU† must be contained in a region which deformation retracts to a curve C′ in the same homology class as C. For example, for a translation-invariant system, translating by a possibly extensive amount realizes such a homology-preserving (but not locality-preserving) automorphism.
More precisely, for B(xk−1, xk+1), the relevant matrices are and is diagonal with entries . Here, F is the F-matrix associated with basis changes on the four-punctured sphere (see Section V A), whereas determines an isomorphism between certain Hilbert spaces associated with the three-punctured sphere. We refer to, e.g., Ref. 39 (p. 48) for a derivation of these expressions.
