Protected gates for topological quantum field theories

2014/09/30 by Michael E. Beverland, Oliver Buerschaper, Robert Koenig +3 · 54 citations
Computer Science · Mathematics · Physics and Astronomy · #Anyon #Combinatorics #Locality #Mathematics #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum and electron transport phenomena #Quantum computer #Quantum gate #Quantum many-body systems #Quantum mechanics #Topological quantum computer #Topology (electrical circuits) #cond-mat.str-el #quant-ph

paper · pdf · doi:10.1063/1.4939783

published in Journal of Mathematical Physics 57(2) (American Institute of Physics) · 50 pages, many figures, v3: updated to match published version

openalex publication_date 2016/01/13 · arxiv created 2016/01/25 · arxiv updated 2016/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

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.

Citations

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