Protected gates for topological quantum field theories

Journal Article · Mon Feb 15 04:00:00 EST 2016 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4939783· OSTI ID:22479628

Beverland, Michael E.; Pastawski, Fernando; Preskill, John ^[1]; Buerschaper, Oliver ^[2]; Koenig, Robert ^[3]; Sijher, Sumit ^[4]

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1. Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125 (United States)
2. Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin (Germany)
3. Institute for Advanced Study and Zentrum Mathematik, Technische Universität München, 85748 Garching (Germany)
4. Institute for Quantum Computing and Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1 (Canada)

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.

• OSTI ID: 22479628
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 2 Vol. 57; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGEBRA
ANYONS
ERRORS
HAMILTONIANS
HILBERT SPACE
LOCALITY
QUANTUM FIELD THEORY
TOPOLOGY