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Protected gates for topological quantum field theories

Beverland, Michael E. and Oliver, Buerschaper and Koenig, Robert and Pastawski, Fernando and Preskill, John and Sijher, Sumit (2016) Protected gates for topological quantum field theories. Journal of Mathematical Physics, 57 (2). Art. No. 022201. ISSN 0022-2488. doi:10.1063/1.4939783.

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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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Additional Information:© 2016 AIP Publishing LLC. Received 10 August 2015; accepted 21 December 2015; published online 13 January 2016. R.K. and S.S. gratefully acknowledge support by NSERC, and M.B., F.P., and J.P. gratefully acknowledge support by NSF Grant Nos. PHY-0803371 and PHY-1125565, NSA/ARO Grant No. W911NF-09-1-0442, and AFOSR/DARPA Grant No. FA8750-12-2-0308. R.K. is supported by the Technische Universität München — Institute for Advanced Study, funded by the German Excellence Initiative and the European Union Seventh Framework Programme under Grant Agreement No. 291763. O.B. gratefully acknowledges support by the ERC (TAQ). The Institute for Quantum Information and Matter (IQIM) is a NSF Physics Frontiers Center with support by the Gordon and Betty Moore Foundation. R.K. and S.S. thank the IQIM for their hospitality. We thank Jeongwan Haah, Olivier Landon-Cardinal, and Beni Yoshida for helpful discussions, and the referees and editors for their comments.
Group:Institute for Quantum Information and Matter
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Army Research Office (ARO)W911NF-09-1-0442
Defense Advanced Research Projects Agency (DARPA)FA8750-12-2-0308
German Excellence InitiativeUNSPECIFIED
European Union FP7291763
European Research Council (ERC)UNSPECIFIED
Gordon and Betty Moore FoundationUNSPECIFIED
Institute for Quantum Informatoin and Matter (IQIM)UNSPECIFIED
Issue or Number:2
Record Number:CaltechAUTHORS:20141209-131847639
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Official Citation:Protected gates for topological quantum field theories Beverland, Michael E. and Buerschaper, Oliver and Koenig, Robert and Pastawski, Fernando and Preskill, John and Sijher, Sumit, Journal of Mathematical Physics, 57, 022201 (2016), DOI:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:52506
Deposited By: Joy Painter
Deposited On:09 Dec 2014 21:56
Last Modified:10 Nov 2021 19:41

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