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Gapped boundaries, group cohomology and fault-tolerant logical gates

Yoshida, Beni (2017) Gapped boundaries, group cohomology and fault-tolerant logical gates. Annals of Physics, 377 . pp. 387-413. ISSN 0003-4916. doi:10.1016/j.aop.2016.12.014.

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This paper attempts to establish the connection among classifications of gapped boundaries in topological phases of matter, bosonic symmetry-protected topological (SPT) phases and fault-tolerantly implementable logical gates in quantum error-correcting codes. We begin by presenting constructions of gapped boundaries for the d-dimensional quantum double model by using d-cocycles functions (d≥2). We point out that the system supports mm-dimensional excitations (m<d), which we shall call fluctuating charges, that are superpositions of point-like electric charges characterized by mm-dimensional bosonic SPT wavefunctions. There exist gapped boundaries where electric charges or magnetic fluxes may not condense by themselves, but may condense only when accompanied by fluctuating charges. Magnetic fluxes and codimension-2 fluctuating charges exhibit non-trivial multi-excitation braiding statistics, involving more than two excitations. The statistical angle can be computed by taking slant products of underlying cocycle functions sequentially. We find that excitations that may condense into a gapped boundary can be characterized by trivial multi-excitation braiding statistics, generalizing the notion of the Lagrangian subgroup. As an application, we construct fault-tolerantly implementable logical gates for the d-dimensional quantum double model by using d-cocycle functions. Namely, corresponding logical gates belong to the dth level of the Clifford hierarchy, but are outside of the (d−1))th level, if cocycle functions have non-trivial sequences of slant products.

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Additional Information:© 2016 Elsevier Inc. Received 17 October 2016, Accepted 6 December 2016, Available online 14 December 2016. I would like to thank Isaac Kim, Fernando Pastawski, John Preskill and Chenjie Wang for helpful discussions and/or comments. Part of the work was completed during the visits to the Kavli Institute for Theoretical Physics. We acknowledge funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation (Grants No. PHY-0803371 and PHY-1125565). I was supported by the David and Ellen Lee Postdoctoral fellowship. This research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
Group:Walter Burke Institute for Theoretical Physics, Institute for Quantum Information and Matter
Funding AgencyGrant Number
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
NSF Physics Frontiers CenterUNSPECIFIED
Gordon and Betty Moore FoundationUNSPECIFIED
David and Ellen Lee Postdoctoral FellowshipUNSPECIFIED
Industry CanadaUNSPECIFIED
Ontario Ministry of Research and InnovationUNSPECIFIED
Subject Keywords:Gapped boundary; Quantum error-correcting code; Topological order; Fault-tolerance
Record Number:CaltechAUTHORS:20151123-103215851
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Official Citation:Beni Yoshida, Gapped boundaries, group cohomology and fault-tolerant logical gates, Annals of Physics, Volume 377, February 2017, Pages 387-413, ISSN 0003-4916, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:62326
Deposited By: Tony Diaz
Deposited On:23 Nov 2015 19:00
Last Modified:10 Nov 2021 23:00

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