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Topological color code and symmetry-protected topological phases

Yoshida, Beni (2015) Topological color code and symmetry-protected topological phases. Physical Review B, 91 (24). Art. No. 245131. ISSN 1098-0121. http://resolver.caltech.edu/CaltechAUTHORS:20150706-143013854

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Abstract

We study (d−1)-dimensional excitations in the d-dimensional color code that are created by transversal application of the R_d phase operators on connected subregions of qubits. We find that such excitations are the superpositions of electric charges and can be characterized by the fixed-point wave functions of (d−1)-dimensional bosonic symmetry-protected topological (SPT) phases with (Z_2)^(⊗d) symmetry. While these SPT excitations are localized on (d−1)-dimensional boundaries, their creation requires operations acting on all qubits inside the boundaries, reflecting the nontriviality of emerging SPT wave functions. Moreover, these SPT excitations can be physically realized as transparent gapped domain walls which exchange excitations in the color code. Namely, in the three-dimensional color code, the domain wall, associated with the transversal R_3 operator, exchanges a magnetic flux and a composite of a magnetic flux and the looplike SPT excitation, revealing rich possibilities of boundaries in higher-dimensional TQFTs. We also find that magnetic fluxes and the looplike SPT excitations exhibit nontrivial three-loop braiding statistics in three dimensions as a result of the fact that the R_3 phase operator belongs to the third level of the Clifford hierarchy. We believe that the connection between SPT excitations, fault-tolerant logical gates and gapped domain walls, established in this paper, can be generalized to a large class of topological quantum codes and TQFTs.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1103/PhysRevB.91.245131DOIArticle
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.91.245131PublisherArticle
Additional Information:© 2015 American Physical Society. Received 7 April 2015; published 15 June 2015. I would like to thank Xie Chen, Isaac Kim, Aleksander Kubica, Iman Marvian, Fernando Pastawski, John Preskill, and Sujeet Shukla for discussions and/or comments. I would like to thank Aleksander Kubica for careful reading of the manuscript. Part of the work was completed during the visit to the Perimeter Institute for theoretical physics. I am supported by the David and Ellen Lee Postdoctoral fellowship. I 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).
Group:IQIM, Institute for Quantum Information and Matter, Walter Burke Institute for Theoretical Physics
Funders:
Funding AgencyGrant Number
David and Ellen Lee Postdoctoral ScholarshipUNSPECIFIED
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
NSFPHY-1125565
NSFPHY-0803371
Gordon and Betty Moore FoundationUNSPECIFIED
Classification Code:PACS: 71.27.+a; 03.67.−a.
Record Number:CaltechAUTHORS:20150706-143013854
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20150706-143013854
Official Citation:Yoshida, B. (2015). Topological color code and symmetry-protected topological phases. Physical Review B, 91(24), 245131.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:58783
Collection:CaltechAUTHORS
Deposited By: Joanne McCole
Deposited On:06 Jul 2015 22:07
Last Modified:06 Jul 2015 22:07

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