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Unfolding the color code

Kubica, Aleksander and Yoshida, Beni and Pastawski, Fernando (2015) Unfolding the color code. New Journal of Physics, 17 (8). Art. No. 083026. ISSN 1367-2630. doi:10.1088/1367-2630/17/8/083026.

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The topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. Here we show that the color code on a d-dimensional closed manifold is equivalent to multiple decoupled copies of the d-dimensional toric code up to local unitary transformations and adding or removing ancilla qubits. Our result not only generalizes the proven equivalence for d = 2, but also provides an explicit recipe of how to decouple independent components of the color code, highlighting the importance of colorability in the construction of the code. Moreover, for the d-dimensional color code with d + 1 boundaries of d + 1 distinct colors, we find that the code is equivalent to multiple copies of the d-dimensional toric code which are attached along a (d - 1)-dimensional boundary. In particular, for d = 2, we show that the (triangular) color code with boundaries is equivalent to the (folded) toric code with boundaries. We also find that the d-dimensional toric code admits logical non-Pauli gates from the dth level of the Clifford hierarchy, and thus saturates the bound by Bravyi and König. In particular, we show that the logical d-qubit control-Z gate can be fault-tolerantly implemented on the stack of d copies of the toric code by a local unitary transformation.

Item Type:Article
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Additional Information:© 2015 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Received 31 March 2015; Revised 9 June 2015; Accepted for publication 25 June 2015; Published 13 August 2015. We would like to thank John Preskill, Olivier Landon-Cardinal and Dan Browne for helpful discussions. We acknowledge funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NFS Grant PHY-1125565) with support of the Gordon and Betty Moore Foundation (GBMF-12500028). BY is supported by the David and Ellen Lee Postdoctoral fellowship.
Group: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 FoundationGBMF-12500028
David and Ellen Lee Postdoctoral ScholarshipUNSPECIFIED
Subject Keywords:quantum codes, transversal gates, fault-tolerant quantum computation, color code, topological order, boundaries and domain walls, classification of quantum phases
Issue or Number:8
Record Number:CaltechAUTHORS:20151001-082745067
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:60647
Deposited By: Tony Diaz
Deposited On:01 Oct 2015 17:00
Last Modified:12 Jul 2022 19:40

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