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Logical-operator tradeoff for local quantum codes

Haah, Jeongwan and Preskill, John (2012) Logical-operator tradeoff for local quantum codes. Physical Review A, 86 (3). Art No. 032308. ISSN 1050-2947. doi:10.1103/PhysRevA.86.032308.

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We study the structure of logical operators in local D-dimensional quantum codes, considering both subsystem codes with geometrically local gauge generators and codes defined by geometrically local commuting projectors. We show that if the code distance is d, then any logical operator can be supported on a set of specified geometry containing ˜d qubits, where ˜dd^(1/(D−1)) = O(n) and n is the code length. Our results place limitations on partially self-correcting quantum memories, in which at least some logical operators are protected by energy barriers that grow with system size. We also show that for any two-dimensional local commuting projector code there is a nontrivial logical “string” operator supported on a narrow strip, where the operator is only slightly entangling across any cut through the strip.

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Additional Information:© 2012 American Physical Society. Received 5 July 2012; published 7 September 2012. We are grateful to Salman Beigi, Alexei Kitaev, Robert König, Olivier Landon-Cardinal, and Norbert Schuch for helpful discussions, and we especially thank David Poulin for useful comments on the manuscript. This research was supported in part by NSF under Grant No. PHY-0803371, by DOE under Grant No. DE-FG03-92-ER40701, by NSA/ARO under Grant No. W911NF-09-1-0442, and by the Korea Foundation for Advanced Studies. The Institute for Quantum Information and Matter (IQIM) is an NSF Physics Frontiers Center with support from the Gordon and Betty Moore Foundation.
Group:Institute for Quantum Information and Matter, Caltech Theory
Funding AgencyGrant Number
Department of Energy (DOE)DE-FG03-92-ER40701
National Security Agency (NSA)/Army Research Office (ARO)W911NF-09-1-0442
Korea Foundation for Advanced StudiesUNSPECIFIED
Gordon and Betty Moore FoundationUNSPECIFIED
Issue or Number:3
Classification Code:PACS: 03.67.Pp, 03.67.Lx
Record Number:CaltechAUTHORS:20121012-142001671
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Official Citation:Haah, J. and J. Preskill (2012). "Logical-operator tradeoff for local quantum codes." Physical Review A 86(3): 032308.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:34880
Deposited On:12 Oct 2012 23:14
Last Modified:09 Nov 2021 23:11

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