Disjointness of Stabilizer Codes and Limitations on Fault-Tolerant Logical Gates
Abstract
Stabilizer codes are among the most successful quantum error-correcting codes, yet they have important limitations on their ability to fault tolerantly compute. Here, we introduce a new quantity, the disjointness of the stabilizer code, which, roughly speaking, is the number of mostly nonoverlapping representations of any given nontrivial logical Pauli operator. The notion of disjointness proves useful in limiting transversal gates on any error-detecting stabilizer code to a finite level of the Clifford hierarchy. For code families, we can similarly restrict logical operators implemented by constant-depth circuits. For instance, we show that it is impossible, with a constant-depth but possibly geometrically nonlocal circuit, to implement a logical non-Clifford gate on the standard two-dimensional surface code.
Additional Information
© 2018 The Author(s). Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. (Received 28 October 2017; revised manuscript received 3 March 2018; published 21 May 2018) The authors would like to thank Ben Brown, Steve Flammia, and Daniel Gottesman for helpful discussions. In particular, we would like to thank Michael Beverland for comments on the manuscript and for also showing us that the transversal gates of all stabilizer and subsystem codes are restricted to the Clifford hierarchy in unpublished work with John Preskill [35]. T. J. acknowledges the support from the Walter Burke Institute for Theoretical Physics in the form of the Sherman Fairchild Fellowship. A. K. acknowledges funding provided by the Simons Foundation through the "It from Qubit" Collaboration, as well as by the Institute for Quantum Information and Matter (IQIM), an NSF Physics Frontiers Center (NSF Grant No. PHY-1125565), with support from the Gordon and Betty Moore Foundation (GBMF-12500028). 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. T. Y. is grateful for support from the Department of Defense (DoD) through the National Defense Science and Engineering Graduate (NDSEG) Fellowship program and also an IBM Ph.D. Fellowship award. The authors acknowledge the MIT Open Access Article Publication Subvention Fund as well as IQIM for support in making this work available for open access publication.Attached Files
Published - PhysRevX.8.021047.pdf
Submitted - 1710.07256.pdf
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Additional details
- Eprint ID
- 86524
- Resolver ID
- CaltechAUTHORS:20180521-152345918
- Walter Burke Institute for Theoretical Physics, Caltech
- Sherman Fairchild Foundation
- Simons Foundation
- Institute for Quantum Information and Matter (IQIM)
- NSF
- PHY-1125565
- Gordon and Betty Moore Foundation
- GBMF-12500028
- Industry Canada
- Ontario Ministry of Research and Innovation
- National Defense Science and Engineering Graduate (NDSEG) Fellowship
- IBM
- MIT Open Access Article Publication Subvention Fund
- Created
-
2018-05-21Created from EPrint's datestamp field
- Updated
-
2021-11-15Created from EPrint's last_modified field
- Caltech groups
- Walter Burke Institute for Theoretical Physics, Institute for Quantum Information and Matter