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Continuous Symmetries and Approximate Quantum Error Correction

Faist, Philippe and Nezami, Sepehr and Albert, Victor V. and Salton, Grant and Pastawski, Fernando and Hayden, Patrick and Preskill, John (2020) Continuous Symmetries and Approximate Quantum Error Correction. Physical Review X, 10 (4). Art. No. 041018. ISSN 2160-3308. https://resolver.caltech.edu/CaltechAUTHORS:20201027-095348367

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Abstract

Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here, we study the compatibility of these two important principles. If a logical quantum system is encoded into n physical subsystems, we say that the code is covariant with respect to a symmetry group G if a G transformation on the logical system can be realized by performing transformations on the individual subsystems. For a G-covariant code with G a continuous group, we derive a lower bound on the error-correction infidelity following erasure of a subsystem. This bound approaches zero when the number of subsystems n or the dimension d of each subsystem is large. We exhibit codes achieving approximately the same scaling of infidelity with n or d as the lower bound. Leveraging tools from representation theory, we prove an approximate version of the Eastin-Knill theorem for quantum computation: If a code admits a universal set of transversal gates and corrects erasure with fixed accuracy, then, for each logical qubit, we need a number of physical qubits per subsystem that is inversely proportional to the error parameter. We construct codes covariant with respect to the full logical unitary group, achieving good accuracy for large d (using random codes) or n (using codes based on W states). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1103/physrevx.10.041018DOIArticle
https://arxiv.org/abs/1902.07714arXivDiscussion Paper
ORCID:
AuthorORCID
Albert, Victor V.0000-0002-0335-9508
Preskill, John0000-0002-2421-4762
Additional Information:© 2020 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 12 April 2019; revised 27 July 2020; accepted 7 September 2020; published 26 October 2020. During the preparation of this work, the authors became aware of an independent effort by Álvaro Alhambra and Mischa Woods to analyze how well the Eastin-Knill theorem can be evaded by allowing for a small recovery error [24]. We thank them for collegially agreeing to synchronize our arXiv posts. The authors also thank Álvaro Alhambra, Galit Anikeeva, Cédric Bény, Fernando Brandão, Elizabeth Crosson, Steve Flammia, Daniel Harlow, Liang Jiang, Tomas Jochym-O’Connor, Aleksander Kubica, Iman Marvian, Hirosi Ooguri, Burak Şahinoğlu, and Michael Walter for discussions. Ph. F. acknowledges support from the Swiss National Science Foundation (SNSF) through the Early PostDoc.Mobility fellowship No. P2EZP2_165239 hosted by the Institute for Quantum Information and Matter (IQIM) at Caltech, from the IQIM which is a National Science Foundation (NSF) Physics Frontiers Center (NSF Grant No. PHY-1733907), from the Department of Energy (DOE) Grant No. DE-SC0018407, and from the Deutsche Forschungsgemeinschaft (DFG) Research Unit FOR 2724. V. V. A. acknowledges support from the Walter Burke Institute for Theoretical Physics at Caltech. G. S. acknowledges support from the IQIM at Caltech and the Stanford Institute for Theoretical Physics. P. H. acknowledges support from CIFAR, AFOSR (FA9550-16-1-0082), DOE (DE-SC0019380), and the Simons Foundation. J. P. acknowledges support from ARO, DOE, IARPA, NSF, and the Simons Foundation. Some of this work was done during the 2017 program on “Quantum Physics of Information” at the Kavli Institute for Theoretical Physics (NSF Grant No. PHY-1748958).
Group:Institute for Quantum Information and Matter, Walter Burke Institute for Theoretical Physics
Funders:
Funding AgencyGrant Number
Swiss National Science Foundation (SNSF)P2EZP2_165239
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
NSFPHY-1733907
Department of Energy (DOE)DE-SC0018407
Deutsche Forschungsgemeinschaft (DFG)FOR 2724
Walter Burke Institute for Theoretical Physics, CaltechUNSPECIFIED
Stanford UniversityUNSPECIFIED
Canadian Institute for Advanced Research (CIFAR)UNSPECIFIED
Air Force Office of Scientific Research (AFOSR)FA9550-16-1-0082
Department of Energy (DOE)DE-SC0019380
Simons FoundationUNSPECIFIED
Army Research Office (ARO)UNSPECIFIED
Intelligence Advanced Research Projects Activity (IARPA)UNSPECIFIED
NSFPHY-1748958
Issue or Number:4
Record Number:CaltechAUTHORS:20201027-095348367
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20201027-095348367
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:106298
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:27 Oct 2020 17:04
Last Modified:27 Oct 2020 17:04

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