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Error Correction of Quantum Reference Frame Information

Hayden, Patrick and Nezami, Sepehr and Popescu, Sandu and Salton, Grant (2021) Error Correction of Quantum Reference Frame Information. PRX Quantum, 2 (1). Art. No. 010326. ISSN 2691-3399. doi:10.1103/prxquantum.2.010326.

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The existence of quantum error-correcting codes is one of the most counterintuitive and potentially technologically important discoveries of quantum-information theory. In this paper, we study a problem called “covariant quantum error correction”, in which the encoding is required to be group covariant. This problem is intimately tied to fault-tolerant quantum computation and the well-known Eastin-Knill theorem. We show that this problem is equivalent to the problem of encoding reference-frame information. In standard quantum error correction, one seeks to protect abstract quantum information, i.e., information that is independent of the physical incarnation of the systems used for storing the information. There are, however, other forms of information that are physical—one of the most ubiquitous being reference-frame information. The basic question we seek to answer is whether or not error correction of physical information is possible and, if so, what limitations govern the process. The main challenge is that the systems used for transmitting physical information, in addition to any actions applied to them, must necessarily obey these limitations. Encoding and decoding operations that obey a restrictive set of limitations need not exist a priori. Equivalently, there may not exist covariant quantum error-correcting codes. Indeed, we prove a no-go theorem showing that no finite-dimensional, group-covariant quantum codes exist for Lie groups with an infinitesimal generator [e.g., U(1), SU(2), and SO(3)]. We then explain how one can circumvent this no-go theorem using infinite-dimensional codes, and we give an explicit example of a covariant quantum error-correcting code using continuous variables for the group U(1). Finally, we demonstrate that all finite groups have finite-dimensional codes, giving both an explicit construction and a randomized approximate construction with exponentially better parameters. Our results imply that one can, in principle, circumvent the Eastin-Knill theorem.

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Salton, Grant0000-0003-3191-0325
Additional Information:© 2021 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 11 February 2019; revised 5 August 2020; accepted 12 January 2021; published 18 February 2021. We thank Dawei Ding, Iman Marvian, Michael Walter, and Beni Yoshida for helpful discussions. S.N. acknowledges support from Stanford Graduate Fellowship. G.S. acknowledges support from a NSERC postgraduate scholarship. This work is supported by the CIFAR and the Simons Foundation. The work of G.S. was performed before joining Amazon Web Services.
Group:AWS Center for Quantum Computing, Institute for Quantum Information and Matter
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Stanford UniversityUNSPECIFIED
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Canadian Institute for Advanced Research (CIFAR)UNSPECIFIED
Simons FoundationUNSPECIFIED
Issue or Number:1
Record Number:CaltechAUTHORS:20210218-151442441
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108108
Deposited By: Tony Diaz
Deposited On:18 Feb 2021 23:37
Last Modified:16 Nov 2021 19:09

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