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Kerdock Codes Determine Unitary 2-Designs

Can, Trung and Rengaswamy, Narayanan and Calderbank, Robert and Pfister, Henry D. (2019) Kerdock Codes Determine Unitary 2-Designs. In: 2019 IEEE International Symposium on Information Theory (ISIT). IEEE , Piscataway, NJ, pp. 2908-2912. ISBN 9781538692912. https://resolver.caltech.edu/CaltechAUTHORS:20191004-100333074

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Abstract

The binary non-linear Kerdock codes are Gray images of ℤ_4-linear Kerdock codes of length N =2^m . We show that exponentiating ı=−√-1 by these ℤ_4-valued codewords produces stabilizer states, which are the common eigenvectors of maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the proof of the classical weight distribution of Kerdock codes. Next, we partition stabilizer states into N +1 mutually unbiased bases and prove that automorphisms of the Kerdock code permute the associated MCS. This automorphism group, represented as symplectic matrices, is isomorphic to the projective special linear group PSL(2,N) and forms a unitary 2-design. The design described here was originally discovered by Cleve et al. (2016), but the connection to classical codes is new. This significantly simplifies the description of the design and its translation to circuits.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
https://doi.org/10.1109/isit.2019.8849504DOIArticle
https://arxiv.org/abs/1904.07842arXivDiscussion Paper
https://resolver.caltech.edu/CaltechAUTHORS:20201012-134120392Related ItemJournal Article
Additional Information:© 2019 IEEE.
Subject Keywords:Heisenberg-Weyl group, Pauli group, quantum computing, Clifford group, symplectic geometry, Kerdock codes, Delsarte-Goethals codes, Gray map, stabilizer states, mutually unbiased bases, unitary t-designs
Record Number:CaltechAUTHORS:20191004-100333074
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20191004-100333074
Official Citation:T. Can, N. Rengaswamy, R. Calderbank and H. D. Pfister, "Kerdock Codes Determine Unitary 2-Designs," 2019 IEEE International Symposium on Information Theory (ISIT), Paris, France, 2019, pp. 2908-2912. doi: 10.1109/ISIT.2019.8849504
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:99077
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:04 Oct 2019 19:39
Last Modified:12 Oct 2020 20:42

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