Kerdock Codes Determine Unitary 2-Designs

Journal Article

The binary non-linear Kerdock codes are Gray images of {\mathbb{Z}-4}-linear Kerdock codes of length N =2m. We show that exponentiating \imath = \sqrt { - 1} by these {\mathbb{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.

Full Text

Duke Authors

Cited Authors

  • Can, T; Rengaswamy, N; Calderbank, R; Pfister, HD

Published Date

  • July 1, 2019

Published In

Volume / Issue

  • 2019-July /

Start / End Page

  • 2908 - 2912

International Standard Serial Number (ISSN)

  • 2157-8095

Digital Object Identifier (DOI)

  • 10.1109/ISIT.2019.8849504

Citation Source

  • Scopus