Subsystem stabilizer codes cannot have a universal set of transversal gates for even one encoded qudit

Xie Chen, Hyeyoun Chung, Andrew W. Cross, Bei Zeng, and Isaac L. Chuang
Phys. Rev. A 78, 012353 – Published 28 July 2008

Abstract

A long-standing open problem in fault-tolerant quantum computation has been to find a universal set of transversal gates. As shown by Zeng et al. (e-print arXiv:0706.1382), such a set does not exist for binary stabilizer codes. Here we generalize our work to show that for subsystem stabilizer codes in d-dimensional Hilbert space, such a universal set of transversal gates cannot exist for even one encoded qudit, for any dimension d, prime or nonprime. This result strongly supports the idea that other primitives, such as quantum teleportation, are necessary for universal fault-tolerant quantum computation, and may be an important factor for fault-tolerance noise thresholds.

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  • Received 15 January 2008

DOI:https://doi.org/10.1103/PhysRevA.78.012353

©2008 American Physical Society

Authors & Affiliations

Xie Chen1, Hyeyoun Chung2, Andrew W. Cross2,3, Bei Zeng1, and Isaac L. Chuang1,2

  • 1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
  • 2Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
  • 3IBM Research Division, T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598, USA

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Vol. 78, Iss. 1 — July 2008

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