Elementary gates for quantum computation

Phys. Rev. A 52, 3457 – Published 1 November 1995
Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter

Abstract

We show that a set of gates that consists of all one-bit quantum gates [U(2)] and the two-bit exclusive-OR gate [that maps Boolean values (x,y) to (x,xy)] is universal in the sense that all unitary operations on arbitrarily many bits n [U(2n)] can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical and of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary n-bit unitary operations.

DOI: http://dx.doi.org/10.1103/PhysRevA.52.3457

  • Received 22 March 1995
  • Published in the issue dated November 1995

© 1995 The American Physical Society

Authors & Affiliations

Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter

  • Clarendon Laboratory, Oxford University, Oxford OX1 3PU, United Kingdom
  • IBM Research, Yorktown Heights, New York 10598 Department of Computer Science, University of Calgary, Calgary, Alberta, Canada T2N 1N4
  • Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • AT&T Bell Laboratories, Murray Hill, New Jersey 07974
  • Physics Department, New York University, New York, New York 10003
  • Physics Department, University of California at Los Angeles, Los Angeles, California 90024
  • Institute for Experimental Physics, University of Innsbruck, A-6020 Innsbruck, Austria

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