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Communications in Mathematical Physics

, Volume 291, Issue 1, pp 257–302 | Cite as

Random Quantum Circuits are Approximate 2-designs

  • Aram W. Harrow
  • Richard A. Low
Article

Abstract

Given a universal gate set on two qubits, it is well known that applying random gates from the set to random pairs of qubits will eventually yield an approximately Haar-distributed unitary. However, this requires exponential time. We show that random circuits of only polynomial length will approximate the first and second moments of the Haar distribution, thus forming approximate 1- and 2-designs. Previous constructions required longer circuits and worked only for specific gate sets. As a corollary of our main result, we also improve previous bounds on the convergence rate of random walks on the Clifford group.

Keywords

Markov Chain Stationary Distribution Haar Measure Quantum Circuit Sobolev Constant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of BristolBristolU.K.

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