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Random Quantum Circuits are Approximate 2-designs

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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.

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Authors and Affiliations

  1. Department of Computer Science, University of Bristol, Bristol, U.K.

    Aram W. Harrow & Richard A. Low

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  1. Aram W. Harrow
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  2. Richard A. Low
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Correspondence to Aram W. Harrow.

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Communicated by A. Connes

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Harrow, A.W., Low, R.A. Random Quantum Circuits are Approximate 2-designs. Commun. Math. Phys. 291, 257–302 (2009). https://doi.org/10.1007/s00220-009-0873-6

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