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arXiv · 0910.0913

Convergence rates for arbitrary statistical moments of random quantum circuits

Winton G. Brown·Lorenza Viola
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Abstract

We consider a class of random quantum circuits where at each step a gate from a universal set is applied to a random pair of qubits, and determine how quickly averages of arbitrary finite-degree polynomials in the matrix elements of the resulting unitary converge to Haar measure averages. This is accomplished by establishing an exact mapping between the superoperator that describes t-order moments on n qubits and a multilevel SU(4^t) Lipkin-Meshkov-Glick Hamiltonian. For arbitrary fixed t, we find that the spectral gap scales as 1/n in the thermodynamic limit. Our results imply that random quantum circuits yield an efficient implementation of ε-approximate unitary t-designs.

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Winton G. Brown, Lorenza Viola. 2010-03-24. Convergence rates for arbitrary statistical moments of random quantum circuits. https://doi.org/10.1103/physrevlett.104.250501

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