We use field-theoretic methods to explore the statistics of eigenfunctions of
    the Floquet operator for a large family of Floquet random quantum circuits. The
    correlation function of the quasienergy eigenstates is calculated and shown to
    exhibit random matrix circular unitary ensemble statistics, which is consistent
    with the analogue of Berry’s conjecture for quantum circuits. This quantity
    determines all key metrics of quantum chaos, such as the spectral form factor
    and thermalizing time-dependence of the expectation value of an arbitrary
    observable. It also allows us to explicitly show that the matrix elements of
    local operators satisfy the eigenstate thermalization hypothesis (ETH); i.e.,
    the variance of the off-diagonal matrix elements of such operators is
    exponentially small in the system size. These results represent a proof of ETH
    for the family of Floquet random quantum circuits at a physical level of rigor.
    An outstanding open question for this and most of other sigma-model
    calculations is a mathematically rigorous proof of the validity of the
    saddle-point approximation in the large-N limit.

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