  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.
