  Quantum circuits have been widely used as a platform to simulate generic
quantum many-body systems. In particular, random quantum circuits provide a
means to probe universal features of many-body quantum chaos and ergodicity.
Some such features have already been experimentally demonstrated in the noisy
intermediate-scale quantum (NISQ) devices. On the theory side, properties of
random quantum circuits have been studied on a case-by-case basis and for
certain specific systems, a hallmark of quantum chaos - universal Wigner-Dyson
level statistics - has been derived. This work develops an effective field
theory for a large class of random quantum circuits. The theory has the form of
a replica sigma model and is similar to the low-energy approach to diffusion in
disordered systems. The method is used to explicitly derive universal random
matrix behavior of a large family of random circuits. In particular, we
rederive Wigner-Dyson spectral statistics of the brickwork circuit model by
Chan, De Luca, and Chalker [Phys. Rev. X 8, 041019 (2018)] and show within the
same calculation that its various permutations and higher-dimensional
generalizations preserve the universal level statistics. Finally, we use the
replica sigma model framework to rederive the Weingarten calculus, which is a
method to evaluate integrals of polynomials of matrix elements with respect to
the Haar measure over compact groups and has many applications in the studies
of quantum circuits. The effective field theory, derived here, provides both a
method to quantitatively characterize quantum dynamics of random Floquet
systems (e.g., calculating operator and entanglement spreading) and also path
to understanding the general fundamental mechanism behind quantum chaos and
thermalization in these systems.
