Quantum circuits have been widely used as a platform to explore universal properties of generic quantum many-body systems. In this talk, I will present our work in which we construct a field theoretical approach to study quantum circuits. We reformulate the sigma model for time periodic Floquet systems using the replica method, and apply it to the study of spectral statistics of the evolution operator of quantum circuits. This approach is applicable to a wide class of Floquet circuits, and allows us to look into the mechanism behind the emergence of universal random matrix behaviors in quantum chaotic systems. In the same replica sigma model framework, we also rederive the Weingarten calculus, which is a method to evaluate integrals of polynomials with respect to Haar measure over compact groups and has wide applications in the studies of quantum circuits.
