  We study the time evolution operator in a family of local quantum circuits
with random fields in a fixed direction. We argue that the presence of quantum
chaos implies that at large times the time evolution operator becomes
effectively a random matrix in the many-body Hilbert space. To quantify this
phenomenon we compute analytically the squared magnitude of the trace of the
evolution operator -- the generalised spectral form factor -- and compare it
with the prediction of Random Matrix Theory (RMT). We show that for the systems
under consideration the generalised spectral form factor can be expressed in
terms of dynamical correlation functions of local observables in the infinite
temperature state, linking chaotic and ergodic properties of the systems. This
also provides a connection between the many-body Thouless time $\tau_{\rm th}$
-- the time at which the generalised spectral form factor starts following the
random matrix theory prediction -- and the conservation laws of the system.
Moreover, we explain different scalings of $\tau_{\rm th}$ with the system
size, observed for systems with and without the conservation laws.
