  We study random quantum circuits with symmetry, where the local 2-site
unitaries are drawn from a quotient or subgroup of the full unitary group
$U(d)$. Random quantum circuits are minimal models of local quantum chaotic
dynamics and can be used to study operator growth and the emergence of
diffusive hydrodynamics. We derive the transition probabilities for the
stochastic process governing the growth of operators in four classes of
symmetric random circuits. We then compute the butterfly velocities and
diffusion constants for a spreading operator by solving a simple random walk in
each class of circuits.
