Approximate t-designs in generic circuit architectures

Unitary t-designs are distributions on the unitary group whose first t moments appear maximally random. Previous work has established several upper bounds on the depths at which certain specific random quantum circuit ensembles approximate t-designs. Here we show that these bounds can be extended to any fixed architecture of Haar-random two-site gates. This is accomplished by relating the spectral gaps of such architectures to those of 1D brickwork architectures. Our bound depends on the details of the architecture only via the typical number of layers needed for a block of the circuit to form a connected graph over the sites. When this quantity is independent of width, the circuit forms an approximate t-design in linear depth. We also give an implicit bound for nondeterministic architectures in terms of properties of the corresponding distribution over fixed architectures.