Approximate t-design in general architectures

A key point of interest in quantum circuit design is understanding the rate at which a random circuit architecture approaches a globally uniform distribution. Previous works have proven that a one-dimensional brickwork architecture becomes an approximate t-design within a depth that is linear in system size. Using a cluster-linking picture, we show how to reduce more general architectures to this one-dimensional case while preserving the linear nature of the unitary t-designs.