Random unitary circuits with constant spectral gap

We prove constant lower bounds for the spectral gap of the following random walks on unitary groups $\mathsf{SU}(2^n)$ on $n$ qubits. (i) Random Pauli Rotation: choose an $n$-qubit Pauli operator $P$ and an angle $θ\in [0,2π)$, both uniformly at random, and apply $e^{\mathrm i θP}$. (ii) Brickwork R