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 $\theta \in \mathbb R / 2\pi \mathbb Z$, both uniformly at random, and apply $e^{\mathrm i \theta P}$.

(ii) Brickwork Random Unitary Circuit: choose $n-1$ unitaries $U_{i}$ uniformly at random from $\mathsf{SU}(4)$ independently, and apply $U_{2j-1}$ on two qubits $2j-1, 2j$ and then $U_{2j}$ on two qubits $2j, 2j+1$.

Importantly, the spectral gaps are independent of $n$ and apply for all finite dimensional unitary representations of $\mathsf{SU}(2^n)$ uniformly, including those that appear in unitary $t$-designs.

We also prove analogous constant gap results for Clifford unitaries, which are indispensable for our result on Brickwork Random Unitary Circuit.