arXiv:2607.20919v1 Announce Type: new Abstract: 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