Exact spectral gaps for random Pauli rotations

We resolve the spectral-gap problem posed by Baer and Haah for random Pauli rotations. In this walk, a nonidentity $n$-qubit Pauli operator $P$ and an angle $θ$ modulo $2π$ are chosen independently and uniformly, and the step is $e^{\mathrm{i}θP}$. Writing $d=2^n$, we prove that the gap on $\mathsf{