Exact spectral gaps for random Pauli rotations

Abstract

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 $\theta$ modulo $2\pi$ are chosen independently and uniformly, and the step is $e^{\mathrm{i}\theta P}$. Writing $d=2^n$, we prove that the gap on $\mathsf{SU}(d)$ is $(d-8)/(8(d-1))$ for $n\ge4$. This disproves their conjectured formula on the full special unitary group. We also prove that the conjectured value, $d(d-3)/(8(d^2-1))$, is exactly the gap on $\mathsf{PU}(d)$ for $n\ge3$. The special-unitary gap is attained by an explicit vector in $\bigwedge^8\mathbb C^d$, constructed from affine three-dimensional subspaces of $\mathbb F_2^n$. Its nontrivial central action explains why balanced tensor representations do not detect this smaller gap. On the projective group, the gap is attained in $U\mapsto U^{\otimes4}\otimes\bar U^{\otimes4}$. The matching lower bounds hold uniformly over all finite-dimensional unitary representations. They combine the spectrum of the graph of anticommuting Pauli operators, a minimum-weight bound for binary polynomials, and a local inequality on a single-qubit Clifford-fixed subspace. The proof is analytic and requires no computational verification.