Spectral gaps and slow modes of Pauli rotations and random quantum circuits

Abstract

We determine which representation sectors control the slowest convergence of random Pauli rotations and several local random quantum circuits. For $d=2^n$, the continuous Pauli walk has exact all-representation gaps $${\Delta }_{\mathsf{SU}(d)}=\frac{d-8}{8(d-1)}(n\ge 4),{\Delta }_{\mathsf{PU}(d)}=\frac{d(d-3)}{8(d^2-1)}(n\ge 3).$$ The special-unitary gap is attained in ${\bigwedge }^8{\mathbb{C}}^d$ and is invisible to balanced moments, while the projective gap is already attained in a balanced fourth-moment representation. We obtain exact discrete and orthogonal Pauli analogues, exact finite-Clifford chain/path gaps, and dimension-independent all-representation Haar brickwork lower bounds; on three qubits every weighted Haar-$\mathsf{SU}(4)$ edge walk reduces to one universal principal-angle constant. For real complete-graph models, the entire ordinary fourth-tensor gap reduces to a quadratic-size occupation operator, is simple for every $n\ge 4$, and has an expansion beginning $${\Delta }_n^{(4)}=\frac{10}{9n}-\frac{88}{105n^2}+O(n^{-3}),$$ with competing branches separated only beyond all algebraic orders. For the two finite real Clifford complete-graph walks, this fourth-tensor mode is the exact all-representation group gap for every $n\ge 144$; in the same range it simultaneously governs every ordinary even tensor degree $k\ge 4$. Thus different representation scopes can have genuinely different slow modes.

arXiv comment79 pages