发表机构
School of Mathematics, Sun Yat-sen University（中山大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文精确求解了随机Pauli旋转的谱隙问题，否证了Baer-Haah在特殊酉群上的猜想，并给出其在射影群上的精确值，方法结合图谱、最小权重界与局部不等式。
AI 中文摘要
我们解决了Baer和Haah针对随机Pauli旋转提出的谱隙问题。在该随机游走中，一个非恒等的$n$量子比特Pauli算符$P$和一个模$2\pi$的角度$\theta$被独立且均匀地选取，步长为$e^{\mathrm{i}\theta P}$。记$d=2^n$，我们证明对于$n\ge4$，在$\mathsf{SU}(d)$上的谱隙为$(d-8)/(8(d-1))$。这否证了他们在整个特殊酉群上的猜想公式。我们还证明，猜想值$d(d-3)/(8(d^2-1))$恰好是$n\ge3$时$\mathsf{PU}(d)$上的谱隙。特殊酉群上的谱隙由一个显式向量在$\bigwedge^8\mathbb C^d$中取得，该向量由$\mathbb F_2^n$的仿射三维子空间构造。其非平凡的中心作用解释了为何平衡张量表示无法检测到这个更小的谱隙。在射影群上，谱隙在$U\mapsto U^{\otimes4}\otimes\bar U^{\otimes4}$中取得。匹配的下界对所有有限维酉表示一致成立。它们结合了反交换Pauli算符图的谱、二元多项式的最小权重界以及单量子比特Clifford固定子空间上的局部不等式。该证明是解析的，不需要计算验证。
英文摘要
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 $$Δ_{\mathsf{SU}(d)}=\frac{d-8}{8(d-1)}\quad(n\ge4),\qquad Δ_{\mathsf{PU}(d)}=\frac{d(d-3)}{8(d^2-1)}\quad(n\ge3).$$ 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 $n\ge4$, and has an expansion beginning $$Δ_n^{(4)}=\frac{10}{9n}-\frac{88}{105n^2}+O(n^{-3}),$$ 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 $n\ge144$; in the same range it simultaneously governs every ordinary even tensor degree $k\ge4$. Thus different representation scopes can have genuinely different slow modes.
Comments79 pages
