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arXiv:2609.40164

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

Ziyuan Dong, Xiang Fan

quant-phmath.PRSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations• Revised Oct 4, 2026 (v2)
v1 · Sep 30, 2026v2 · Oct 4, 2026
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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 $$Δ_{\mathsf{SU}(d)}=\frac{d-8}{8(d-1)}\quad(n\ge4),\qquad Δ_{\mathsf{PU}(d)}=\frac{d(d-3)}{8(d^2-1)}\quad(n\ge3).$$ 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.

DOI
10.5281/zenodo.23099545
Comments
79 pages

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We determine which representation sectors control the slowest convergence of random Pauli rotations and several local random quantum circuits. For d=2nd=2^nd=2n, the continuous Pauli walk has exact all-representation gaps ΔSU(d)=d−88(d−1)(n≥4),ΔPU(d)=d(d−3)8(d2−1)(n≥3).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).ΔSU(d)​=8(d−1)d−8​(n≥4),ΔPU(d)​=8(d2−1)d(d−3)​(n≥3). The special-unitary gap is attained in ⋀8Cd\bigwedge^8\mathbb{C}^d⋀8Cd 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-SU(4)\mathsf{SU}(4)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≥4n\ge4n≥4, and has an expansion beginning Δn(4)=109n−88105n2+O(n−3),Δ_n^{(4)}=\frac{10}{9n}-\frac{88}{105n^2}+O(n^{-3}),Δn(4)​=9n10​−105n288​+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≥144n\ge144n≥144; in the same range it simultaneously governs every ordinary even tensor degree k≥4k\ge4k≥4. Thus different representation scopes can have genuinely different slow modes.
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We construct random walks on simple Lie groups that quickly converge to the Haar measure for all moments up to orderttt. Specifically, a step of the walk on the unitary or orthogonal group of dimension2n2^{\mathrm{n}}2nis a random Pauli rotationeiθP/2e^{\mathrm{i}\theta P/2}eiθP/2. The spectral gap of this random walk is shown to beΩ(1/t)\Omega(1/t)Ω(1/t), which coincides with the best previously known bound for a random walk on the permutation group on{0,1}n\{0,1\}^{\mathrm{n}}{0,1}n. This implies that the walk gives anε\varepsilonε-approximate unitary t-design in depthO(nt2+tlog⁡1ε)d\mathcal{O}(\mathrm{n}t^{2}+t\log\frac{1}{\varepsilon})dO(nt2+tlogε1​)dwhered=O(log⁡n)d=\mathrm{O}(\log \mathrm{n})d=O(logn)is the circuit depth to implementeiθP/2e^{\mathrm{i}\theta P/2}eiθP/2. Our simple proof uses quadratic Casimir operators of Lie algebras.
We construct random walks on simple Lie groups that quickly converge to the Haar measure for all moments up to order ttt. Specifically, a step of the walk on the unitary or orthognoal group of dimension 2n2^{\mathsf n}2n is a random Pauli rotation eiθP/2e^{\mathrm i θP /2}eiθP/2. The spectral gap of this random walk is shown to be Ω(1/t)Ω(1/t)Ω(1/t), which coincides with the best previously known bound for a random walk on the permutation group on {0,1}n\{0,1\}^{\mathsf n}{0,1}n. This implies that the walk gives an ε\varepsilonε-approximate unitary ttt-design in depth O(nt2+tlog⁡1/ε)dO(\mathsf n t^2 + t \log 1/\varepsilon)dO(nt2+tlog1/ε)d where d=O(log⁡n)d=O(\log \mathsf n)d=O(logn) is the circuit depth to implement eiθP/2e^{\mathrm i θP /2}eiθP/2. Our simple proof uses quadratic Casimir operators of Lie algebras.
Entangling measurements can reduce the cost of learning many-body correlations. We consider classical shadows in which a fixed unitary on a kkk-qubit region is sandwiched between two layers of random single-qubit Clifford gates, and ask which middle unitary best estimates Pauli strings with full support on the region. Combining symplectic support counting with a constraint from operator-size parity, we prove that within this architecture the minimum squared shadow norm over all Clifford middle unitaries is Ckopt=4⋅9k/(3⋅5k+3k−6)C_k^{\rm opt}=4\cdot 9^k/(3\cdot 5^k+3^k-6)Ckopt​=4⋅9k/(3⋅5k+3k−6) for odd kkk and Ckopt=4⋅9k/(3⋅5k−3k+2)C_k^{\rm opt}=4\cdot 9^k/(3\cdot 5^k-3^k+2)Ckopt​=4⋅9k/(3⋅5k−3k+2) for even kkk. A dual-axis rotation circuit U∗U_*U∗​ attains this minimum for every kkk. Asymptotically Ckopt≃(4/3)⋅(9/5)kC_k^{\rm opt}\simeq(4/3)\cdot(9/5)^kCkopt​≃(4/3)⋅(9/5)k: the exponential base of Wu et al.'s contractive unitary is optimal, and the prefactor improves from 222 to 4/34/34/3. For 1≤k≤71\leq k\leq71≤k≤7, exact certificates show that U∗U_*U∗​ is optimal over the entire unitary group U(2k)\mathrm U(2^k)U(2k), and we conjecture that this holds for every kkk.
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Abstract Parametrized and random unitary (or orthogonal) n -qubit circuits play a central role in quantum information. As such, one could naturally assume that circuits implementing symplectic transformations would attract similar attention. However, this is not the case, as S P ( d / 2 ) —the group of d × d unitary symplectic matrices—has thus far been overlooked. In this work, we aim at starting to fill this gap. We begin by presenting a universal set of generators G for the symplectic algebra sp ( d / 2 ) , consisting of one- and two-qubit Pauli operators acting on neighboring sites in a one-dimensional lattice. Here, we uncover two critical differences between such set, and equivalent ones for unitary and orthogonal circuits. Namely, we find that the operators in G cannot generate arbitrary local symplectic unitaries and that they are not translationally invariant. We then review the Schur–Weyl duality between the symplectic group and the Brauer algebra, and use tools from Weingarten calculus to prove that Pauli measurements at the output of Haar random symplectic circuits can converge to Gaussian processes. As a by-product, such analysis provides us with concentration bounds for Pauli measurements in circuits that form t -designs over S P ( d / 2 ) . To finish, we present tensor-network tools to analyze shallow random symplectic circuits, and we use these to numerically show that computational-basis measurements anti-concentrate at logarithmic depth.