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=2n, the continuous Pauli walk has exact all-representation gaps ΔSU(d)=(d−8)/(8(d−1)) (n≥4), ΔPU(d)=(d(d−3))/(8(d2−1)) (n≥3). The special-unitary gap is attained in bigwedge8ℂd 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) 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≥4, and has an expansion beginning Δn(4)=10/9n−88/105n2+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≥144; in the same range it simultaneously governs every ordinary even tensor degree k≥4. Thus different representation scopes can have genuinely different slow modes.
Versions
Related
- Random unitary circuits with constant spectral gap
- One Gate at a Time: Complexity Growth in Random Quantum Circuits
- Bayesian Selection of Edge Spectral Modes for Crash Counts on Urban Road Networks with Applications in Barcelona and Bogotá
- Spectral properties of deterministic matrices multiplied by rotationally invariant random non-Hermitian ensembles
- Smooth spectral statistics of random perturbations of Schrödinger operators near Anosov energy levels
BibTeX
@misc{dong2026spectral,
title = {{Spectral gaps and slow modes of Pauli rotations and random quantum circuits}},
author = {Ziyuan Dong and Xiang Fan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.40164},
eprint = {2609.40164},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40164},
}
RIS
TY - UNPB
TI - Spectral gaps and slow modes of Pauli rotations and random quantum circuits
AU - Ziyuan Dong
AU - Xiang Fan
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2609.40164
UR - https://arxiv.org/abs/2609.40164
AB - 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_SU(d)=(d−8)/(8(d−1)) (n≥4), Δ_PU(d)=(d(d−3))/(8(d^2−1)) (n≥3). The special-unitary gap is attained in bigwedge^8ℂ^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-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≥4, and has an expansion beginning Δ_n^(4)=10/9n−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≥144; in the same range it simultaneously governs every ordinary even tensor degree k≥4. Thus different representation scopes can have genuinely different slow modes.
ER -