Lie-Algebraic Analysis of Generators: Approximation-Error Bounds and Barren-Plateau Heuristics

Lie algebras provide a useful framework for theoretical analysis in quantum machine learning, particularly in hybrid quantum-classical learning. From the viewpoint of function approximation, expectation values of parameterized quantum circuits can be viewed as trigonometric polynomials whose accessible Fourier modes are determined by the spectra of the generators. In this study, we describe: (1) a minimax lower bound on the L²-approximation error over a Sobolev ball when the circuit's effective frequency set is contained in a radius-K ball, which yields a scaling law of the form Ω\(K^{frac{d}{2} - r}\) for r > frac{d}{2} assuming the target function belongs to the Sobolev space $ W₂^r(mathbb{T}^d $), and we also derive a Jackson-type upper bound on the approximation error of quantum circuits under Sobolev regularity of the target function, expressed in terms of an effective bandwidth determined by generator spectral gaps; (2) a generator-selection rule motivated by enlarging the effective frequency set via non-commuting generators; and (3) a simple heuristic metric based on the trace component of generators, aimed at characterizing training behaviors related to barren plateaus. Simulation experiments on toy problems illustrate the practical implications of the frequency-spectrum perspective and the proposed heuristics.