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

TL;DR

This paper uses Lie algebra to analyze quantum circuit approximation errors, generator selection, and barren plateau heuristics, providing bounds and insights for quantum machine learning performance.

Contribution

It introduces a spectral analysis framework for quantum circuits, deriving error bounds, generator selection rules, and heuristics for barren plateau mitigation.

Findings

Minimax lower bound on approximation error based on generator spectra

Effective frequency set scaling law for Sobolev functions

Heuristic metric for training behavior related to barren plateaus

Abstract

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 $ L^{2} $-approximation error over a Sobolev ball when the circuit's effective frequency set is contained in a radius-$K$ ball, which yields a scaling law of the form $ \Omega(K^{\frac{d}{2} - r}) $ for $ r > \frac{d}{2} $ (assuming the target function belongs to the Sobolev space $ W_2^{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.