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arXiv · 2609.04462

A Representation-Theoretic Framework for Characterizing Barren Plateaus

Pedro Alcântara·Leandro Morais·Rafael Chaves
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Abstract

The scalability of variational quantum algorithms is fundamentally limited by the barren plateau effect, where the cost-function variance vanishes with system size, rendering optimization impractical. Recent Lie-algebraic approaches for deep parameterized have enabled a unified analytical understanding of this challenge but require either the initial state or the measurement observable to belong to the dynamical Lie algebra generated by the circuit. Here, we introduce a representation-theoretic framework under $2$-design hypothesis showing that variational quantum landscapes admit a natural decomposition into irreducible representation channels. This yields exact expressions and analytical bounds for the cost-function variance applicable to arbitrary initial states and observables, with previous Lie-algebraic results emerging as a special case. We illustrate the framework by analyzing the energy landscape of the one-dimensional ANNNI model for several circuit architectures, revealing trainability regimes inaccessible to existing methods. Our results establish a general representation-theoretic framework for analyzing variational quantum landscapes, substantially extending the analytical theory of barren plateaus.

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Pedro Alcântara, Leandro Morais, Rafael Chaves. 2026-09-03. A Representation-Theoretic Framework for Characterizing Barren Plateaus. https://arxiv.org/abs/2609.04462

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