A Representation-Theoretic Framework for Characterizing Barren Plateaus

• URL: http://arxiv.org/abs/2609.04462v1
• Date: Thu, 03 Sep 2026 20:44:53 GMT
• Title: A Representation-Theoretic Framework for Characterizing Barren Plateaus
• Abstract summary: We introduce a representation-theoretic framework under $2$-design hypothesis showing that variational quantum landscapes admit a natural decomposition into irreducible representation channels.<n>This yields exact expressions and analytical bounds for the cost-function variance applicable to arbitrary initial states and observables.
• Score: 0.13999481573773068
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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