Random-Matrix-Induced Simplicity Bias in Over-parameterized Variational Quantum Circuits

2026-01-05

AI Summary

This work addresses the expressivity collapse in overparameterized, unstructured variational quantum circuits, which—despite their high representational capacity—induce function classes that degenerate into near-constant mappings due to universality properties of random matrices. This phenomenon underlies vanishing gradients and poor generalization. The paper introduces the concept of “simplicity bias” to unify the understanding of barren plateaus, limited expressivity, and generalization failure. Leveraging tools from random matrix theory and concentration of measure, the authors rigorously analyze the behavior of hypothesis classes induced by quantum circuits. They prove that unstructured circuits, with high probability, produce degenerate outputs on large datasets, whereas structured designs—such as those based on tensor networks—preserve output diversity and non-degenerate gradients, thereby mitigating expressivity collapse.

Technology Category

• Machine Learning: Quantum Machine Learning
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Over-parameterization is commonly used to increase the expressivity of variational quantum circuits (VQCs), yet deeper and more highly parameterized circuits often exhibit poor trainability and limited generalization. In this work, we provide a theoretical explanation for this phenomenon from a function-class perspective. We show that sufficiently expressive, unstructured variational ansatze enter a Haar-like universality class in which both observable expectation values and parameter gradients concentrate exponentially with system size. As a consequence, the hypothesis class induced by such circuits collapses with high probability to a narrow family of near-constant functions, a phenomenon we term simplicity bias, with barren plateaus arising as a consequence rather than the root cause. Using tools from random matrix theory and concentration of measure, we rigorously characterize this universality class and establish uniform hypothesis-class collapse over finite datasets. We further show that this collapse is not unavoidable: tensor-structured VQCs, including tensor-network-based and tensor-hypernetwork parameterizations, lie outside the Haar-like universality class. By restricting the accessible unitary ensemble through bounded tensor rank or bond dimension, these architectures prevent concentration of measure, preserve output variability for local observables, and retain non-degenerate gradient signals even in over-parameterized regimes. Together, our results unify barren plateaus, expressivity limits, and generalization collapse under a single structural mechanism rooted in random-matrix universality, highlighting the central role of architectural inductive bias in variational quantum algorithms.

Problem

• over-parameterization
• variational quantum circuits
• simplicity bias
• barren plateaus
• expressivity

Innovation

• simplicity bias
• Haar-like universality
• tensor-structured VQCs
• concentration of measure
• barren plateaus