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

TL;DR

This paper explains why highly over-parameterized variational quantum circuits often become untrainable and generalize poorly, showing that their function class collapses to simple, near-constant functions due to a random matrix universality phenomenon.

Contribution

It introduces the concept of simplicity bias caused by Haar-like universality in unstructured VQCs and demonstrates how tensor-structured architectures can avoid this collapse.

Findings

Unstructured VQCs exhibit exponential concentration leading to function collapse.

Tensor-structured VQCs avoid the universality class, maintaining expressivity.

Barren plateaus are a consequence, not the cause, of the universality phenomenon.

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.