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

arXiv: https://arxiv.org/abs/2601.01877 · HTML: https://mlsanity.com/paper/2601.01877/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 34 references that are in the map.
- First posted on arXiv 5 Jan 2026.

## Builds on (most cited first, 13 of 13)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Quantum circuit architecture search for variational quantum algorithms](https://mlsanity.com/paper/2010.10217/) (2020)
- [Towards Quantum Machine Learning with Tensor Networks](https://mlsanity.com/paper/1803.11537/) (2018)
- [Quantum noise protects quantum classifiers against adversaries](https://mlsanity.com/paper/2003.09416/) (2020)
- [Quantum Convolutional Neural Networks for High Energy Physics Data Analysis](https://mlsanity.com/paper/2012.12177/) (2020)
- [Theory for Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.08566/) (2022)
- [An end-to-end trainable hybrid classical-quantum classifier](https://mlsanity.com/paper/2102.02416/) (2021)
- [Towards provably efficient quantum algorithms for large-scale machine-learning models](https://mlsanity.com/paper/2303.03428/) (2023)
- [Theoretical Error Performance Analysis for Variational Quantum Circuit Based Functional Regression](https://mlsanity.com/paper/2206.04804/) (2022)
- [TensorHyper-VQC: A Tensor-Train-Guided Hypernetwork for Robust and Scalable Variational Quantum Computing](https://mlsanity.com/paper/2508.01116/) (2025)
- [Nonlinear L'evy Processes and their Characteristics](https://mlsanity.com/paper/1401.7253/) (2014)

## Abstract (arXiv)

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.