# A Convergence Theory for Over-parameterized Variational Quantum Eigensolvers

arXiv: https://arxiv.org/abs/2205.12481 · HTML: https://mlsanity.com/paper/2205.12481/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 39 references that are in the map.
- First posted on arXiv 25 May 2022.

## Builds on (most cited first, 16 of 16)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Stronger generalization bounds for deep nets via a compression approach](https://mlsanity.com/paper/1802.05296/) (2018)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [On the Computational Efficiency of Training Neural Networks](https://mlsanity.com/paper/1410.1141/) (2014)
- [Representation Learning via Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2111.04225/) (2021)
- [Exponentially Many Local Minima in Quantum Neural Networks](https://mlsanity.com/paper/2110.02479/) (2021)
- [Analytic theory for the dynamics of wide quantum neural networks](https://mlsanity.com/paper/2203.16711/) (2022)
- [Learning Unitaries by Gradient Descent](https://mlsanity.com/paper/2001.11897/) (2020)
- [Layer VQE: A Variational Approach for Combinatorial Optimization on Noisy Quantum Computers](https://mlsanity.com/paper/2102.05566/) (2021)
- [Quantum tangent kernel](https://mlsanity.com/paper/2111.02951/) (2021)
- [Quantum Control Landscapes Are Almost Always Trap Free](https://mlsanity.com/paper/1608.06198/) (2016)

## Built on by (most cited first, 15 of 15)

- [Quantum Deep Hedging](https://mlsanity.com/paper/2303.16585/) (2023)
- [Let Quantum Neural Networks Choose Their Own Frequencies](https://mlsanity.com/paper/2309.03279/) (2023)
- [Symmetric Pruning in Quantum Neural Networks](https://mlsanity.com/paper/2208.14057/) (2022)
- [Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classification](https://mlsanity.com/paper/2301.01597/) (2022)
- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Laziness, Barren Plateau, and Noise in Machine Learning](https://mlsanity.com/paper/2206.09313/) (2022)
- [Generalization of Quantum Machine Learning Models Using Quantum Fisher Information Metric](https://mlsanity.com/paper/2303.13462/) (2023)
- [Effects of noise on the overparametrization of quantum neural networks](https://mlsanity.com/paper/2302.05059/) (2023)
- [Double descent in quantum kernel methods](https://mlsanity.com/paper/2501.10077/) (2025)
- [Analyzing Convergence in Quantum Neural Networks: Deviations from Neural Tangent Kernels](https://mlsanity.com/paper/2303.14844/) (2023)
- [QuACK: Accelerating Gradient-Based Quantum Optimization with Koopman Operator Learning](https://mlsanity.com/paper/2211.01365/) (2022)
- [MG-Net: Learn to Customize QAOA with Circuit Depth Awareness](https://mlsanity.com/paper/2409.18692/) (2024)
- [Dynamical transition in controllable quantum neural networks with large depth](https://mlsanity.com/paper/2311.18144/) (2023)
- [Fragmentation is Efficiently Learnable by Quantum Neural Networks](https://mlsanity.com/paper/2512.00751/) (2025)
- [Provably Efficient Self-Calibrating Quantum Fault Tolerance](https://mlsanity.com/paper/2608.05686/) (2026)

## Abstract (arXiv)

The Variational Quantum Eigensolver (VQE) is a promising candidate for quantum applications on near-term Noisy Intermediate-Scale Quantum (NISQ) computers. Despite a lot of empirical studies and recent progress in theoretical understanding of VQE's optimization landscape, the convergence for optimizing VQE is far less understood. We provide the first rigorous analysis of the convergence of VQEs in the over-parameterization regime. By connecting the training dynamics with the Riemannian Gradient Flow on the unit-sphere, we establish a threshold on the sufficient number of parameters for efficient convergence, which depends polynomially on the system dimension and the spectral ratio, a property of the problem Hamiltonian, and could be resilient to gradient noise to some extent. We further illustrate that this overparameterization threshold could be vastly reduced for specific VQE instances by establishing an ansatz-dependent threshold paralleling our main result. We showcase that our ansatz-dependent threshold could serve as a proxy of the trainability of different VQE ansatzes without performing empirical experiments, which hence leads to a principled way of evaluating ansatz design. Finally, we conclude with a comprehensive empirical study that supports our theoretical findings.