# Exponentially Many Local Minima in Quantum Neural Networks

arXiv: https://arxiv.org/abs/2110.02479 · HTML: https://mlsanity.com/paper/2110.02479/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 62 references that are in the map.
- First posted on arXiv 6 Oct 2021.

## Builds on (most cited first, 30 of 30)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [The Loss Surfaces of Multilayer Networks](https://mlsanity.com/paper/1412.0233/) (2014)
- [Deep Learning without Poor Local Minima](https://mlsanity.com/paper/1605.07110/) (2016)
- [Transfer learning in hybrid classical-quantum neural networks](https://mlsanity.com/paper/1912.08278/) (2019)
- [Spurious Local Minima are Common in Two-Layer ReLU Neural Networks](https://mlsanity.com/paper/1712.08968/) (2017)
- [On the Power of Over-parametrization in Neural Networks with Quadratic Activation](https://mlsanity.com/paper/1803.01206/) (2018)
- [Recovery Guarantees for One-hidden-layer Neural Networks](https://mlsanity.com/paper/1706.03175/) (2017)
- [The loss surface of deep and wide neural networks](https://mlsanity.com/paper/1704.08045/) (2017)
- [Continuous-variable quantum neural networks](https://mlsanity.com/paper/1806.06871/) (2018)
- [Stochastic gradient descent for hybrid quantum-classical optimization](https://mlsanity.com/paper/1910.01155/) (2019)
- [Equilibrated adaptive learning rates for non-convex optimization](https://mlsanity.com/paper/1502.04390/) (2015)
- [Comparing Dynamics: Deep Neural Networks versus Glassy Systems](https://mlsanity.com/paper/1803.06969/) (2018)
- [Reliably Learning the ReLU in Polynomial Time](https://mlsanity.com/paper/1611.10258/) (2016)
- [No bad local minima: Data independent training error guarantees for multilayer neural networks](https://mlsanity.com/paper/1605.08361/) (2016)
- [Small nonlinearities in activation functions create bad local minima in neural networks](https://mlsanity.com/paper/1802.03487/) (2018)
- [Enabling Multi-programming Mechanism for Quantum Computing in the NISQ Era](https://mlsanity.com/paper/2102.05321/) (2021)
- [On the Optimization Landscape of Tensor Decompositions](https://mlsanity.com/paper/1706.05598/) (2017)
- [Spurious Valleys in Two-layer Neural Network Optimization Landscapes](https://mlsanity.com/paper/1802.06384/) (2018)
- [Learning Unitaries by Gradient Descent](https://mlsanity.com/paper/2001.11897/) (2020)
- [Explorations on high dimensional landscapes](https://mlsanity.com/paper/1412.6615/) (2014)
- [Improved Learning of One-hidden-layer Convolutional Neural Networks with Overlaps](https://mlsanity.com/paper/1805.07798/) (2018)
- [Quantum Control Landscapes Are Almost Always Trap Free](https://mlsanity.com/paper/1608.06198/) (2016)
- [Sub-Optimal Local Minima Exist for Neural Networks with Almost All Non-Linear Activations](https://mlsanity.com/paper/1911.01413/) (2019)

## Built on by (most cited first, 22 of 22)

- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Does provable absence of barren plateaus imply classical simulability?](https://mlsanity.com/paper/2312.09121/) (2023)
- [Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits](https://mlsanity.com/paper/2203.09376/) (2022)
- [On the relation between trainability and dequantization of variational quantum learning models](https://mlsanity.com/paper/2406.07072/) (2024)
- [An exponentially-growing family of universal quantum circuits](https://mlsanity.com/paper/2212.00736/) (2022)
- [A Convergence Theory for Over-parameterized Variational Quantum Eigensolvers](https://mlsanity.com/paper/2205.12481/) (2022)
- [Graph neural network initialisation of quantum approximate optimisation](https://mlsanity.com/paper/2111.03016/) (2021)
- [Analyzing Convergence in Quantum Neural Networks: Deviations from Neural Tangent Kernels](https://mlsanity.com/paper/2303.14844/) (2023)
- [Expressive variational quantum circuits provide inherent privacy in federated learning](https://mlsanity.com/paper/2309.13002/) (2023)
- [Scalable Variational Quantum Circuits for Autoencoder-based Drug Discovery](https://mlsanity.com/paper/2112.12563/) (2021)
- [Quantum Federated Learning with Entanglement Controlled Circuits and Superposition Coding](https://mlsanity.com/paper/2212.01732/) (2022)
- [MG-Net: Learn to Customize QAOA with Circuit Depth Awareness](https://mlsanity.com/paper/2409.18692/) (2024)
- [Predictive Performance of Deep Quantum Data Re-uploading Models](https://mlsanity.com/paper/2505.20337/) (2025)
- [Bayesian Learning of Parameterised Quantum Circuits](https://mlsanity.com/paper/2206.07559/) (2022)
- [Dynamical transition in controllable quantum neural networks with large depth](https://mlsanity.com/paper/2311.18144/) (2023)
- [Superposed parameterised quantum circuits](https://mlsanity.com/paper/2506.08749/) (2025)
- [Statistical Analysis of Quantum State Learning Process in Quantum Neural Networks](https://mlsanity.com/paper/2309.14980/) (2023)
- [The curse of random quantum data](https://mlsanity.com/paper/2408.09937/) (2024)
- [Enhancing Circuit Trainability with Selective Gate Activation Strategy](https://mlsanity.com/paper/2503.12738/) (2025)
- [Quantum automated learning with provable and explainable trainability](https://mlsanity.com/paper/2502.05264/) (2025)
- [Trainable Quantum Neural Network for Multiclass Image Classification with the Power of Pre-trained Tree Tensor Networks](https://mlsanity.com/paper/2504.14995/) (2025)
- [The effect of the number of parameters and the number of local feature patches on loss landscapes in distributed quantum neural networks](https://mlsanity.com/paper/2504.19239/) (2025)

## Abstract (arXiv)

Quantum Neural Networks (QNNs), or the so-called variational quantum circuits, are important quantum applications both because of their similar promises as classical neural networks and because of the feasibility of their implementation on near-term intermediate-size noisy quantum machines (NISQ). However, the training task of QNNs is challenging and much less understood. We conduct a quantitative investigation on the landscape of loss functions of QNNs and identify a class of simple yet extremely hard QNN instances for training. Specifically, we show for typical under-parameterized QNNs, there exists a dataset that induces a loss function with the number of spurious local minima depending exponentially on the number of parameters. Moreover, we show the optimality of our construction by providing an almost matching upper bound on such dependence. While local minima in classical neural networks are due to non-linear activations, in quantum neural networks local minima appear as a result of the quantum interference phenomenon. Finally, we empirically confirm that our constructions can indeed be hard instances in practice with typical gradient-based optimizers, which demonstrates the practical value of our findings.