# Fundamental limitations on optimization in variational quantum algorithms

arXiv: https://arxiv.org/abs/2205.05056 · HTML: https://mlsanity.com/paper/2205.05056/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 78 references that are in the map.
- First posted on arXiv 10 May 2022.

## Builds on (most cited first, 15 of 15)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Absence of Barren Plateaus in Quantum Convolutional Neural Networks](https://mlsanity.com/paper/2011.02966/) (2020)
- [Entanglement Induced Barren Plateaus](https://mlsanity.com/paper/2010.15968/) (2020)
- [Robust data encodings for quantum classifiers](https://mlsanity.com/paper/2003.01695/) (2020)
- [Layerwise learning for quantum neural networks](https://mlsanity.com/paper/2006.14904/) (2020)
- [Effect of barren plateaus on gradient-free optimization](https://mlsanity.com/paper/2011.12245/) (2020)
- [Noisy intermediate-scale quantum (NISQ) algorithms](https://mlsanity.com/paper/2101.08448/) (2021)
- [Learning to learn with quantum neural networks via classical neural networks](https://mlsanity.com/paper/1907.05415/) (2019)
- [Equivalence of quantum barren plateaus to cost concentration and narrow gorges](https://mlsanity.com/paper/2104.05868/) (2021)
- [On barren plateaus and cost function locality in variational quantum algorithms](https://mlsanity.com/paper/2011.10530/) (2020)
- [Variational Quantum Singular Value Decomposition](https://mlsanity.com/paper/2006.02336/) (2020)
- [Variational Quantum Eigensolver for Frustrated Quantum Systems](https://mlsanity.com/paper/2005.00544/) (2020)

## Built on by (most cited first, 1 of 1)

- [Quantum Machine Learning Approaches for Coordinated Stealth Attack Detection in Distributed Generation Systems](https://mlsanity.com/paper/2601.00873/) (2025)

## Abstract (arXiv)

Exploring quantum applications of near-term quantum devices is a rapidly growing field of quantum information science with both theoretical and practical interests. A leading paradigm to establish such near-term quantum applications is variational quantum algorithms (VQAs). These algorithms use a classical optimizer to train a parameterized quantum circuit to accomplish certain tasks, where the circuits are usually randomly initialized. In this work, we prove that for a broad class of such random circuits, the variation range of the cost function via adjusting any local quantum gate within the circuit vanishes exponentially in the number of qubits with a high probability. This result can unify the restrictions on gradient-based and gradient-free optimizations in a natural manner and reveal extra harsh constraints on the training landscapes of VQAs. Hence a fundamental limitation on the trainability of VQAs is unraveled, indicating the essential mechanism of the optimization hardness in the Hilbert space with exponential dimension. We further showcase the validity of our results with numerical simulations of representative VQAs. We believe that these results would deepen our understanding of the scalability of VQAs and shed light on the search for near-term quantum applications with advantages.