# Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits

arXiv: https://arxiv.org/abs/2001.00550 · HTML: https://mlsanity.com/paper/2001.00550/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jan 2020.

## Built on by (most cited first, 26 of 26)

- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Group-Invariant Quantum Machine Learning](https://mlsanity.com/paper/2205.02261/) (2022)
- [Enhancing Generative Models via Quantum Correlations](https://mlsanity.com/paper/2101.08354/) (2021)
- [On quantum backpropagation, information reuse, and cheating measurement collapse](https://mlsanity.com/paper/2305.13362/) (2023)
- [Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus](https://mlsanity.com/paper/2102.01828/) (2021)
- [projUNN: efficient method for training deep networks with unitary matrices](https://mlsanity.com/paper/2203.05483/) (2022)
- [Variational inference with a quantum computer](https://mlsanity.com/paper/2103.06720/) (2021)
- [Quantum Machine Learning on Near-Term Quantum Devices: Current State of Supervised and Unsupervised Techniques for Real-World Applications](https://mlsanity.com/paper/2307.00908/) (2023)
- [Learning quantum data with the quantum Earth Mover's distance](https://mlsanity.com/paper/2101.03037/) (2021)
- [Variational quantum compiling with double Q-learning](https://mlsanity.com/paper/2103.11611/) (2021)
- [Quantum activation functions for quantum neural networks](https://mlsanity.com/paper/2201.03700/) (2022)
- [Optimizing Quantum Variational Circuits with Deep Reinforcement Learning](https://mlsanity.com/paper/2109.03188/) (2021)
- [Long-time simulations with high fidelity on quantum hardware](https://mlsanity.com/paper/2102.04313/) (2021)
- [Variational Quantum Classifiers Through the Lens of the Hessian](https://mlsanity.com/paper/2105.10162/) (2021)
- [Policy Gradient Approach to Compilation of Variational Quantum Circuits](https://mlsanity.com/paper/2111.10227/) (2021)
- [Enhancing variational quantum algorithms by balancing training on classical and quantum hardware](https://mlsanity.com/paper/2503.16361/) (2025)
- [Quantum machine learning with adaptive linear optics](https://mlsanity.com/paper/2102.04579/) (2021)
- [Entanglement Diagnostics for Efficient Quantum Computation](https://mlsanity.com/paper/2102.12534/) (2021)
- [Mitigating Noise-Induced Gradient Vanishing in Variational Quantum Algorithm Training](https://mlsanity.com/paper/2111.13209/) (2021)
- [Quantum Inception Score](https://mlsanity.com/paper/2311.12163/) (2023)
- [Taming Barren Plateaus in Arbitrary Parameterized Quantum Circuits without Sacrificing Expressibility](https://mlsanity.com/paper/2511.13408/) (2025)
- [Towards Efficient Ansatz Architecture for Variational Quantum Algorithms](https://mlsanity.com/paper/2111.13730/) (2021)
- [Enhancing Quantum Variational Algorithms with Zero Noise Extrapolation via Neural Networks](https://mlsanity.com/paper/2403.07025/) (2024)
- [Sculpting Quantum Landscapes: Fubini-Study Metric Conditioning for Geometry Aware Learning in Parameterized Quantum Circuits](https://mlsanity.com/paper/2506.21940/) (2025)
- [The Signal Horizon: Local Blindness and the Contraction of Pauli-Weight Spectra in Noisy Quantum Encodings](https://mlsanity.com/paper/2602.14735/) (2026)
- [Soft-Quantum Algorithms](https://mlsanity.com/paper/2604.06523/) (2026)

## Abstract (arXiv)

Variational quantum algorithms (VQAs) optimize the parameters vecθ of a parametrized quantum circuit V(vecθ) to minimize a cost function C. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming V(vecθ) is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining C in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when V(vecθ) is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining C with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of V(vecθ) is O(log n). Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.