# An Empirical Comparison of Optimizers for Quantum Machine Learning with SPSA-based Gradients

arXiv: https://arxiv.org/abs/2305.00224 · HTML: https://mlsanity.com/paper/2305.00224/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 48 references that are in the map.
- First posted on arXiv 27 Apr 2023.

## Builds on (most cited first, 8 of 8)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [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)
- [On Empirical Comparisons of Optimizers for Deep Learning](https://mlsanity.com/paper/1910.05446/) (2019)
- [Deep Reinforcement Learning with Enhanced Safety for Autonomous Highway Driving](https://mlsanity.com/paper/1910.12905/) (2019)
- [Iteration Complexity of Variational Quantum Algorithms](https://mlsanity.com/paper/2209.10615/) (2022)

## Built on by (most cited first, 14 of 14)

- [Better than classical? The subtle art of benchmarking quantum machine learning models](https://mlsanity.com/paper/2403.07059/) (2024)
- [Just One Byte (per gradient): A Note on Low-Bandwidth Decentralized Language Model Finetuning Using Shared Randomness](https://mlsanity.com/paper/2306.10015/) (2023)
- [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)
- [Qiskit-Torch-Module: Fast Prototyping of Quantum Neural Networks](https://mlsanity.com/paper/2404.06314/) (2024)
- [Guided-SPSA: Simultaneous Perturbation Stochastic Approximation assisted by the Parameter Shift Rule](https://mlsanity.com/paper/2404.15751/) (2024)
- [Fourier Analysis of Variational Quantum Circuits for Supervised Learning](https://mlsanity.com/paper/2411.03450/) (2024)
- [STIQ: Safeguarding Training and Inferencing of Quantum Neural Networks from Untrusted Cloud](https://mlsanity.com/paper/2405.18746/) (2024)
- [Security Concerns in Quantum Machine Learning as a Service](https://mlsanity.com/paper/2408.09562/) (2024)
- [Hybrid Quantum Graph Neural Network for Molecular Property Prediction](https://mlsanity.com/paper/2405.05205/) (2024)
- [DyPP: Dynamic Parameter Prediction to Accelerate Convergence of Variational Quantum Algorithms](https://mlsanity.com/paper/2307.12449/) (2023)
- [Quantum Neural Networks for Solving Power System Transient Simulation Problem](https://mlsanity.com/paper/2405.11427/) (2024)
- [An Efficient Quantum Classifier Based on Hamiltonian Representations](https://mlsanity.com/paper/2504.10542/) (2025)
- [CutReg: A loss regularizer for enhancing the scalability of QML via adaptive circuit cutting](https://mlsanity.com/paper/2506.14858/) (2025)
- [Trainability-Oriented Hybrid Quantum Regression via Geometric Preconditioning and Curriculum Optimization](https://mlsanity.com/paper/2601.11942/) (2026)

## Abstract (arXiv)

VQA have attracted a lot of attention from the quantum computing community for the last few years. Their hybrid quantum-classical nature with relatively shallow quantum circuits makes them a promising platform for demonstrating the capabilities of NISQ devices. Although the classical machine learning community focuses on gradient-based parameter optimization, finding near-exact gradients for VQC with the parameter-shift rule introduces a large sampling overhead. Therefore, gradient-free optimizers have gained popularity in quantum machine learning circles. Among the most promising candidates is the SPSA algorithm, due to its low computational cost and inherent noise resilience. We introduce a novel approach that uses the approximated gradient from SPSA in combination with state-of-the-art gradient-based classical optimizers. We demonstrate numerically that this outperforms both standard SPSA and the parameter-shift rule in terms of convergence rate and absolute error in simple regression tasks. The improvement of our novel approach over SPSA with stochastic gradient decent is even amplified when shot- and hardware-noise are taken into account. We also demonstrate that error mitigation does not significantly affect our results.