# Guided-SPSA: Simultaneous Perturbation Stochastic Approximation assisted by the Parameter Shift Rule

arXiv: https://arxiv.org/abs/2404.15751 · HTML: https://mlsanity.com/paper/2404.15751/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 63 references that are in the map.
- First posted on arXiv 24 Apr 2024.

## Builds on (most cited first, 27 of 27)

- [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)
- [OpenAI Gym](https://mlsanity.com/paper/1606.01540/) (2016)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [Generalization in quantum machine learning from few training data](https://mlsanity.com/paper/2111.05292/) (2021)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [Quantum Natural Gradient](https://mlsanity.com/paper/1909.02108/) (2019)
- [Better than classical? The subtle art of benchmarking quantum machine learning models](https://mlsanity.com/paper/2403.07059/) (2024)
- [Reinforcement learning for optimization of variational quantum circuit architectures](https://mlsanity.com/paper/2103.16089/) (2021)
- [Quantum machine learning for image classification](https://mlsanity.com/paper/2304.09224/) (2023)
- [A Survey on Quantum Reinforcement Learning](https://mlsanity.com/paper/2211.03464/) (2022)
- [Variational Quantum Reinforcement Learning via Evolutionary Optimization](https://mlsanity.com/paper/2109.00540/) (2021)
- [Deep Reinforcement Learning with Enhanced Safety for Autonomous Highway Driving](https://mlsanity.com/paper/1910.12905/) (2019)
- [Quantum Policy Gradient Algorithm with Optimized Action Decoding](https://mlsanity.com/paper/2212.06663/) (2022)
- [Uncovering Instabilities in Variational-Quantum Deep Q-Networks](https://mlsanity.com/paper/2202.05195/) (2022)
- [An Empirical Comparison of Optimizers for Quantum Machine Learning with SPSA-based Gradients](https://mlsanity.com/paper/2305.00224/) (2023)
- [Incremental Data-Uploading for Full-Quantum Classification](https://mlsanity.com/paper/2205.03057/) (2022)
- [Qiskit-Torch-Module: Fast Prototyping of Quantum Neural Networks](https://mlsanity.com/paper/2404.06314/) (2024)
- [Quantum Natural Policy Gradients: Towards Sample-Efficient Reinforcement Learning](https://mlsanity.com/paper/2304.13571/) (2023)
- [Neutral Atom Quantum Computing Hardware: Performance and End-User Perspective](https://mlsanity.com/paper/2304.14360/) (2023)
- [BCQQ: Batch-Constraint Quantum Q-Learning with Cyclic Data Re-uploading](https://mlsanity.com/paper/2305.00905/) (2023)
- [1-2-3 Reproducibility for Quantum Software Experiments](https://mlsanity.com/paper/2201.12031/) (2022)
- [QPU-System Co-Design for Quantum HPC Accelerators](https://mlsanity.com/paper/2208.11449/) (2022)
- [Iteration Complexity of Variational Quantum Algorithms](https://mlsanity.com/paper/2209.10615/) (2022)
- [Influence of HW-SW-Co-Design on Quantum Computing Scalability](https://mlsanity.com/paper/2306.04246/) (2023)

## Built on by (most cited first, 4 of 4)

- [Fourier Analysis of Variational Quantum Circuits for Supervised Learning](https://mlsanity.com/paper/2411.03450/) (2024)
- [Scaling Recurrent Neural Networks to a Billion Parameters with Zero-Order Optimization](https://mlsanity.com/paper/2505.17852/) (2025)
- [Hype or Heuristic? Quantum Reinforcement Learning for Join Order Optimisation](https://mlsanity.com/paper/2405.07770/) (2024)
- [CutReg: A loss regularizer for enhancing the scalability of QML via adaptive circuit cutting](https://mlsanity.com/paper/2506.14858/) (2025)

## Abstract (arXiv)

The study of variational quantum algorithms (VQCs) has received significant attention from the quantum computing community in recent years. These hybrid algorithms, utilizing both classical and quantum components, are well-suited for noisy intermediate-scale quantum devices. Though estimating exact gradients using the parameter-shift rule to optimize the VQCs is realizable in NISQ devices, they do not scale well for larger problem sizes. The computational complexity, in terms of the number of circuit evaluations required for gradient estimation by the parameter-shift rule, scales linearly with the number of parameters in VQCs. On the other hand, techniques that approximate the gradients of the VQCs, such as the simultaneous perturbation stochastic approximation (SPSA), do not scale with the number of parameters but struggle with instability and often attain suboptimal solutions. In this work, we introduce a novel gradient estimation approach called Guided-SPSA, which meaningfully combines the parameter-shift rule and SPSA-based gradient approximation. The Guided-SPSA results in a 15% to 25% reduction in the number of circuit evaluations required during training for a similar or better optimality of the solution found compared to the parameter-shift rule. The Guided-SPSA outperforms standard SPSA in all scenarios and outperforms the parameter-shift rule in scenarios such as suboptimal initialization of the parameters. We demonstrate numerically the performance of Guided-SPSA on different paradigms of quantum machine learning, such as regression, classification, and reinforcement learning.