# Efficient Gradient Estimation for Parameterized Quantum Systems with Lie Algebraic Symmetries

arXiv: https://arxiv.org/abs/2404.05108 · HTML: https://mlsanity.com/paper/2404.05108/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 48 references that are in the map.
- First posted on arXiv 7 Apr 2024.

## Builds on (most cited first, 13 of 13)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [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)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [Stochastic gradient descent for hybrid quantum-classical optimization](https://mlsanity.com/paper/1910.01155/) (2019)
- [On quantum backpropagation, information reuse, and cheating measurement collapse](https://mlsanity.com/paper/2305.13362/) (2023)
- [Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms](https://mlsanity.com/paper/1901.05374/) (2019)
- [A Theoretical Framework for Learning from Quantum Data](https://mlsanity.com/paper/2107.06406/) (2021)
- [Toward Physically Realizable Quantum Neural Networks](https://mlsanity.com/paper/2203.12092/) (2022)
- [Learning k-qubit Quantum Operators via Pauli Decomposition](https://mlsanity.com/paper/2102.05209/) (2021)

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

- [Efficient Quantum Gradient and Higher-order Derivative Estimation via Generalized Hadamard Test](https://mlsanity.com/paper/2408.05406/) (2024)

## Abstract (arXiv)

Gradient estimation is a central challenge in training parameterized quantum circuits (PQCs) for hybrid quantum-classical optimization and learning problems. This difficulty arises from several factors, including the exponential dimensionality of the Hilbert spaces and the information loss in quantum measurements. Existing estimators, such as finite difference and the parameter shift rule, often fail to adequately address these challenges for certain classes of PQCs. In this work, we propose a novel gradient estimation framework that leverages the underlying Lie algebraic structure of PQCs, combined with the Hadamard test. By analyzing the differential of the matrix exponential, we derive an expression for the gradient as a linear combination of expectation values obtained via Hadamard tests. The coefficients in this decomposition depend solely on the circuit's parameterization and can be estimated using state-of-the-art shadow tomography techniques. Hence, our approach enables efficient gradient estimation, requiring a number of measurement shots that scales logarithmically with the number of parameters, and with polynomial classical and quantum time. This is an exponential reduction in the measurement cost and a polynomial speed-up in time compared to existing works.