# Quantum Neural Estimation of Entropies

arXiv: https://arxiv.org/abs/2307.01171 · HTML: https://mlsanity.com/paper/2307.01171/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 107 references that are in the map.
- First posted on arXiv 3 Jul 2023.

## Builds on (most cited first, 30 of 30)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [YOLOv7: Trainable bag-of-freebies sets new state-of-the-art for real-time object detectors](https://mlsanity.com/paper/2207.02696/) (2022)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [Generalization and Equilibrium in Generative Adversarial Nets (GANs)](https://mlsanity.com/paper/1703.00573/) (2017)
- [Noisy intermediate-scale quantum (NISQ) algorithms](https://mlsanity.com/paper/2101.08448/) (2021)
- [Exploiting symmetry in variational quantum machine learning](https://mlsanity.com/paper/2205.06217/) (2022)
- [Theory for Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.08566/) (2022)
- [Efficient quantum tomography II](https://mlsanity.com/paper/1612.00034/) (2016)
- [On the Discrimination-Generalization Tradeoff in GANs](https://mlsanity.com/paper/1711.02771/) (2017)
- [Universal expressiveness of variational quantum classifiers and quantum kernels for support vector machines](https://mlsanity.com/paper/2207.05865/) (2022)
- [Quantum Hamiltonian-Based Models and the Variational Quantum Thermalizer Algorithm](https://mlsanity.com/paper/1910.02071/) (2019)
- [Neural Estimation of Statistical Divergences](https://mlsanity.com/paper/2110.03652/) (2021)
- [Learning Unitaries by Gradient Descent](https://mlsanity.com/paper/2001.11897/) (2020)
- [Variational Representations and Neural Network Estimation of R'enyi Divergences](https://mlsanity.com/paper/2007.03814/) (2020)
- [Non-Asymptotic Performance Guarantees for Neural Estimation of mathsff-Divergences](https://mlsanity.com/paper/2103.06923/) (2021)
- [Quantum query complexity of entropy estimation](https://mlsanity.com/paper/1710.06025/) (2017)
- [Distributional property testing in a quantum world](https://mlsanity.com/paper/1902.00814/) (2019)
- [Variational Quantum Algorithms for Trace Distance and Fidelity Estimation](https://mlsanity.com/paper/2012.05768/) (2020)
- [Variational Quantum Algorithms for Semidefinite Programming](https://mlsanity.com/paper/2112.08859/) (2021)
- [Mitigating barren plateaus of variational quantum eigensolvers](https://mlsanity.com/paper/2205.13539/) (2022)
- [Estimating distinguishability measures on quantum computers](https://mlsanity.com/paper/2108.08406/) (2021)
- [Multivariate trace estimation in constant quantum depth](https://mlsanity.com/paper/2206.15405/) (2022)

## Built on by (most cited first, 6 of 6)

- [Quantum Boltzmann machine learning of ground-state energies](https://mlsanity.com/paper/2410.12935/) (2024)
- [Multimodal deep representation learning for quantum cross-platform verification](https://mlsanity.com/paper/2311.03713/) (2023)
- [Performance Guarantees for Quantum Neural Estimation of Entropies](https://mlsanity.com/paper/2511.19289/) (2025)
- [Estimating quantum relative entropies on quantum computers](https://mlsanity.com/paper/2501.07292/) (2025)
- [The Average Relative Entropy and Transpilation Depth determines the noise robustness in Variational Quantum Classifiers](https://mlsanity.com/paper/2603.21300/) (2026)
- [Entropy Estimation in Multi-Qutrit Systems via Variational and Classical Neural Networks](https://mlsanity.com/paper/2606.20504/) (2026)

## Abstract (arXiv)

Entropy measures quantify the amount of information and correlation present in a quantum system. In practice, when the quantum state is unknown and only copies thereof are available, one must resort to the estimation of such entropy measures. Here we propose a variational quantum algorithm for estimating the von Neumann and R'enyi entropies, as well as the measured relative entropy and measured R'enyi relative entropy. Our approach first parameterizes a variational formula for the measure of interest by a quantum circuit and a classical neural network, and then optimizes the resulting objective over parameter space. Numerical simulations of our quantum algorithm are provided, using a noiseless quantum simulator. The algorithm provides accurate estimates of the various entropy measures for the examples tested, which renders it as a promising approach for usage in downstream tasks.