# Performance Guarantees for Quantum Neural Estimation of Entropies

arXiv: https://arxiv.org/abs/2511.19289 · HTML: https://mlsanity.com/paper/2511.19289/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 103 references that are in the map.
- First posted on arXiv 24 Nov 2025.

## Builds on (most cited first, 25 of 25)

- [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)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Generalization and Equilibrium in Generative Adversarial Nets (GANs)](https://mlsanity.com/paper/1703.00573/) (2017)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [Layerwise learning for quantum neural networks](https://mlsanity.com/paper/2006.14904/) (2020)
- [Does provable absence of barren plateaus imply classical simulability?](https://mlsanity.com/paper/2312.09121/) (2023)
- [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)
- [Neural Estimation of Statistical Divergences](https://mlsanity.com/paper/2110.03652/) (2021)
- [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)
- [Quantum Neural Estimation of Entropies](https://mlsanity.com/paper/2307.01171/) (2023)
- [Measuring Quantum Entropy](https://mlsanity.com/paper/1711.00814/) (2017)
- [A Quantum Algorithm Framework for Discrete Probability Distributions with Applications to R'enyi Entropy Estimation](https://mlsanity.com/paper/2212.01571/) (2022)
- [Estimating quantum relative entropies on quantum computers](https://mlsanity.com/paper/2501.07292/) (2025)
- [Semi-definite optimization of the measured relative entropies of quantum states and channels](https://mlsanity.com/paper/2406.19060/) (2024)

## Built on by (most cited first, 3 of 3)

- [Generative modeling using evolved quantum Boltzmann machines](https://mlsanity.com/paper/2512.02721/) (2025)
- [Quantum Maximum Likelihood Prediction via Hilbert Space Embeddings](https://mlsanity.com/paper/2602.18364/) (2026)
- [Unifying quantum measurement constructions via a relative-entropy minimum change principle](https://mlsanity.com/paper/2608.04055/) (2026)

## Abstract (arXiv)

Estimating quantum entropies and divergences is an important problem in quantum physics, information theory, and machine learning. Quantum neural estimators (QNEs), which utilize a hybrid classical-quantum architecture, have recently emerged as an appealing computational framework for estimating these measures. Such estimators combine classical neural networks with parametrized quantum circuits, and their deployment typically entails tedious tuning of hyperparameters controlling the sample size, network architecture, and circuit topology. This work initiates the study of formal guarantees for QNEs of measured (R'enyi) relative entropies in the form of non-asymptotic error risk bounds. We further establish exponential tail bounds showing that the error is sub-Gaussian and thus sharply concentrates about the ground truth value. For an appropriate sub-class of density operator pairs on a space of dimension d with bounded Thompson metric, our theory establishes a copy complexity of O(|Theta(U)|d/ε2) for QNE with a quantum circuit parameter set Theta(U), which has minimax optimal dependence on the accuracy ε. Additionally, if the density operator pairs are permutation invariant, we improve the dimension dependence above to O(|Theta(U)|polylog(d)/ε2). Our theory aims to facilitate principled implementation of QNEs for measured relative entropies and guide hyperparameter tuning in practice.