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 (Rényi) 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 $O(|Θ(\mathcal{U})|d/ε^2)$ for QNE with a quantum circuit parameter set $Θ(\mathcal{U})$, 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 $O(|Θ(\mathcal{U})|\mathrm{polylog}(d)/ε^2)$. Our theory aims to facilitate principled implementation of QNEs for measured relative entropies and guide hyperparameter tuning in practice.

翻译：在量子物理学、信息理论和机器学习中，估计量子熵与散度是一个重要问题。最近，利用混合经典-量子架构的量子神经估计器（QNEs）作为一种有吸引力的计算框架，被用于估计这些度量指标。此类估计器将经典神经网络与参数化量子电路相结合，其部署通常需要对控制样本量、网络架构和电路拓扑的超参数进行繁琐调整。本文首次为实测（Rényi）相对熵的QNEs建立了形式化保证，形式为非渐近误差风险界。我们进一步建立了指数尾界，表明误差是次高斯的，因此会围绕真值高度集中。对于维度为$d$且具有有界Thompson度量的空间中的密度算子对的一个适当子类，我们的理论证明：对于量子电路参数集$Θ(\mathcal{U})$的QNE，其副本复杂度为$O(|Θ(\mathcal{U})|d/ε^2)$，该复杂度在精度$ε$上具有极小极大最优依赖性。此外，若密度算子对是置换不变的，我们将上述维度依赖性改进为$O(|Θ(\mathcal{U})|\mathrm{polylog}(d)/ε^2)$。我们的理论旨在促进针对实测相对熵的QNEs的原理性实现，并为实际中的超参数调优提供指导。