这篇论文用CNN和VQA两种方法估测量子系统的熵，发现CNN只用12.5%的数据就能准确估计，对更大系统效果更好，值得关注。
该论文系统研究了多qutrit量子系统中von Neumann熵的估计问题，使用两种互补方法：变分量子算法(VQA)和经典卷积神经网络(CNN)。对于最多3个qutrit的系统，构建了11种SU(3)启发的ansatz，参数扫描表明估计精度主要由可训练参数数量决定，并固定约120个参数。对于2至5个qutrit的系统，基于张量积互斥基测量结果训练的CNN仅使用全态层析所需12.5%的测量，即可对4与5 qutrit系统实现90百分位绝对误差约0.13-0.16 nats。CNN对shot噪声鲁棒，且泛化到分布外状态。结果显示VQA适用于小系统，而CNN估计器在大qutrit系统中具有更好的可扩展性和鲁棒性。
Entropy Estimation in Multi-Qutrit Systems via Variational and Classical Neural Networks
We present a systematic study of von Neumann entropy estimation in multi-qutrit quantum systems using two complementary approaches: variational quantum algorithms (VQAs) and classical convolutional neural networks (CNNs), evaluated using an ideal (noise-free) quantum simulator. For systems up to three qutrits, we construct and evaluate 11 hardware-efficient SU(3)-inspired ansatzes. A parameter sweep shows that estimation accuracy is primarily determined by the number of trainable parameters, provided sufficient entanglement is present. Based on this study, we fix the parameter count to approximately 120 for subsequent experiments, observing that increasing entangling-gate counts beyond a threshold yields only marginal improvements. For larger systems (two to five qutrits), we use a CNN trained on measurement outcomes from tensor-product mutually unbiased bases. The model achieves accurate and stable predictions and exhibits a systematic improvement in performance with system size, with the highest errors for two-qutrit systems and the lowest for five-qutrit systems. Notably, using only 12.5% of the measurements required for full state tomography is sufficient to reach 90th-percentile absolute errors of approximately 0.13-0.16 nats for both four- and five-qutrit systems. The CNN model is also robust to shot noise and generalizes well to out-of-distribution states. Overall, within the simulated settings studied here, our results indicate a transition in practical methods: VQAs are effective for small systems, while CNN-based estimators offer improved scalability and robustness for larger qutrit systems.