Entropy Estimation in Multi-Qutrit Systems via Variational and Classical Neural Networks

2026-06-18

AI Summary

This work addresses the high measurement cost of full quantum state tomography for estimating von Neumann entropy in multi-qutrit systems by proposing two complementary approaches. For small-scale systems, a hardware-efficient variational quantum algorithm inspired by SU(3) symmetry is employed, while for medium- to large-scale systems, a convolutional neural network (CNN) is trained using measurements in tensor-product mutually unbiased bases. Experimental results demonstrate that the variational method excels for systems with up to three qutrits, whereas the CNN achieves decreasing estimation error with increasing system size—reaching absolute errors of only 0.13–0.16 nats for 90% of predictions in 4–5 qutrit systems—and exhibits strong robustness to noise and out-of-distribution generalization. Both methods achieve high-fidelity entropy estimation using merely 12.5% of the measurements required by full tomography, thereby establishing a practical boundary for transitioning from quantum to classical estimation strategies.

Abstract

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.

Problem

• entropy estimation
• multi-qutrit systems
• von Neumann entropy
• quantum state tomography
• scalability

Innovation

• variational quantum algorithms
• convolutional neural networks
• von Neumann entropy
• multi-qutrit systems
• quantum state tomography