Researchers have explored two complementary methodologies for estimating von Neumann entropy in multi-qutrit quantum systems: variational quantum algorithms (VQAs) and classical convolutional neural networks (CNNs). The study, conducted using an ideal (noise-free) quantum simulator, evaluated the effectiveness of these techniques for quantifying entanglement and information in quantum systems that use qutrits, three-state quantum information units (unlike two-state qubits).

For small systems, up to three qutrits, 11 hardware-efficient SU(3)-inspired ansatzes were constructed and evaluated. Results indicated that estimation accuracy is primarily determined by the number of trainable parameters, provided sufficient entanglement is present. A parameter count of approximately 120 was fixed for subsequent experiments, observing that increasing entangling-gate counts beyond a threshold yielded only marginal improvements. For larger systems, from two to five qutrits, a CNN trained on measurement outcomes from tensor-product mutually unbiased bases was employed.

The CNN model demonstrated accurate and stable predictions, with performance improving with system size. The highest errors were observed for two-qutrit systems and the lowest for five-qutrit systems. Notably, using only 12.5% of the measurements required for full state tomography was sufficient to reach 90th-percentile absolute errors of approximately 0.13-0.16 nats for both four- and five-qutrit systems. Furthermore, the CNN model proved robust to shot noise and generalized well to out-of-distribution states.

These findings suggest a transition in practical methods for entropy estimation: VQAs are effective for small systems, while CNN-based estimators offer improved scalability and robustness for larger qutrit systems. This advance is crucial for the development of qutrit-based quantum computing, which could offer advantages in certain quantum architectures and algorithms by allowing for greater information density per quantum unit.