Sample Complexity for Distance Estimation between Ensembles of States

The generation of ensembles of quantum states is a common task in quantum information processing, such as quantum cryptography, quantum machine learning, and quantum thermalization. Recent works on the generative learning of ensembles of states suggest distance metrics such as the Wasserstein distance as cost functions to characterize deviations between ensembles of states. Here we propose k-moment maximum mean discrepancy (k-MMD) besides the Wasserstein distance, and analyze the sample complexity of estimating these quantities. As k increases, k-MMD forms a hierarchy of cost functions with increasing discriminative power for learning, and at the same time the cost of estimation increases. We explore this trade-off using numerical simulations and analytical calculations. As the number of states N increases, the sample complexity of estimating k-MMD remains constant for small N and scales as N^{2-2/k} for large N, while it scales as N^2 log N for the Wasserstein distance. As an application, we use k-MMD to train the quantum denoising diffusion probabilistic model (QuDDPM). Our results show that k-MMD can learn ensembles previously thought to require the Wasserstein distance.