Recent advancements highlight the pivotal role of quantum machine learning (QML) [4, 13] in processing quantum data derived from quantum systems [14]. A fundamental task in QML is generating quantum data by learning the underlying distribution, essential for understanding quantum systems, synthesizing new samples, and advancing applications in quantum chemistry and materials science. However, extending classical generative approaches to quantum data presents significant challenges, as quantum distributions exhibit superposition, entanglement, and non-locality that classical models struggle to replicate efficiently. Quantum generative models such as quantum generative adversarial networks [24, 42] and quantum variational autoencoders [20, 38] can be used to prepare a fixed single quantum state [21, 28, 37], but are inefficient for generating ensembles of quantum states [3] due to the need for training deep parameterized quantum circuits (PQCs). The quantum denoising diffusion probabilistic model [40] offers a promising approach that employs intermediate training steps to smoothly interpolate between the target distribution and noise, thereby enabling efficient training.

Many time series can be modeled as a sequence of segments representing high-level discrete states, such as running and walking in a human activity application. Flexible models should describe the system state and observations in stationary ``pure-state'' periods as well as transition periods between adjacent segments, such as a gradual slowdown between running and walking. However, most prior work assumes instantaneous transitions between pure discrete states. We propose a dynamical Wasserstein barycentric (DWB) model that estimates the system state over time as well as the data-generating distributions of pure states in an unsupervised manner. Our model assumes each pure state generates data from a multivariate normal distribution, and characterizes transitions between states via displacement-interpolation specified by the Wasserstein barycenter. The system state is represented by a barycentric weight vector which evolves over time via a random walk on the simplex. Parameter learning leverages the natural Riemannian geometry of Gaussian distributions under the Wasserstein distance, which leads to improved convergence speeds. Experiments on several human activity datasets show that our proposed DWB model accurately learns the generating distribution of pure states while improving state estimation for transition periods compared to the commonly used linear interpolation mixture models.

Quantum theory provides non-classical principles, such as superposition and entanglement, that inspires promising paradigms in machine learning. However, most existing quantum-inspired fusion models rely solely on unitary or unitary-like transformations to generate quantum entanglement. While theoretically expressive, such approaches often suffer from training instability and limited generalizability. In this work, we propose a Quantum-inspired Neural Network with Quantum Jump (QiNN-QJ) for multimodal entanglement modelling. Each modality is firstly encoded as a quantum pure state, after which a differentiable module simulating the QJ operator transforms the separable product state into the entangled representation. By jointly learning Hamiltonian and Lindblad operators, QiNN-QJ generates controllable cross-modal entanglement among modalities with dissipative dynamics, where structured stochasticity and steady-state attractor properties serve to stabilize training and constrain entanglement shaping. The resulting entangled states are projected onto trainable measurement vectors to produce predictions. In addition to achieving superior performance over the state-of-the-art models on benchmark datasets, including CMU-MOSI, CMU-MOSEI, and CH-SIMS, QiNN-QJ facilitates enhanced post-hoc interpretability through von-Neumann entanglement entropy. This work establishes a principled framework for entangled multimodal fusion and paves the way for quantum-inspired approaches in modelling complex cross-modal correlations.

Many time series can be modeled as a sequence of segments representing high-level discrete states, such as running and walking in a human activity application.

Many time series can be modeled as a sequence of segments representing high-level discrete states, such as running and walking in a human activity application. Flexible models should describe the system state and observations in stationary pure-state'' periods as well as transition periods between adjacent segments, such as a gradual slowdown between running and walking. However, most prior work assumes instantaneous transitions between pure discrete states. We propose a dynamical Wasserstein barycentric (DWB) model that estimates the system state over time as well as the data-generating distributions of pure states in an unsupervised manner. Our model assumes each pure state generates data from a multivariate normal distribution, and characterizes transitions between states via displacement-interpolation specified by the Wasserstein barycenter.

Quantum state tomography, the task of learning an unknown quantum state, is a fundamental problem in quantum information. In standard settings, the complexity of this problem depends significantly on the type of quantum state that one is trying to learn, with pure states being substantially easier to learn than general mixed states. A natural question is whether this separation holds for any quantum state learning setting. In this work, we consider the online learning framework and prove the surprising result that learning pure states in this setting is as hard as learning mixed states. More specifically, we show that both classes share almost the same sequential fat-shattering dimension, leading to identical regret scaling under the $L_1$-loss. We also generalize previous results on full quantum state tomography in the online setting to learning only partially the density matrix, using smooth analysis.

Many time series can be modeled as a sequence of segments representing high-level discrete states, such as running and walking in a human activity application. Flexible models should describe the system state and observations in stationary pure-state'' periods as well as transition periods between adjacent segments, such as a gradual slowdown between running and walking. However, most prior work assumes instantaneous transitions between pure discrete states. We propose a dynamical Wasserstein barycentric (DWB) model that estimates the system state over time as well as the data-generating distributions of pure states in an unsupervised manner. Our model assumes each pure state generates data from a multivariate normal distribution, and characterizes transitions between states via displacement-interpolation specified by the Wasserstein barycenter.

Learning faithful representations of quantum states is crucial to fully characterizing the variety of many-body states created on quantum processors. While various tomographic methods such as classical shadow and MPS tomography have shown promise in characterizing a wide class of quantum states, they face unique limitations in detecting topologically ordered two-dimensional states. To address this problem, we implement and study a heuristic tomographic method that combines variational optimization on tensor networks with randomized measurement techniques. Using this approach, we demonstrate its ability to learn the ground state of the surface code Hamiltonian as well as an experimentally realizable quantum spin liquid state. In particular, we perform numerical experiments using MPS ans\"atze and systematically investigate the sample complexity required to achieve high fidelities for systems of sizes up to $48$ qubits. In addition, we provide theoretical insights into the scaling of our learning algorithm by analyzing the statistical properties of maximum likelihood estimation. Notably, our method is sample-efficient and experimentally friendly, only requiring snapshots of the quantum state measured randomly in the $X$ or $Z$ bases. Using this subset of measurements, our approach can effectively learn any real pure states represented by tensor networks, and we rigorously prove that random-$XZ$ measurements are tomographically complete for such states.