Quantum computers are next-generation devices that hold promise to perform calculations beyond the reach of classical computers. A leading method towards achieving this goal is through quantum machine learning, especially quantum generative learning. Due to the intrinsic probabilistic nature of quantum mechanics, it is reasonable to postulate that quantum generative learning models (QGLMs) may surpass their classical counterparts. As such, QGLMs are receiving growing attention from the quantum physics and computer science communities, where various QGLMs that can be efficiently implemented on near-term quantum machines with potential computational advantages are proposed. In this paper, we review the current progress of QGLMs from the perspective of machine learning. Particularly, we interpret these QGLMs, covering quantum circuit Born machines, quantum generative adversarial networks, quantum Boltzmann machines, and quantum variational autoencoders, as the quantum extension of classical generative learning models. In this context, we explore their intrinsic relations and their fundamental differences. We further summarize the potential applications of QGLMs in both conventional machine learning tasks and quantum physics. Last, we discuss the challenges and further research directions for QGLMs.

Isolating past information relevant for future prediction is central to quantitative science. Quantum models offer a promising approach, enabling statistically faithful modeling while using less past information than any classical counterpart. Here we introduce a class of phase-enhanced quantum models, representing the most general means of simulating a stochastic process unitarily in causal order. The resulting constructions surpass previous state-of-art methods—both in reducing the information they need to store about the past and in the minimal memory dimension they require to store this information. Moreover, these two features are generally competing factors in optimization—leading to an ambiguity in optimal modeling that is unique to the quantum regime. Our results simultaneously offer quantum advantages for stochastic simulation and illustrate further qualitative differences between classical and quantum notions of complexity.

Shadow estimation is an efficient method for predicting many observables of a quantum state with a statistical guarantee. In the multi-shot scenario, one performs projective measurement on the sequentially prepared state for K times after the same unitary evolution, and repeats this procedure for M rounds of random sampled unitary. As a result, there are MK times measurements in total. Here we analyze the performance of shadow estimation in this multi-shot scenario, which is characterized by the variance of estimating the expectation value of some observable O. We find that in addition to the shadow-norm ‖O‖shadow introduced in \cite{huang2020predicting}, the variance is also related to another norm, and we denote it as the cross-shadow-norm ‖O‖Xshadow. For both random Pauli and Clifford measurements, we analyze and show the upper bounds of ‖O‖Xshadow. In particular, we figure out the exact variance formula for Pauli observable under random Pauli measurements. Our work gives theoretical guidance for the application of multi-shot shadow estimation.

Heisenberg’s uncertainty principle implies fundamental constraints on what properties of a quantum system we can simultaneously learn. However, it typically assumes that we probe these properties via measurements at a single point in time. In contrast, inferring causal dependencies in complex processes often requires interactive experimentation—multiple rounds of interventions where we adaptively probe the process with different inputs to observe how they affect outputs. Here, we demonstrate universal uncertainty principles for general interactive measurements involving arbitrary rounds of interventions. As a case study, we show that they imply an uncertainty trade-off between measurements compatible with different causal dependencies.