Sampling from high-dimensional and structured probability distributions is a fundamental challenge in computational physics, particularly in the context of lattice field theory (LFT), where generating field configurations efficiently is critical, yet computationally intensive. In this work, we apply a previously developed hybrid quantum-classical normalizing flow model to explore quantum-enhanced sampling in such regimes. Our approach embeds parameterized quantum circuits within a classical normalizing flow architecture, leveraging amplitude encoding and quantum entanglement to enhance expressivity in the generative process. The quantum circuit serves as a trainable transformation within the flow, while classical networks provide adaptive coupling and compensate for quantum hardware imperfections. This design enables efficient density estimation and sample generation, potentially reducing the resources required compared to purely classical methods. While LFT provides a representative and physically meaningful application for benchmarking, our focus is on improving the sampling efficiency of generative models through quantum components. This work contributes toward the development of quantum-enhanced generative modeling frameworks that address the sampling bottlenecks encountered in physics and beyond.
Sampling from high-dimensional and structured probability distributions is a fundamental challenge in computational physics, particularly in the context of lattice field theory (LFT), where generating field configurations efficiently is critical, yet computationally intensive. In this work, we apply a previously developed hybrid quantum-classical normalizing flow model to explore quantum-enhanced sampling in such regimes. Our approach embeds parameterized quantum circuits within a classical normalizing flow architecture, leveraging amplitude encoding and quantum entanglement to enhance expressivity in the generative process. The quantum circuit serves as a trainable transformation within the flow, while classical networks provide adaptive coupling and compensate for quantum hardware imperfections. This design enables efficient density estimation and sample generation, potentially reducing the resources required compared to purely classical methods. While LFT provides a representative and physically meaningful application for benchmarking, our focus is on improving the sampling efficiency of generative models through quantum components. This work contributes toward the development of quantum-enhanced generative modeling frameworks that address the sampling bottlenecks encountered in physics and beyond.
With the rapid development of quantum computing technology, we have entered the era of noisy intermediate-scale quantum (NISQ) computers. Therefore, designing quantum algorithms that adapt to the hardware conditions of current NISQ devices and can preliminarily solve some practical problems has become the focus of researchers. In this paper, we focus on quantum generative models in the field of quantum machine learning, and propose a hybrid quantum-classical normalizing flow (HQCNF) model based on parameterized quantum circuits. Based on the ideas of classical normalizing flow models and the characteristics of parameterized quantum circuits, we cleverly design the form of the ansatz and the hybrid method of quantum and classical computing, and derive the form of the loss function in the case that quantum computing is involved. We test our model on the image generation problem. Experimental results show that our model is capable of generating images of good quality. Compared with other quantum generative models, such as quantum generative adversarial networks (QGAN), our model achieves lower (better) Fréchet inception distance (FID) score, and compared with classical generative models, we can complete the image generation task with significantly fewer parameters. These results prove the advantage of our proposed model.
Learning many-body quantum systems is of fundamental importance in quantum information processing, however, it is a challenging task which typically requires estimating quantum distributions of dimensionality exponentially scaling to the system size. As generative models have shown a great scalability to learn high-dimensional distributions and found wide applications in the domain of image and text, they can be a powerful tool to facilitate us to accomplish the challenging quantum tasks. In this work, we propose using normalizing flow (NF) models with fast sampling to learn discrete quantum distributions for quantum state tomography. Particularly, three NF models, including denoising flow, argmax flow, and tree flow, are first adapted to the task of explicit quantum probability density estimation. We then perform extensive experiments on a large scale of quantum systems, and our numerical results demonstrate that these discrete NFs admit an excellent sampling efficiency in the sense that they are insensitive to the system size to learn the high-dimensional quantum distributions, without compromising the learning performance. Finally, in comparison to the other generative models, such as autoregressive, the NFs avoid the problem of slow sequential sampling.
The convergence of quantum and classical computing, particularly in the realm of quantum machine learning, has gained substantial attention. Hybrid quantum algorithms, merging the strengths of both computing paradigms, offer a promising avenue, especially in the era of NISQ (noisy intermediate-scale quantum) computing. Leveraging these hybrid approaches allows researchers to employ quantum algorithms on existing quantum processors while harnessing classical methods to handle complex computational challenges. A core component of these hybrid systems is the Parameterized Quantum Circuit Element (PQCE). These circuits, defined by parameterized unitary operations, possess exceptional expressiveness, effectively modeling intricate distributions. This study introduces a novel application of Parameterized Quantum Circuits in Generative Flow Networks for object generation tasks. This model demonstrates efficiency and quality on par with classical deep neural network-based approaches. Despite longer training times, the quantum model achieves comparable results with significantly fewer parameters (151), emphasizing its potential in scenarios prioritizing parameter efficiency.
Quantum generative modeling has emerged as a promising application of quantum computers, aiming to model complex probability distributions beyond the reach of classical methods. In practice, however, training such models often requires costly gradient estimation performed directly on the quantum hardware. Crucially, for certain structured quantum circuits, expectation values of local observables can be efficiently evaluated on a classical computer, enabling classical training without calls to the quantum hardware in the optimization loop. In these models, sampling from the resulting circuits can still be classically hard, so inference must be performed on a quantum device, yielding a potential computational advantage. In this work, we introduce a photonic quantum generative model built on parametrized Gaussian Boson Sampling circuits. The training is based on the efficient classical evaluation of expectation values enabled by the Gaussian structure of the state, allowing scalable optimization of the model parameters through the maximum mean discrepancy loss function. We demonstrate the effectiveness of the approach through numerical experiments on photonic systems with up to 805 modes and over a million trainable parameters, highlighting its scalability and suitability for near-term photonic quantum devices.
This paper explores the application of adaptive quantum sampling to probability density estimation, a critical task in various fields including physics, chemistry, and finance. Classical methods struggle with high-dimensional probability distributions, often requiring computationally intensive sampling techniques. We introduce a novel quantum algorithm that dynamically adjusts its sampling strategy based on observed data, significantly enhancing the accuracy of probability density estimation compared to traditional methods. The core mechanism leverages quantum circuits to implement a probabilistic sampling process, ensuring a more robust and efficient estimation of distributions. We detail the algorithm's architecture, key quantum operations, and demonstrate its effectiveness through simulations and illustrative examples. The resulting quantum algorithm holds the potential to unlock deeper insights into complex probability distributions, offering a significant advancement in statistical analysis.
We propose a hybrid variational framework that enhances Neural Quantum States (NQS) with a Normalising Flow-based sampler to improve the expressivity and trainability of quantum many-body wavefunctions. Our approach decouples the sampling task from the variational ansatz by learning a continuous flow model that targets a discretised, amplitude-supported subspace of the Hilbert space. This overcomes limitations of Markov Chain Monte Carlo (MCMC) and autoregressive methods, especially in regimes with long-range correlations and volume-law entanglement. Applied to the transverse-field Ising model with both short- and long-range interactions, our method achieves comparable ground state energy errors with state-of-the-art matrix product states and lower energies than autoregressive NQS. For systems up to 50 spins, we demonstrate high accuracy and robust convergence across a wide range of coupling strengths, including regimes where competing methods fail. Our results showcase the utility of flow-assisted sampling as a scalable tool for quantum simulation and offer a new approach toward learning expressive quantum states in high-dimensional Hilbert spaces.