As an alternative but unified and more fundamental description for quantum physics, Feynman path integrals generalize the classical action principle to a probabilistic perspective, under which the physical observables' estimation translates into a weighted sum over all possible paths. The underlying difficulty is to tackle the whole path manifold from finite samples that can effectively represent the Feynman propagator dictated probability distribution. Modern generative models in machine learning can handle learning and representing probability distribution with high computational efficiency. In this study, we propose a Fourier-flow generative model to simulate the Feynman propagator and generate paths for quantum systems. As demonstration, we validate the path generator on the harmonic and anharmonic oscillators. The latter is a double-well system without analytic solutions. To preserve the periodic condition for the system, the Fourier transformation is introduced into the flow model to approach a Matsubara representation. With this novel development, the ground-state wave function and low-lying energy levels are estimated accurately. Our method offers a new avenue to investigate quantum systems with machine learning assisted Feynman Path integral solving.
As an alternative but unified and more fundamental description for quantum physics, Feynman path integrals generalize the classical action principle to a probabilistic perspective, under which the physical observables' estimation translates into a weighted sum over all possible paths. The underlying difficulty is to tackle the whole path manifold from finite samples that can effectively represent the Feynman propagator dictated probability distribution. Modern generative models in machine learning can handle learning and representing probability distribution with high computational efficiency. In this study, we propose a Fourier-flow generative model to simulate the Feynman propagator and generate paths for quantum systems. As demonstration, we validate the path generator on the harmonic and anharmonic oscillators. The latter is a double-well system without analytic solutions. To preserve the periodic condition for the system, the Fourier transformation is introduced into the flow model to approach a Matsubara representation. With this novel development, the ground-state wave function and low-lying energy levels are estimated accurately. Our method offers a new avenue to investigate quantum systems with machine learning assisted Feynman Path integral solving.
As an alternative but unified and more fundamental description for quantum physics, Feynman path integrals generalize the classical action principle to a probabilistic perspective, under which the physical observables' estimation translates into a weighted sum over all possible paths. The underlying difficulty is to tackle the whole path manifold from finite samples that can effectively represent the Feynman propagator dictated probability distribution. Modern generative models in machine learning can handle learning and representing probability distribution with high computational efficiency. In this study, we propose a Fourier-flow generative model to simulate the Feynman propagator and generate paths for quantum systems. As a demonstration, we validate the path generator on the harmonic and anharmonic oscillators. The latter is a double-well system without analytic solutions. To preserve the periodic condition for the system, the Fourier transformation is introduced into the flow model to approach a Matsubara representation. With this novel development, the ground-state wave function and low-lying energy levels are estimated accurately. Our method offers a new avenue to investigate quantum systems with machine learning assisted Feynman path integral solving.
Feynman path integrals provide an elegant, classically inspired representation for the quantum propagator and the quantum dynamics, through summing over a huge manifold of all possible paths. From computational and simulational perspectives, the ergodic tracking of the whole path manifold is a hard problem. Machine learning can help, in an efficient manner, to identify the relevant subspace and the intrinsic structure residing at a small fraction of the vast path manifold. In this work, we propose the Feynman path generator for quantum mechanical systems, which efficiently generates Feynman paths with fixed endpoints, from a (low-dimensional) latent space and by targeting a desired density of paths in the Euclidean space-time. With such path generators, the Euclidean propagator as well as the ground-state wave function can be estimated efficiently for a generic potential energy. Our work provides an alternative approach for calculating the quantum propagator and the ground-state wave function, paves the way toward generative modeling of quantum mechanical Feynman paths, and offers a different perspective to understand the quantum-classical correspondence through deep learning.
Flow models are a cornerstone of modern machine learning. They are generative models that progressively transform probability distributions according to learned dynamics. Specifically, they learn a continuous-time Markov process that efficiently maps samples from a simple source distribution into samples from a complex target distribution. We show that these models are naturally related to the Schrödinger equation, for an unusual Hamiltonian on continuous variables. Moreover, we prove that the dynamics generated by this Hamiltonian can be efficiently simulated on a quantum computer. Together, these results give a quantum algorithm for preparing coherent encodings (a.k.a., qsamples) for a vast family of probability distributions--namely, those expressible by flow models--by reducing the task to an existing classical learning problem, plus Hamiltonian simulation. For statistical problems defined by flow models, such as mean estimation and property testing, this enables the use of quantum algorithms tailored to qsamples, which may offer advantages over classical algorithms based only on samples from a flow model. More broadly, these results reveal a close connection between state-of-the-art machine learning models, such as flow matching and diffusion models, and one of the main expected capabilities of quantum computers: simulating quantum dynamics.
Machine learning has had a significant impact on multiple areas of science, technology, health, and computer and information sciences. Through the advent of quantum computing, quantum machine learning has developed as a new and important avenue for the study of complex learning problems. Yet there is substantial debate and uncertainty in regard to the foundations of machine learning. Here, we provide a detailed exposition of the mathematical connections between a general machine learning approach called Boltzmann machines and Feynman’s description of quantum and statistical mechanics. In Feynman’s description, quantum phenomena arise from an elegant, weighted sum over (or superposition of) paths. Our analysis shows that Boltzmann machines and neural networks have a similar mathematical structure. This allows the interpretation that the hidden layers in Boltzmann machines and neural networks are discrete versions of path elements and allows a path integral interpretation of machine learning similar to that in quantum and statistical mechanics. Since Feynman paths are a natural and elegant depiction of interference phenomena and the superposition principle germane to quantum mechanics, this analysis allows us to interpret the goal in machine learning as finding an appropriate combination of paths, and accumulated path-weights, through a network, that cumulatively captures the correct properties of an x -to- y map for a given mathematical problem. We are forced to conclude that neural networks are naturally related to Feynman path-integrals and hence may present one avenue to be considered as quantum problems. Consequently, we provide general quantum circuit models applicable to both Boltzmann machines and Feynman path integrals.
The path integral formulation of quantum mechanics generalizes the action principle of classical mechanics. The Feynman path integral is, roughly speaking, a sum over all possible paths that a particle can take between fixed endpoints, where each path contributes to the sum by a phase factor involving the action for the path. The resulting sum gives the probability amplitude of propagation between the two endpoints, a quantity called the propagator. Solutions of the Feynman path integral formula exist, however, only for a small number of simple systems, and modifications need to be made when dealing with more complicated systems involving singular potentials, including the Coulomb potential. We derive a generalized path integral formula, that can be used in these cases, for a quantity called the pseudo-propagator from which we obtain the fixed-energy amplitude, related to the propagator by a Fourier transform. The new path integral formula is then successfully solved for the Hydrogen atom in two and three dimensions, and we obtain integral representations for the fixed-energy amplitude.
Studying the dynamics of open quantum systems can enable breakthroughs both in fundamental physics and applications to quantum engineering and quantum computation. Since the density matrix ρ, which is the fundamental description for the dynamics of such systems, is high-dimensional, customized deep generative neural networks have been instrumental in modeling ρ. However, the complex-valued nature and normalization constraints of ρ, as well as its complicated dynamics, prohibit a seamless connection between open quantum systems and the recent advances in deep generative modeling. Here we lift that limitation by utilizing a reformulation of open quantum system dynamics to a partial differential equation (PDE) for a corresponding probability distribution Q, the Husimi Q function. Thus, we model the Q function seamlessly with off-the-shelf deep generative models such as normalizing flows. Additionally, we develop novel methods for learning normalizing flow evolution governed by high-dimensional PDEs based on the Euler method and the application of the time-dependent variational principle. We name the resulting approach Q-Flow and demonstrate the scalability and efficiency of Q-Flow on open quantum system simulations, including the dissipative harmonic oscillator and the dissipative bosonic model. Q-Flow is superior to conventional PDE solvers and state-of-the-art physics-informed neural network solvers, especially in high-dimensional systems.