Estimating the Euclidean quantum propagator with deep generative modeling of Feynman paths

Yanming Che, Clemens Gneiting, and Franco Nori

Phys. Rev. B 105, 214205 – Published 15 June 2022

Abstract

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.

Received 8 February 2022
Revised 1 June 2022
Accepted 1 June 2022

DOI:https://doi.org/10.1103/PhysRevB.105.214205

Physics Subject Headings (PhySH)

Machine learning

Deep learning Path-integral methods

Quantum Information, Science & TechnologyStatistical Physics & ThermodynamicsGeneral Physics

Authors & Affiliations

Yanming Che ^1,2,*, Clemens Gneiting^2,3,†, and Franco Nori ^1,2,3,‡

¹Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
²Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
³RIKEN Center for Quantum Computing (RQC), 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan

^*yanmingche01@gmail.com
^†clemens.gneiting@riken.jp
^‡fnori@riken.jp

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 105, Iss. 21 — 1 June 2022

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Schematics of Feynman paths and path integrals in (a) Lorentzian and (b) Euclidean space-times. The times $τ$ and $t$ are connected via the Wick rotation $i t \to τ$ . The integrands of the propagators are plotted with the respective representative action curves $S [x (t)]$ and $S_{E} [x (τ)]$ . In (a), $Re$ and $Im$ denote the real and imaginary parts, respectively, and $x_{c l} (t)$ stands for the classical path (stationary path), which satisfies $δ S = 0$ [3]. In (b), $x_{c l} (τ)$ is the path with the least Euclidean action. It can be seen that such paths and their neighborhoods contribute dominantly to the propagators, while large deviations away from them cancel each other through rapid oscillations in (a), and are exponentially suppressed in (b). In (b), the exponential decay factor defines a probability distribution function for the Feynman paths in Euclidean space-time. In the case when $x_{c l} (τ)$ is identical to the classical Euclidean path, the red cloud around it represents quantum fluctuations.
Reuse & Permissions
Figure 2
(a) A variational realization of the Feynman path generator (VFPG) with a recurrent neural network (RNN). The input of the VFPG is a two-dimensional (2D) latent vector $z$ sampled from the standard Gaussian distribution $N (z; 0, I)$ , followed by a repeat operation (Repeat) that extends the dimension to match the input shape of the RNN. The output sequence is fed into densely connected layers (Dense) to produce the variational parameters $ϕ$ for the path distribution $q_{ϕ} [x (τ)]$ , which is modeled, at each time stamp, by a Gaussian mixture (see Appendix pp1 for details). The network is trained to target the path distribution in Eq. (6). (b) The structure of the RNN in (a), where $X (τ)$ and $Y (τ)$ are the input and output sequences, respectively; the blue squares on the right denote the RNN units, and the blue arrows denote the hidden state transfer [55]. In this work we use the long short-term memory (LSTM) [56] for the RNN.
Reuse & Permissions
Figure 3
Estimating the Euclidean quantum propagator through the variational Feynman path generator (VFPG) for a harmonic oscillator, where $V (x) = \frac{1}{2} m ω^{2} x^{2}$ . (a) The trace-normalized propagator vs $x_{f}$ , with $x_{i} = 0$ , $ω = 1$ , and $T = 0.5$ . (b) The ground-state probability density obtained from the Euclidean propagator. The blue dashed line is fit from the VFPG observations (blue dots), and we have used $ω = 5$ and $T = 1$ . In (a) and (b), the error bars denote the two standard deviations of the plotted quantity, evaluated from the training noise (standard deviation) of $F_{ϕ}$ . See the main text. (c) An example of the distribution of $10^{4}$ generated Feynman paths in the plane of the variational log probability and the action $S_{E}$ , where both axes are shifted, with $S_{E \min}$ the minimum action and $b = (- F_{ϕ} + S_{E \min}) / ℏ$ . The red solid line represents the exact exponential distribution (6). It is also the location of Feynman paths satisfying $F_{ϕ} [x (τ)] = F_{ϕ}$ , and the color (from blue to red) indicates the distance between $F_{ϕ} [x (τ)]$ and its expectation value $F_{ϕ}$ . (d) An example of the free energy $F_{ϕ}$ (blue line) during the training process for estimating the quantum propagator in (a), where the width of the red shaded area denotes its two standard deviations in the generated path ensemble. The parameters used in (c) and (d) are the same as that in (a), except that the final positions are fixed at $x_{f} = 1$ . Other representative parameters used are $N_{τ} = 32$ , $ℏ = 1$ , and $m = 1$ , respectively.
Reuse & Permissions
Figure 4
Double-well potential $V (x) = α x^{4} + β x^{2}$ . (a) Ground state wave function obtained from the Euclidean quantum propagator estimated via the variational Feynman path generator (VFPG), where the red dashed line is the result of the exact diagonalization. The green dashed line denotes the shifted and rescaled potential energy $\tilde{V} (x) = [V (x) + 5] / 100$ . (b) An example of the distribution of $10^{4}$ generated Feynman paths (with $x_{i} = x_{f} = x = 3$ ) in the plane of the log-probability and the action $S_{E}$ , where both axes are shifted, with $S_{E \min}$ the minimum action and $b = (- F_{ϕ} + S_{E \min}) / ℏ$ . The red solid line represents the training target (6). It is also the location of Feynman paths that exactly satisfy $F_{ϕ} [x (τ)] = F_{ϕ}$ , and the color (from blue to red) indicates the distance $d = | F_{ϕ} [x (τ)] - F_{ϕ} |$ from the exact distribution. Representative parameters used are: $α = 0.05$ , $β = - 1$ , $N_{τ} = 32$ , $ℏ = 1$ , $m = 1$ and $T = 2$ , respectively.
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics