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Physical Review A

Machine learning quantum mechanical ground states based on stochastic mechanics

Kai-Hendrik Henk, Wolfgang Paul

Published Dec 12, 2023• 2 citations
Publisher

Abstract

The Rayleigh-Ritz variation principle is a proven way to find ground states and energies for bound quantum systems in the Schr\"odinger picture. Advances in machine learning and neural networks make it possible to extend it from an analytical search from a subspace of the complete Hilbert space to the a numerical search in the almost complete Hilbert space. In this paper, we extend the Rayleigh-Ritz principle to Nelson's stochastic mechanics formulation of nonrelativistic quantum mechanics and propose an algorithm to find the osmotic velocities $u(x)$, which contain the information of a quantum systems in this picture. As a proof of concept, we calculated $u(x)$ for one-dimensional systems, the harmonic oscillator, the double well and the P\"oschl-Teller potential. To obtain exited states, we calculate ground states of supersymmetrical partner Hamiltonians for each of these potentials. We will show that this method is more efficient than the stochastic optimal control algorithm that was the usual method to obtain osmotic velocities without going back to the Schr\"odinger equation.

DOI
10.1103/physreva.108.062412

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The Rayleigh-Ritz variation principle is a proven way to find ground states and energies for bound quantum systems in the Schr"odinger picture. Advances in machine learning and neural networks make it possible to extend it from an analytical search from a subspace of the complete Hilbert space to the a numerical search in the almost complete Hilbert space. In this paper, we extend the Rayleigh-Ritz principle to Nelson's stochastic mechanics formulation of nonrelativistic quantum mechanics and propose an algorithm to find the osmotic velocities u(x)u(x)u(x), which contain the information of a quantum systems in this picture. As a proof of concept, we calculated u(x)u(x)u(x) for one-dimensional systems, the harmonic oscillator, the double well and the P"oschl-Teller potential. To obtain exited states, we calculate ground states of supersymmetrical partner Hamiltonians for each of these potentials. We will show that this method is more efficient than the stochastic optimal control algorithm that was the usual method to obtain osmotic velocities without going back to the Schr"odinger equation.
Quantum mechanics is one of the most successful theories of modern physics. Nelson’s stochastic mechanics description offers an alternative approach to non-relativistic quantum mechanics based on Newtonian mechanics. In this thesis, we derive the stochastic mechanical equivalent to the quantum-mechanical Rayleigh-Ritz principle. This principle is then used to build a genetic algorithm that calculates the osmotic velocity together with the ground state energy by first minimizing the derived energy functional and then solving a Riccati equation. This new and efficient algorithm is then used to solve the ground state of two tweezer potentials used in levitodynamics. These are then compared to each other and with the harmonic oscillator by using methods from time series analysis. Additionally, we use the Itô-formula to derive two coupled stochastic differential equations, which yields a phase space description of quantum mechanics without violating the Heisenberg uncertainty principle.
On the basis of Nelson’s stochastic mechanics derivation of the Schrödinger equation, a formal mathematical structure of non-relativistic quantum mechanics equivalent to the one in classical analytical mechanics has been established in the literature. We recently were able to augment this structure by deriving quantum Hamilton equations of motion by finding the Nash equilibrium of a stochastic optimal control problem, which is the generalization of Hamilton’s principle of classical mechanics to quantum systems. We showed that these equations allow a description and numerical determination of the ground state of quantum problems without using the Schrödinger equation. We extend this approach here to deliver the complete discrete energy spectrum and related eigenfunctions for bound states of one-dimensional stationary quantum systems. We exemplify this analytically for the one-dimensional harmonic oscillator and numerically by analyzing a quartic double-well potential, a model of broad importance in many areas of physics. We furthermore point out a relation between the tunnel splitting of such models and mean first passage time concepts applied to Nelson’s diffusion paths in the ground state.
We present a moment equation Rayleigh - Ritz (RR) variational theory for the ground-state energy. An extended Fourier transform space can be defined through the moment equation corresponding to a given Schrödinger Hamiltonian. Within this extended space, we can implement a variational ansatz with respects to configurations of the form in which the energy-dependent functions : (i) satisfy the momentum space Schrödinger equation; (ii) are uniquely prescribed; (iii) are (most likely) non-integrable; and (iv) yield the physical solution for physical E and values. On the basis of this representation, one can minimize the energy expectation value with respects to the E and missing moment variables, . The proposed approach is in sharp contrast to traditional configuration-space RR implementations in which the selection of a variational basis is not manifest a priori, particularly for problems of spatial dimension greater than one. The analysis of one- and two-dimension problems is presented.
The potential energy surface (PES) of molecules with respect to their nuclear positions is a primary tool in understanding chemical reactions from first principles. However, obtaining this information is complicated by the fact that sampling a large number of ground states over a high-dimensional PES can require a vast number of state preparations. In this work, we propose using a generative quantum machine learning model to prepare quantum states at arbitrary points on the PES. The model is trained using quantum data consisting of ground-state wavefunctions associated with different classical nuclear coordinates. Our approach uses a classical neural network to convert the nuclear coordinates of a molecule into quantum parameters of a variational quantum circuit. The model is trained using a fidelity loss function to optimize the neural network parameters. We show that gradient evaluation is efficient and numerically demonstrate our method's ability to prepare wavefunctions on the PES of hydrogen chains, water, and beryllium hydride. In all cases, we find that a small number of training points are needed to achieve very high overlap with the groundstates in practice. From a theoretical perspective, we further prove limitations on these protocols by showing that if we were able to learn across an avoided crossing using a small number of samples, then we would be able to violate Grover's lower bound. Additionally, we prove lower bounds on the amount of quantum data needed to learn a locally optimal neural network function using arguments from quantum Fisher information. This work further identifies that quantum chemistry can be an important use case for quantum machine learning.
Quantum theory has been remarkably successful in providing an understanding of physical systems at foundational scales. Solving the Schrödinger equation provides full knowledge of all dynamical quantities of the physical system. However closed form solutions to this equation are only available for a few systems and approximation methods are typically used to find solutions. In this paper we address the problem of solving the time independent Schrödinger equation for the ground state solution of physical systems. We propose using end-to-end deep learning approach in a variational optimization scheme for approximating the ground state energies and wave functions of these systems. A neural network realizes a universal trial wave function and is trained in an unsupervised learning framework by optimizing the expectation value of the Hamiltonian of a physical system. The proposed approach is evaluated on physical systems consisting of a particle in a box with and without a perturbation. We demonstrate that our approach obtains approximations of ground state energies and wave functions that are highly accurate, which makes it a potentially plausible candidate for solving more complex physical systems for which analytical solutions are beyond reach.
Finding the ground state of a quantum mechanical system can be formulated as an optimal control problem. In this formulation, the drift of the optimally controlled process is chosen to match the distribution of paths in the Feynman--Kac (FK) representation of the solution of the imaginary time Schr\"odinger equation. This provides a variational principle that can be used for reinforcement learning of a neural representation of the drift. Our approach is a drop-in replacement for path integral Monte Carlo, learning an optimal importance sampler for the FK trajectories. We demonstrate the applicability of our approach to several problems of one-, two-, and many-particle physics.