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arXiv:2308.16848 · Science

Accurate computation of quantum excited states with neural networks

David Pfau, Simon Axelrod, Halvard Sutterud, Ingrid von Glehn, James S. Spencer

Published Aug 22, 2024• 69 citations
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Abstract

We present an algorithm to estimate the excited states of a quantum system by variational Monte Carlo, which has no free parameters and requires no orthogonalization of the states, instead transforming the problem into that of finding the ground state of an expanded system. Arbitrary observables can be calculated, including off-diagonal expectations, such as the transition dipole moment. The method works particularly well with neural network ansätze, and by combining this method with the FermiNet and Psiformer ansätze, we can accurately recover excitation energies and oscillator strengths on a range of molecules. We achieve accurate vertical excitation energies on benzene-scale molecules, including challenging double excitations. Beyond the examples presented in this work, we expect that this technique will be of interest for atomic, nuclear, and condensed matter physics.

DOI
10.1126/science.adn0137

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We present an algorithm to estimate the excited states of a quantum system by variational Monte Carlo, which has no free parameters and requires no orthogonalization of the states, instead transforming the problem into that of finding the ground state of an expanded system. Arbitrary observables can be calculated, including off-diagonal expectations, such as the transition dipole moment. The method works particularly well with neural network ansätze, and by combining this method with the FermiNet and Psiformer ansätze, we can accurately recover excitation energies and oscillator strengths on a range of molecules. We achieve accurate vertical excitation energies on benzene-scale molecules, including challenging double excitations. Beyond the examples presented in this work, we expect that this technique will be of interest for atomic, nuclear, and condensed matter physics.
Obtaining accurate ground and low-lying excited states of electronic systems is crucial in a multitude of important applications. One ab initio method for solving the Schrödinger equation that scales favorably for large systems is variational quantum Monte Carlo (QMC). The recently introduced deep QMC approach uses ansatzes represented by deep neural networks and generates nearly exact ground-state solutions for molecules containing up to a few dozen electrons, with the potential to scale to much larger systems where other highly accurate methods are not feasible. In this paper, we extend one such ansatz (PauliNet) to compute electronic excited states. We demonstrate our method on various small atoms and molecules and consistently achieve high accuracy for low-lying states. To highlight the method's potential, we compute the first excited state of the much larger benzene molecule, as well as the conical intersection of ethylene, with PauliNet matching results of more expensive high-level methods.
Variational ab initio methods in quantum chemistry stand out among other methods in providing direct access to the wave function. This allows, in principle, straightforward extraction of any other observable of interest, besides the energy, but, in practice, this extraction is often technically difficult and computationally impractical. Here, we consider the electron density as a central observable in quantum chemistry and introduce a novel method to obtain accurate densities from real-space many-electron wave functions by representing the density with a neural network that captures known asymptotic properties and is trained from the wave function by score matching and noise-contrastive estimation. We use variational quantum Monte Carlo with deep-learning Ansätze to obtain highly accurate wave functions free of basis set errors and from them, using our novel method, correspondingly accurate electron densities, which we demonstrate by calculating dipole moments, nuclear forces, contact densities, and other density-based properties.
Artificial neural networks have been recently introduced as a general ansatz to represent many-body wave functions. In conjunction with variational Monte Carlo calculations, this ansatz has been applied to find Hamiltonian ground states and their energies. Here, we provide extensions of this method to study excited states, a central task in several many-body quantum calculations. First, we give a prescription that allows us to target eigenstates of a (nonlocal) symmetry of the Hamiltonian. Second, we give an algorithm to compute low-lying excited states without symmetries. We demonstrate our approach with both restricted Boltzmann machines and feed-forward neural networks. Results are shown for the one-dimensional spin-1/2 Heisenberg model, and for the one-dimensional Bose-Hubbard model. When comparing to exact results, we obtain good agreement for a large range of excited-states energies. Interestingly, we find that deep networks typically outperform shallow architectures for high-energy states.
Artificial neural networks have been recently introduced as a general ansatz to represent many-body wave functions. In conjunction with variational Monte Carlo calculations, this ansatz has been applied to find Hamiltonian ground states and their energies. Here, we provide extensions of this method to study excited states, a central task in several many-body quantum calculations. First, we give a prescription that allows us to target eigenstates of a (nonlocal) symmetry of the Hamiltonian. Second, we give an algorithm to compute low-lying excited states without symmetries. We demonstrate our approach with both restricted Boltzmann machines and feed-forward neural networks. Results are shown for the one-dimensional spin-1/2 Heisenberg model, and for the one-dimensional Bose-Hubbard model. When comparing to exact results, we obtain good agreement for a large range of excited-states energies. Interestingly, we find that deep networks typically outperform shallow architectures for high-energy states.
We present the first variational Monte Carlo study of nuclear and hypernuclear excited states within the neural-network quantum states (NQS) framework. We implement both the overlap penalty (OP) and natural excited state (NES) methods to compute low-lying excitation spectra. To address the spin contamination in hypernuclear calculations, we propose a quantum number targeting (QNT) technique for the OP method. Both the OP-QNT and NES methods can reproduce diagonal observables, such as energies and spatial structures, in excellent agreement with rigorous benchmarks. We further provide, to our knowledge, the first ab initio calculation of the M 1 transition strength for Λ 4 H . The calculated transition strength is consistent with the weak-coupling limit, exhibiting a ∼ 1.3% suppression. This work demonstrates that NQS can be elevated from ground-state solvers to practical tools for nuclear and hypernuclear spectroscopy.
The complexity of many-body quantum wave functions is a central aspect of several fields of physics and chemistry where nonperturbative interactions are prominent. Artificial neural networks (ANNs) have proven to be a flexible tool to approximate quantum many-body states in condensed matter and chemistry problems. In this work we introduce a neural-network quantum state ansatz to model the ground-state wave function of light nuclei, and approximately solve the nuclear many-body Schrödinger equation. Using efficient stochastic sampling and optimization schemes, our approach extends pioneering applications of ANNs in the field, which present exponentially scaling algorithmic complexity. We compute the binding energies and point-nucleon densities of A≤4 nuclei as emerging from a leading-order pionless effective field theory Hamiltonian. We successfully benchmark the ANN wave function against more conventional parametrizations based on two- and three-body Jastrow functions, and virtually exact Green's function Monte Carlo results.