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arXiv:2608.00178

Neural Quantum States for Nuclear Magnetic Resonance Spectroscopy

Bharadwaj Chowdary Mummaneni, Bo Xing

Published Jul 31, 2026• 0 citations
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Abstract

Predicting a nuclear magnetic resonance (NMR) spectrum from first principles requires propagating a quantum state of dimension $2^N$ for $N$ coupled spins, which becomes intractable beyond larger $N$. We benchmark Neural Quantum States (NQS), a class of variational quantum states expressed as an artificial neural network, as an alternative representation for this problem. Using two propagation methods, the Time-Dependent Variational Principle (TDVP) and projected time-dependent Variational Monte Carlo (p-tVMC), we compute the $^1$H spectra of four ($2 \to 5$ spins) experimentally parameterized molecules. TDVP reproduces all line positions and intensities with average spectral mean squared errors of $<10^{-3}$; p-tVMC reproduces the same features, with accuracy determined by its per-step optimization parameters. One dominant obstacle to larger systems is the steep growth of the number of integration steps with spectral bandwidth, which can be removed by propagating in the interaction frame of the chemical-shifted Hamiltonian. Retaining the same accuracy, this reduces the number of integration steps roughly eightfold for the 3-spin system and by at least an order of magnitude for the 4- and 5-spin systems, and it enables a 14-spin molecule (sucrose) to be accurately propagated via Monte Carlo sampling.

DOI
10.48550/arxiv.2608.00178

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Predicting a nuclear magnetic resonance (NMR) spectrum from first principles requires propagating a quantum state of dimension 2N2^N2N for NNN coupled spins, which becomes intractable beyond larger NNN. We benchmark Neural Quantum States (NQS), a class of variational quantum states expressed as an artificial neural network, as an alternative representation for this problem. Using two propagation methods, the Time-Dependent Variational Principle (TDVP) and projected time-dependent Variational Monte Carlo (p-tVMC), we compute the 1^11H spectra of four (2→52 \to 52→5 spins) experimentally parameterized molecules. TDVP reproduces all line positions and intensities with average spectral mean squared errors of <10−3<10^{-3}<10−3; p-tVMC reproduces the same features, with accuracy determined by its per-step optimization parameters. One dominant obstacle to larger systems is the steep growth of the number of integration steps with spectral bandwidth, which can be removed by propagating in the interaction frame of the chemical-shifted Hamiltonian. Retaining the same accuracy, this reduces the number of integration steps roughly eightfold for the 3-spin system and by at least an order of magnitude for the 4- and 5-spin systems, and it enables a 14-spin molecule (sucrose) to be accurately propagated via Monte Carlo sampling.
Neural quantum states (NQS) provide a flexible and highly expressive parameterization of wave functions for strongly correlated problems in quantum chemistry. Despite rapid advances in network architectures, the evaluation of electronic energies remains almost exclusively based on variational Monte Carlo (VMC). While VMC is effective for structured systems such as spin chains, its accuracy and efficiency for electronic Hamiltonians are hindered by sharply peaked distributions, stochastic gradient noise, and slow convergence with sample size. In this letter, we assess the capability of NQS-VMC to efficiently capture correlation in electronic ground states by comparing it to a recently developed NQS-based selected configuration (NQS-SC) approach. We set up a systematic comparison of the ground-state optimizations obtained with NQS-VMC and NQS-SC for molecular systems dominated by either static or dynamical correlation. The comparison demonstrates a clear advantage of NQS-SC over NQS-VMC in both energy accuracy and wave-function coefficients, particularly for statically correlated molecules. Moreover, NQS-SC exhibits robust systematic improvability, whereas NQS-VMC does not. These findings position NQS-SC as the new default approach over NQS-VMC for electronic structure calculations. We further observe that neither NQS-SC nor NQS-VMC can efficiently capture dynamical correlation, highlighting the need for future hybrid methods, such as multiconfigurational perturbation theories built on top of NQS solutions.
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Neural quantum states (NQS) provide a flexible and highly expressive parametrization of wave functions for strongly correlated problems in quantum chemistry. Despite rapid advances in network architectures, the evaluation of electronic energies remains almost exclusively based on variational Monte Carlo (VMC). While VMC is effective for structured systems such as spin chains, its accuracy and efficiency for electronic Hamiltonians are hindered by sharply peaked distributions, stochastic gradient noise, and slow convergence with sample size. In this Letter, we assess the capability of NQS-VMC to efficiently capture correlation in electronic ground states by comparing it to a recently developed NQS-based selected configuration (NQS-SC) approach. We set up a systematic comparison of the ground-state optimizations obtained with NQS-VMC and NQS-SC for molecular systems dominated by either static or dynamical correlation. The comparison demonstrates a clear advantage of NQS-SC over NQS-VMC in both energy accuracy and wave function coefficients, particularly for statically correlated molecules. Moreover, NQS-SC exhibits robust systematic improvability, whereas NQS-VMC does not. These findings position NQS-SC as the new default approach over NQS-VMC for electronic structure calculations. We further observe that neither NQS-SC nor NQS-VMC can efficiently capture dynamical correlation, highlighting the need for future hybrid methods, such as multiconfigurational perturbation theories built on top of NQS solutions.
Variational Monte Carlo calculations have recently reached state-of-the-art accuracy in the approximation of ground state properties of quantum many-body systems. Making use of flexible neural quantum states and automatic differentiation has bypassed traditional computational obstacles such as reliance on basis sets. In this paper, we propose a neural quantum state architecture capable of representing symmetric bosonic wavefunctions in Fock space, enabling the study of systems with variable particle number. By supplementing our variational state with Monte Carlo sampling and geometric optimization, we demonstrate competitive variational energies across an array of one- and two-dimensional systems, converging to the physical boson number under a set chemical potential. Our approach enables accurate estimates of one-body reduced density matrices, opening access to observables such as condensate fractions and radial density profiles from first principles. Our method opens the door to numerical predictions of key measurable quantities in practical grand canonical systems.
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