A Deterministic Framework for Neural Network Quantum States in Quantum Chemistry

TL;DR

The paper tackles stochastic optimization challenges of Neural Network Quantum States in discrete Fock spaces by introducing a deterministic framework that projects the neural amplitudes onto dynamically adaptive configuration subspaces and augments them with Epstein–Nesbet PT2 corrections. A neural network backflow ansatz parameterizes amplitudes through a configuration-dependent orbital matrix, enabling an exact, noise-free energy evaluation via deterministic summation. Three optimization modes—Asymmetric, Proxy, and Variational—together with a self-consistent inner/outer loop and Heat-Bath screening enable scalable exploration of large Hilbert spaces, while a hybrid CPU-GPU implementation delivers practical performance. Benchmarks on H$_2$O, N$_2$, and Cr$_2$ demonstrate that the method can attain micro-Hartree-level accuracy and sub-linear wall-time scaling in systems with Hilbert spaces exceeding $10^{23}$, highlighting a promising route for high-accuracy quantum chemistry beyond traditional stochastic NQS approaches. The framework sets the stage for future extensions including orbital optimization, symmetry enforcement, and hybrid deterministic-stochastic schemes to further close the gap with state-of-the-art solvers.

Abstract

Stochastic optimization of Neural Network Quantum States (NQS) in discrete Fock spaces is limited by sampling variance and slow mixing. We present a deterministic framework that optimizes a neural backflow ansatz within dynamically adaptive configuration subspaces, corrected by second-order perturbation theory. This approach eliminates Monte Carlo noise and, through a hybrid CPU-GPU implementation, exhibits sub-linear scaling with respect to subspace size. Benchmarks on bond dissociation in H2O and N2, and the strongly correlated chromium dimer Cr2, validate the method's accuracy and stability in large Hilbert spaces.

Figures (5)

• Figure 1: Convergence of deterministic objectives for H2O (6-31G). (a) Energy error $|\Delta E|$ relative to FCI versus wall time for a fixed variational set $|\mathcal{V}|=4096$. (b) Final energy error versus variational set size $\log_2|\mathcal{V}|$.
• Figure 2: Absolute energy errors $|\Delta E|$ relative to FCI/CDFCI benchmarks for the symmetric dissociation of H2O (left) and N2 (right) in the cc-pVDZ basis set. Results are shown for the Variational mode (circles) and the Variational+PT2 scheme (stars). The color gradient indicates the size of the variational set $|\mathcal{V}|$, ranging from $2^{13}$ (yellow) to $2^{17}$ (dark purple). Baselines including UHF, MRCISD, UCCSDT, and DMRG are plotted for comparison.
• Figure 3: Convergence of variational (circles) and PT2-corrected total energies (stars) for Cr2 as a function of the variational set size $|\mathcal{V}|$. (a) CAS(24e, 30o) and (b) CAS(48e, 42o). Reference data include HCI (orange squares), SC-RBM (green triangles), Transformer (pink dashed line), and CCSDTQ (gray dotted line).
• Figure 4: Wall-time decomposition of a single outer-loop iteration versus the variational set size $|\mathcal{V}|$ for (a) H2O, (b) N2, (c) Cr2 (24e, 30o), and (d) Cr2 (48e, 42o). The stacked bars (left axis) represent the relative percentage of each computational phase, while the diamond markers (right axis) track the total wall-time in seconds. The total scaling follows a power law $T \propto |\mathcal{V}|^\alpha$, with fitted exponents $\alpha$ provided in the insets.
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