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Autoregressive neural-network wavefunctions for ab initio quantum chemistry

A preprint version of the article is available at arXiv.


In recent years, neural-network quantum states have emerged as powerful tools for the study of quantum many-body systems. Electronic structure calculations are one such canonical many-body problem that have attracted sustained research efforts spanning multiple decades, whilst only recently being attempted with neural-network quantum states. However, the complex non-local interactions and high sample complexity are substantial challenges that call for bespoke solutions. Here, we parameterize the electronic wavefunction with an autoregressive neural network that permits highly efficient and scalable sampling, whilst also embedding physical priors reflecting the structure of molecular systems without sacrificing expressibility. This allows us to perform electronic structure calculations on molecules with up to 30 spin orbitals—at least an order of magnitude more Slater determinants than previous applications of conventional neural-network quantum states—and we find that our ansatz can outperform the de facto gold-standard coupled-cluster methods even in the presence of strong quantum correlations. With a highly expressive neural network for which sampling is no longer a computational bottleneck, we conclude that the barriers to further scaling are not associated with the wavefunction ansatz itself, but rather are inherent to any variational Monte Carlo approach.

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Fig. 1: The high-level architecture of the ARN implementing our wavefunction ansatz.
Fig. 2: The operation of a single conditional wavefunction subnetwork.
Fig. 3: Comparison of the energies obtained using a NAQS and traditional QC approaches for the diatomic nitrogen molecule, as a function of the nuclear separation.
Fig. 4: The variational energies obtained over the course of optimization for the NAQS model described in the text (Standard) and the associated ablations of both restricting the optimization space to physically viable determinants (No mask) and the spin-flip symmetries of the final wavefunction (No spin sym.).

Data availability

The molecular geometries used in this work are in the STO-3G basis as returned from the PubChem42 database by OpenFermion43. OpenFermion was also used to generate qubit Hamiltonians of the form of equation (4), with the backend calculations and baseline QC methods—Hartree–Fock, configuration interaction, CCSD, CCSD(T)—implemented using Psi444. The exact molecular data generated, along with a notebook to reproduce these steps, can be found in the supporting code at and published on Zenodo45.

Code availability

Source code to reproduce the reported results can be found at and published on Zenodo45.


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We are grateful to G. Carleo for his insights regarding RBMs, and to M. Sapova for her assistance with quantum chemical calculations. A.I.L.’s research is partially supported by the Russian Science Foundation (19-71-10092).

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Authors and Affiliations



T.D.B. conceived the research, wrote the code, performed the experiments and cowrote the paper. A.M. assisted in theoretical analysis of the system and in preparing the manuscript. A.I.L. oversaw the entire project, helped interpret the results and cowrote the paper.

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Correspondence to Thomas D. Barrett or A. I. Lvovsky.

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Nature Machine Intelligence thanks Jan Hermann, Rongxin Xia and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Tables 1–4, Figs. 1 and 2 and extended analysis of wall-clock timings.

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Barrett, T.D., Malyshev, A. & Lvovsky, A.I. Autoregressive neural-network wavefunctions for ab initio quantum chemistry. Nat Mach Intell 4, 351–358 (2022).

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