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Fermionic neural-network states for ab-initio electronic structure
Kenny Choo, Antonio Mezzacapo, Giuseppe Carleo
TL;DR
Electronic-structure calculations need expressive treatments of fermionic many-body wave functions beyond restrictive variational forms and costly determinant expansions. The paper maps fermionic degrees of freedom to spins and models the resulting Hamiltonian with neural-network quantum states, achieving near-complete correlation energy recovery on minimal-basis diatomics while identifying sampling as a bottleneck for larger systems.
Problem
Fermionic neural-network quantum states remain underdeveloped, while exact determinant methods scale factorially and existing variational forms can have restrictive expressive power or unfavorable scaling.
Method
The approach maps the fermionic molecular Hamiltonian to an equivalent spin Hamiltonian and models its ground state with spin-based neural-network quantum states, examining multiple encodings.
Results
On model diatomic molecules, RBM states capture almost the entirety of electronic excitations and improve upon CCSD(T) and Jastrow wave functions, reaching chemical accuracy or better.
Takeaways & Limitations
Shallow neural networks can compactly encode molecular electronic wave functions with high precision within the tested model systems.
Takeaways & Limitations
Uniform sampling from |ΨM(σ)|2 is inefficient for larger molecules and basis sets because large sample sizes and very low Metropolis acceptance probabilities may be required.
Abstract
from arXiv · showhide
Neural-network quantum states have been successfully used to study a variety of lattice and continuous-space problems. Despite a great deal of general methodological developments, representing fermionic matter is however still early research activity. Here we present an extension of neural-network quantum states to model interacting fermionic problems. Borrowing techniques from quantum simulation, we directly map fermionic degrees of freedom to spin ones, and then use neural-network quantum states to perform electronic structure calculations. For several diatomic molecules in a minimal basis set, we benchmark our approach against widely used coupled cluster methods, as well as many-body variational states. On the test molecules, we recover almost the entirety of the correlation energy. We systematically improve upon coupled cluster methods and Jastrow wave functions, reaching levels of chemical accuracy or better. Finally, we discuss routes for future developments and improvements of the methods presented.
Appendix A: Geometries for diatomic molecules
The appendix lists equilibrium configurations used for the ground-state calculations, with coordinates reported in angstroms.
- Table II gives the equilibrium configurations used for the ground-state calculations in the main text.
- The listed coordinates are organized as x, y, and z values and reported in angstroms (Å).
Appendix B: Computing matrix elements
The appendix describes efficient evaluation of spin-Hamiltonian matrix elements for the stochastic variational Monte Carlo procedure. Pauli-string structure restricts which spin configurations contribute and enables direct computation of the resulting matrix elements.
- Efficiently computing ⟨σ′|Hq|σ⟩ is required because these matrix elements enter the local energy used in stochastic variational Monte Carlo.
- Because Hq is a sum of Pauli-operator products, each matrix element is nonzero only for a specific transformed configuration σ′.
- The appendix then gives a direct expression for the resulting matrix element after identifying the relevant configuration.
- The phase or sign contribution depends on n_y, the total number of σy operators in the Pauli string.