Boosting Quantum Machine Learning Models with a Multilevel Combination Technique: Pople Diagrams Revisited

Inspired by Pople diagrams popular
in quantum chemistry, we introduce
a hierarchical scheme, based on the multilevel combination (C) technique,
to combine various levels of approximations made when molecular energies
are calculated. When combined with quantum
machine learning (QML) models, the resulting CQML model is a generalized
unified recursive kernel ridge regression that exploits correlations
implicitly encoded in training data composed of multiple levels in
multiple dimensions. Here, we have investigated up to three dimensions:
chemical space, basis set, and electron correlation treatment. Numerical
results have been obtained for atomization energies of a set of ∼7000
organic molecules with up to 7 atoms (not counting hydrogens) containing
CHONFClS, as well as for ∼6000 constitutional isomers of C7H10O2. CQML learning curves for atomization
energies suggest a dramatic reduction in necessary training samples
calculated with the most accurate and costly method. In order to generate
millisecond estimates of CCSD­(T)/cc-pvdz atomization energies with
prediction errors reaching chemical accuracy (∼1 kcal/mol),
the CQML model requires only ∼100 training instances at CCSD­(T)/cc-pvdz
level, rather than thousands within conventional QML, while more training
molecules are required at lower levels. Our results suggest a possibly
favorable trade-off between various hierarchical approximations whose
computational cost scales differently with electron number.