  Inspired by Pople diagrams popular in quantum chemistry, we introduce a
hierarchical scheme, based on the multi-level combination (C) technique, to
combine various levels of approximations made when calculating molecular
energies within quantum chemistry. When combined with quantum machine learning
(QML) models, the resulting CQML model is a generalized unified recursive
kernel ridge regression which exploits correlations implicitly encoded in
training data comprised 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 $\sim$7'000 organic molecules with up to 7
atoms (not counting hydrogens) containing CHONFClS, as well as for $\sim$6'000
constitutional isomers of C$_7$H$_{10}$O$_2$. CQML learning curves for
atomization energies suggest a dramatic reduction in necessary training samples
calculated with the most accurate and costly method. In order to generate
milli-second estimates of CCSD(T)/cc-pvdz atomization energies with prediction
errors reaching chemical accuracy ($\sim$1 kcal/mol), the CQML model requires
only $\sim$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 favourable trade-off between
various hierarchical approximations whose computational cost scales differently
with electron number.
