  We study the Wasserstein metric to measure distances between molecules
represented by the atom index dependent adjacency "Coulomb" matrix, used in
kernel ridge regression based supervised learning. Resulting quantum machine
learning models exhibit improved training efficiency and result in smoother
predictions of molecular distortions. We first demonstrate smoothness for the
continuous extraction of an atom from some organic molecule. Learning curves,
quantifying the decay of the atomization energy's prediction error as a
function of training set size, have been obtained for tens of thousands of
organic molecules drawn from the QM9 data set. In comparison to conventionally
used metrics ($L_1$ and $L_2$ norm), our numerical results indicate systematic
improvement in terms of learning curve off-set for random as well as sorted (by
norms of row) atom indexing in Coulomb matrices. Our findings suggest that this
metric corresponds to a favorable similarity measure which introduces
index-invariance in any kernel based model relying on adjacency matrix
representations.
