Fig. 4: Performance of the CPD backflow and other models on the water molecule.

Relative ground state correlation energy error (compared to exact diagonalization) for CPD backflow ansätze (Ψ^CPD) on the water molecule (6-31G basis, equilibrium geometry as specified in ref. ⁴) as the number of configurational samples N_S in each Markov chain is increased in the parameter optimization steps. Two support dimensions corresponding to M = 1 (blue circles) and M = 4 (green circles) are shown. Comparison energies for the Gaussian process state augmented by a symmetry-broken Pfaffian (\({\Psi }^{GPS}\times \left\vert \,{\mbox{Pf}}\,\right\rangle\), red diamond) and restricted Boltzmann machine NQS (Ψ^RBM, orange squares) are taken from refs. ⁴,¹⁵, respectively, while the CCSD energy (solid line) is calculated with PySCF^65,66. CPD backflow energies are obtained as averages over 50 independent evaluations using the optimized parameters and a sample size of 2¹⁶, with error bars represented by the standard error across these evaluations.