Fig. 5: Performance of the CPD backflow and other methods on the hydrogen lattice.

a Ground state energy per atom of a 6 × 6 square hydrogen lattice in a STO-6G basis for increasing lattice constants. Shown are RHF (orange line) and UCCSD energies (violet line), while the horizontal gray line indicates the exact energy of the fully dissociated limit in this basis. The CPD backflow ansatz (Ψ^CPD) with support dimension M = 1 (solid black line) is compared to a GPS ansatz augmented with a Slater determinant (\({\Psi }^{GPS}\times \left\vert \Phi \right\rangle\)) with support dimension M = 72¹⁵ (dotted black line), as well as energies obtained with DMRG (red dots and line) using a bond dimension of M = 1024. A further CPD backflow ansatz with support dimension M = 1 and a truncation in the backflow subspace to K = 5 orbitals is shown as a dashed black line. The backflow CPD (GPS augmented by Slater determinant) VMC ansätze were optimized with 4096 (10,000) samples, while DMRG results were obtained using the block2 package^80,81. b Energy difference per atom relative to DMRG of the CPD backflow ansätze and the GPS times Slater determinant ansatz. To aid the comparison, we have removed the outlier DMRG energy at 2.5 Å. The energies of the CPD backflow and GPS times Slater determinant ansätze are obtained as averages over 50 independent evaluations using the optimized parameters and a sample size of 2¹⁴, with error bars represented by the standard error across these evaluations.