Clifford augmented density matrix renormalization group for \textit{ab initio} quantum chemistry

TL;DR

Numerical results show that CA-DMRG can reach higher accuracy than DMRG using the same bond dimension, pointing out a promising route to push the boundary of solving \textit{ab initio} quantum chemistry with strong static correlations.

Abstract

The recently proposed Clifford augmented density matrix renormalization group (CA-DMRG) method seamlessly integrates Clifford circuits with matrix product states, and takes advantage of the expression power from both. CA-DMRG has been shown to be able to achieve higher accuracy than standard DMRG on commonly used lattice models, with only moderate computational overhead compared to the latter. In this work, we propose an efficient scheme in CA-DMRG to deal with \textit{ab initio} quantum chemistry Hamiltonians, and apply it to study several molecular systems. Our numerical results show that CA-DMRG can reach higher accuracy than DMRG using the same bond dimension, pointing out a promising route to push the boundary of solving \textit{ab initio} quantum chemistry with strong static correlations.

Figures (8)

• Figure 1: (a) Local minimization and SVD truncation in CA-DMRG, which contains 4 major steps: (a1) building the two-site effective Hamiltonian $H_{\rm eff}$; (a2) performing eigenvalue decomposition to obtain the lowest eigenvalue and eigenvector $\Psi$ of $H_{\rm eff}$; (a3) selecting a two-site Clifford circuit and apply it onto $\Psi$; (a4) performing SVD truncation on the resulting tensor. Steps (a3, a4) are repeated for all the possible two-site Clifford circuits to obtain the optimal Clifford circuit which minimizes the thrown weight during SVD truncation. Only steps (a1, a2, a4) are required in standard two-site DMRG. (b) Updating the local MPO tensors according to $\mathcal{C}\hat{H}\mathcal{C}^{\dagger}$ after one obtains the optimal two-site Clifford circuit $\mathcal{C}$ from step (a), which is done in $5$ steps as shown in panels (b1-b5). The dashed blue box in (b2) and (b4) means that the tensors inside are contracted first.
• Figure 2: The errors between the ground state energies and the corresponding FCI energies for (a) H$_2$O, (b) NH$_3$, (c) C$_2$ and (d) N$_2$ as functions of the inverse bond dimension. The CA-DMRG and two-site DMRG results are represented by the red line with circle and by the blue line with square respectively. We have used the STO-3G basis set for all the molecules.
• Figure 3: Potential energy curve calculated for the N$_2$ molecular system in the STO-3G basis set. The blue and red squares represent two-site DMRG and CA-DMRG results calculated with $\chi=60$, while the blue and red empty triangles represent DMRG and CA-DMRG results calculated with $\chi=80$. The restricted Hartree Fock (RHF, the orange dashed line) and FCI (the gray solid line) energies are also shown as reference. The inset shows the errors against the corresponding FCI energies.
• Figure 4: The run time per sweep for the C$_2$ molecule as a function of the inverse bond dimension. The CA-DMRG and two-site DMRG results are represented by the red line with empty circle and by the blue line with empty square respectively. The purple squares in the inset show the ratio between the run time of CA-DMRG and that of DMRG as a function of the inverse bond dimension.
• Figure 5: The errors between the ground state energies and the corresponding FCI energies for (a) H$_2$O with $\chi=40$, (b) NH$_3$ with $\chi=100$, (c) C$_2$ with $\chi=200$ and (d) N$_2$ with $\chi=100$ as functions of the number of sweeps. The CA-DMRG and DMRG results are represented by the red line with circle and by the blue line with square respectively.