Density matrix-based variational quantum Monte Carlo providing an asymptotically linear scaling behavior for the local energy

We present a reformulation of the variational quantum Monte Carlo (VQMC) method using the N-particle density matrix (N-PDM VQMC), which allows us to reduce the conventional cubic scaling behavior to linear for the calculation of the local energy with the number of electrons for systems with a nonvanishing highest occupied molecular orbital–lowest unoccupied molecular orbital gap. The “perturbation” of the electron distribution, caused by the truncation of the self-consistent-field (SCF) one-electron density matrix, is compensated by the use of the McWeeny purification transformation in the presented scheme. The performance and accuracy of the presented method are illustrated through exemplary calculations.