Optimizing large parameter sets in variational quantum Monte Carlo

Abstract

We present a technique for optimizing hundreds of thousands of variational parameters in variational quantum Monte Carlo. By introducing iterative Krylov subspace solvers and by multiplying by the Hamiltonian and overlap matrices as they are sampled, we remove the need to construct and store these matrices and thus bypass the most expensive steps of the stochastic reconfiguration and linear method optimization techniques. We demonstrate the effectiveness of this approach by using stochastic reconfiguration to optimize a correlator product state wavefunction with a pfaffian reference for four example systems. In two examples on the two dimensional Fermionic Hubbard model, we study 16 and 64 site lattices, recovering energies accurate to 1% in the smaller lattice and predicting particle-hole phase separation in the larger. In two examples involving an ab initio Hamiltonian, we investigate the potential energy curve of a symmetrically dissociated 4x4 hydrogen lattice as well as the singlet-triplet gap in free base porphin. In the hydrogen system we recover 98% or more of the correlation energy at all geometries, while for porphin we compute the gap in a 24 orbital active space to within 0.02 eV of the exact result. The number of variational parameters in these examples ranges from 4 × 10 3 to 5 × 10 5 .

Variational Monte Carlo method for fermionic models combined with tensor networks and applications to the hole-doped two-dimensional Hubbard model

“…To reduce the cost, one can solve the SR equation iteratively by conjugate gradient (CG) method, so that the explicit construction of S kl is not required 33 . We only need to realize the matrix-vector multiplication As a result, the computational cost is reduced from O n s n 2 p + n 3 p to O (n s n p n iter ), and the memory cost is reduced from O n s n p + n 2 p to O (n s n p ).…”

“…In order to optimize large number of parameters, one can solve the SR equation iteratively by conjugate gradient (CG) method 33 . The detailed implementation is described in Appendix B.…”

“…Another advantage of employing the Monte Carlo sampling into tensor network methods is that it is possible to choose various types of suitable reference basis beyond the real space basis. In the VMC study of correlator product states 32 , a Pfaffian pairing wave function has been used 33 . In the study of one dimen-sional fermionic system, the free fermion Slater determinant has been used as the reference wave function of the MPS 34 , which achieves higher accuracy than the conventional MPS method with the same bond dimension.…”

Multireference symmetry-projected variational approaches for ground and excited states of the one-dimensional Hubbard model

“…1,9 For larger systems, several other methods have been extensively used to study the 1D and 2D Hubbard models as well as their strong coupling versions. 10 Among such approximations, we have the quantum Monte Carlo, 11,12 the variational Monte Carlo, 13 the density matrix renormalization group [14][15][16][17] as well as approximations based on matrix product and tensor network states. 18 Both the dynamical mean field theory and its cluster extensions [19][20][21][22][23][24][25][26] have made important contributions to our present knowledge of the Hubbard model.…”

A transformed framework for dynamic correlation in multireference problems

“…In principle, this can be easily cured by particle number projection, which converts the Bogoliubov vacuum into an antisymmetrized geminal power. [57][58][59][60] Practically, the performance of CT-MP2 at long distances can be improved by adding a level shift to the diagonal part ofĤ 0 , such that no negative eigenvalues appear in Eq. (22).…”