Although the linear method is one of the most robust algorithms for optimizing nonlinearly parametrized wavefunctions in variational Monte Carlo, it suffers from a memory bottleneck due to the fact that at each optimization step, a generalized eigenvalue problem is solved in which the Hamiltonian and overlap matrices are stored in memory. Here, we demonstrate that by applying the Jacobi-Davidson algorithm, one can solve the generalized eigenvalue problem iteratively without having to build and store the matrices in question. The resulting direct linear method greatly lowers the cost and improves the scaling of the algorithm with respect to the number of parameters. To further improve the efficiency of optimization for wavefunctions with a large number of parameters, we use the first order method AMSGrad far from the minimum as it is very inexpensive and only switch to the direct linear method near the end of the optimization where methods such as AMSGrad have long convergence tails. We apply this improved optimizer to wavefunctions with real and orbital space Jastrow factors applied to a symmetry-projected generalized Hartree-Fock reference. Systems addressed include atomic systems such as beryllium and neon, molecular systems such as the carbon dimer and iron(ii) porphyrin, and model systems such as the Hubbard model and hydrogen chains.
Although the linear method is one of the most robust algorithms for optimizing nonlinearly parametrized wavefunctions in variational Monte Carlo, it suffers from a memory bottleneck due to the fact that at each optimization step, a generalized eigenvalue problem is solved in which the Hamiltonian and overlap matrices are stored in memory. Here, we demonstrate that by applying the Jacobi-Davidson algorithm, one can solve the generalized eigenvalue problem iteratively without having to build and store the matrices in question. The resulting direct linear method greatly lowers the cost and improves the scaling of the algorithm with respect to the number of parameters. To further improve the efficiency of optimization for wavefunctions with a large number of parameters, we use the first order method AMSGrad far from the minimum as it is very inexpensive and only switch to the direct linear method near the end of the optimization where methods such as AMSGrad have long convergence tails. We apply this improved optimizer to wavefunctions with real and orbital space Jastrow factors applied to a symmetry-projected generalized Hartree-Fock reference. Systems addressed include atomic systems such as beryllium and neon, molecular systems such as the carbon dimer and iron(ii) porphyrin, and model systems such as the Hubbard model and hydrogen chains.
We present a comparison between a number of recently introduced low-memory wave function optimization methods for variational Monte Carlo in which we find that first and second derivative methods possess strongly complementary relative advantages. While we find that low-memory variants of the linear method are vastly more efficient at bringing wave functions with disparate types of nonlinear parameters to the vicinity of the energy minimum, accelerated descent approaches are then able to locate the precise minimum with less bias and lower statistical uncertainty. By constructing a simple hybrid approach that combines these methodologies, we show that all of these advantages can be had at once when simultaneously optimizing large determinant expansions, molecular orbital shapes, traditional Jastrow correlation factors, and more nonlinear many-electron Jastrow factors.
In this work, we introduce three algorithmic improvements to reduce the cost and improve the scaling of orbital space variational Monte Carlo (VMC). First, we show that, by appropriately screening the one- and two-electron integrals of the Hamiltonian, one can improve the efficiency of the algorithm by several orders of magnitude. This improved efficiency comes with the added benefit that the cost of obtaining a constant error per electron scales as the second power of the system size O ( N 2 ), down from the fourth power O ( N 4 ). Using numerical results, we demonstrate that the practical scaling obtained is, in fact, O ( N 1.5 ) for a chain of hydrogen atoms. Second, we show that, by using the adaptive stochastic gradient descent algorithm called AMSGrad, one can optimize the wave function energies robustly and efficiently. Remarkably, AMSGrad is almost as inexpensive as the simple stochastic gradient descent but delivers a convergence rate that is comparable to that of the Stochastic Reconfiguration algorithm, which is significantly more expensive and has a worse scaling with the system size. Third, we introduce the use of the rejection-free continuous time Monte Carlo (CTMC) to sample the determinants. Unlike the first two improvements, CTMC does come at an overhead that the local energy must be calculated at every Monte Carlo step. However, this overhead is mitigated to a large extent because of the reduced scaling algorithm, which ensures that the asymptotic cost of calculating the local energy is equal to that of updating the walker. The resulting algorithm allows us to calculate the ground state energy of a chain of 160 hydrogen atoms using a wave function containing ∼2 × 10 5 variational parameters with an accuracy of 1 mE h /particle at a cost of just 25 CPU h, which when split over 2 nodes of 24 processors each amounts to only about half hour of wall time. This low cost coupled with embarrassing parallelizability of the VMC algorithm and great freedom in the forms of usable wave functions, represents a highly effective method for calculating the electronic structure of model and ab initio systems.
We study three wave function optimization methods based on energy minimization in a variational Monte Carlo framework: the Newton, linear, and perturbative methods. In the Newton method, the parameter variations are calculated from the energy gradient and Hessian, using a reduced variance statistical estimator for the latter. In the linear method, the parameter variations are found by diagonalizing a nonsymmetric estimator of the Hamiltonian matrix in the space spanned by the wave function and its derivatives with respect to the parameters, making use of a strong zero-variance principle. In the less computationally expensive perturbative method, the parameter variations are calculated by approximately solving the generalized eigenvalue equation of the linear method by a nonorthogonal perturbation theory. These general methods are illustrated here by the optimization of wave functions consisting of a Jastrow factor multiplied by an expansion in configuration state functions (CSFs) for the C2 molecule, including both valence and core electrons in the calculation. The Newton and linear methods are very efficient for the optimization of the Jastrow, CSF, and orbital parameters. The perturbative method is a good alternative for the optimization of just the CSF and orbital parameters. Although the optimization is performed at the variational Monte Carlo level, we observe for the C2 molecule studied here, and for other systems we have studied, that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improves monotonically, implying that the nodal hypersurface also improves monotonically.
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 wave function 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 $4\ifmmode\times\else\texttimes\fi{}4$ 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 $4\ifmmode\times\else\texttimes\fi{}{10}^{3}$ to $5\ifmmode\times\else\texttimes\fi{}{10}^{5}$.
We introduce an algorithm that can be used to perform stochastic perturbation theory (sPT) to correct any non-linearly parametrized wavefunction that can be optimized using orbital space Variational Monte Carlo (VMC). Although the variational method gaurantees that the VMC energy can be systematically improved the cost of doing so in practice is often prohibitive. The sPT algorithm presented in this work represents an efficient way to improve the VMC energies with a relatively small computational overhead. We demonstrate that for the carbon dimer and Fe-porphyrin the sPT algorithm is able to capture >97% and >60% respectively of the correlation energy missing from the zeroth order wavefunction. Further, the sPT algorithm is also ideally suited for massively parallel computations because it delivers super-linear speedup with an increasing number of processors.
In this work, we introduce three algorithmic improvements to reduce the cost and improve the scaling of orbital space variational Monte Carlo (VMC). First, we show that by appropriately screening the one- and two-electron integrals of the Hamiltonian one can improve the efficiency of the algorithm by several orders of magnitude. This improved efficiency comes with the added benefit that the resulting algorithm scales as the second power of the system size O(N2), down from the fourth power O(N4). Using numerical results, we demonstrate that the practical scaling obtained is in fact O(N1.5) for a chain of Hydrogen atoms, and O(N1.2) for the Hubbard model. Second, we introduce the use of the rejection-free continuous time Monte Carlo (CTMC) to sample the determinants. CTMC is usually prohibitively expensive because of the need to calculate a large number of intermediates. Here, we take advantage of the fact that these intermediates are already calculated during the evaluation of the local energy and consequently, just by storing them one can use the CTCM algorithm with virtually no overhead. Third, we show that by using the adaptive stochastic gradient descent algorithm called AMSGrad one can optimize the wavefunction energies robustly and efficiently. The combination of these three improvements allows us to calculate the ground state energy of a chain of 160 hydrogen atoms using a wavefunction containing ∼2×105 variational parameters with an accuracy of 1 mEh/particle at a cost of just 25 CPU hours, which when split over 2 nodes of 24 processors each amounts to only about half hour of wall time. This low cost coupled with embarrassing parallelizability of the VMC algorithm and great freedom in the forms of usable wavefunctions, represents a highly effective method for calculating the electronic structure of model and \emph{ab initio} systems.