We propose a scheme to implement the self-consistent optimization of the trial wave function within the constrained path auxiliary field quantum Monte Carlo (CP-AFQMC) method in the framework of natural orbitals. In this scheme, a new trial wave function in the form of a Slater determinant is constructed from the CP-AFQMC results by diagonalizing the mixed estimator of the one-body reduced density matrix. We compare two ways (from real and mixed estimators in CP-AFQMC) to calculate the one-body reduced density matrix in the self-consistent process and study the ground state of a doped two-dimensional Hubbard model to test the accuracy of the two schemes. By comparing the local density, occupancy, and ground state energy we find the scheme in which a one-body reduced density matrix is calculated from the mixed estimator is computationally more efficient and provides more accurate results with less fluctuation. The local densities from the mixed estimator scheme agree well with the numerically exact values. This scheme provides a useful tool for studying strongly correlated electron systems.
We propose a scheme to implement the self-consistent optimization of the trial wave function within the constrained path auxiliary field quantum Monte Carlo (CP-AFQMC) method in the framework of natural orbitals. In this scheme, a new trial wave function in the form of a Slater determinant is constructed from the CP-AFQMC results by diagonalizing the mixed estimator of the one-body reduced density matrix. We compare two ways (from real and mixed estimators in CP-AFQMC) to calculate the one-body reduced density matrix in the self-consistent process and study the ground state of a doped two-dimensional Hubbard model to test the accuracy of the two schemes. By comparing the local density, occupancy, and ground state energy we find the scheme in which a one-body reduced density matrix is calculated from the mixed estimator is computationally more efficient and provides more accurate results with less fluctuation. The local densities from the mixed estimator scheme agree well with the numerically exact values. This scheme provides a useful tool for studying strongly correlated electron systems.
We present an extension of constrained-path auxiliary-field quantum Monte Carlo (CP-AFQMC) for the treatment of correlated electronic systems coupled to phonons. The algorithm follows the standard CP-AFQMC approach for description of the electronic degrees of freedom while phonons are described in first quantization and propagated via a diffusion Monte Carlo approach. Our method is tested on the one- and two-dimensional Holstein and Hubbard-Holstein models. With a simple semiclassical trial wave function, our approach is remarkably accurate for $\ensuremath{\omega}/(2dt\ensuremath{\lambda})<1$ for all parameters in the Holstein model considered in this study where d is the dimensionality, $\ensuremath{\omega}$ is the phonon frequency, t is the electronic hopping strength, and $\ensuremath{\lambda}$ is the dimensionless electron-phonon coupling strength. In addition, we empirically show that the autocorrelation timescales as $1/\ensuremath{\omega}$ for $\ensuremath{\omega}/t\ensuremath{\lesssim}1$, which is an improvement over the $1/{\ensuremath{\omega}}^{2}$ scaling of the conventional determinant quantum Monte Carlo algorithm. In the Hubbard-Holstein model, the accuracy of our algorithm is found to be consistent with that of standard CP-AFQMC for the Hubbard model when the Hubbard U term dominates the physics of the model, and is nearly exact when the ground state is dominated by the electron-phonon coupling scale $\ensuremath{\lambda}$. The ap- proach developed in this work should be valuable for understanding the complex physics arising from the interplay between electrons and phonons in both model lattice problems and ab initio systems.
We introduce an efficient approach to implement correlated many-body trial wave functions in auxiliary-field quantum Monte Carlo (AFQMC). To control the sign/phase problem in AFQMC, a constraint is derived from an exact gauge condition but is typically imposed approximately through a trial wave function or trial density matrix, whose quality can affect the accuracy of the method. Furthermore, the trial wave function can also affect the efficiency through importance sampling. The most natural form of the trial wave function has been single Slater determinants or their linear combinations. More sophisticated forms, for example, with the inclusion of a Jastrow factor or other explicit correlations, have been challenging to use, and their implementation is often assumed to require a quantum computer. In this work, we demonstrate that a large class of correlated wave functions, written in the general form of multi-dimensional integrals over hidden or auxiliary variables times Slater determinants, can be implemented as a trial wave function by coupling the random walkers to a generalized Metropolis sampling. We discuss the fidelity of AFQMC with stochastically sampled trial wave functions, which are relevant to both quantum and classical algorithms. We illustrate the method and show that an efficient implementation can be achieved, which preserves the computational scaling of AFQMC. We test our method in molecules under bond stretching and in transition metal diatomics. Significant improvements are seen in both accuracy and efficiency over typical trial wave functions, and the method yields total ground-state energies systematically within chemical accuracy. The method can be useful for incorporating other advanced wave functions, for example, neural quantum state wave functions optimized from machine learning techniques, or for other forms of fermion quantum Monte Carlo.
For optimal accuracy, auxiliary-field quantum Monte Carlo (AFQMC) requires trial states consisting of multiple Slater determinants. We develop an efficient algorithm to select the determinants from an AFQMC random walk eliminating the need for other methods. When determinants contribute significantly to the non-orthogonal configuration interaction energy, we include them in the trial state. These refined trial wave functions significantly reduce the phaseless bias and sampling variance of the local energy estimator. With 100 to 200 determinants, we lower the error of AFQMC by up to a factor of 10 for second row elements that are not accurately described with a Hartree-Fock trial wave function. For the HEAT set, we improve the average error to within the chemical accuracy. For benzene, the largest studied system, we reduce AFQMC error by 80% with 214 Slater determinants and find a 10-fold increase of the time to solution. We show that the remaining error of the method prevails in systems with static correlation or strong spin contamination.
For optimal accuracy, auxiliary-field quantum Monte Carlo (AFQMC) requires trial states consisting of multiple Slater determinants. We develop an efficient algorithm to select the determinants from an AFQMC random walk eliminating the need for other methods. When determinants contribute significantly to the nonorthogonal configuration interaction energy, we include them in the trial state. These refined trial wave functions significantly reduce the phaseless bias and sampling variance of the local energy estimator. With 100 to 200 determinants, we lower the error of AFQMC by up to a factor of 10 for second-row elements that are not accurately described with a Hartree-Fock trial wave function. For the HEAT set, we improve the average error to within chemical accuracy. For benzene, the largest studied system, we reduce AFQMC error by 80% with 214 Slater determinants and find a 10-fold increase of the time to solution. We show that phaseless errors prevail in systems with static correlation or strong spin contamination. For such systems, improved trial states enable stable free-projection AFQMC calculations, achieving chemical accuracy even in the strongly correlated regime.
We present a method to construct pseudo-BCS wave functions from the one-body density matrix.The resulting many-body wave function, which can be produced for any fermion systems, including those with purely repulsive interactions, has the form of a number-projected BCS form, or antisymmetrized geminal power (AGP).Such wave functions provide a better ansatz for correlated fermion systems than a single Slater determinant, and often better than a linear combination of Slater determinants (for example from a truncated active space calculation).We describe a procedure to build such a wave function conveniently from a given reduced density matrix of the system, rather than from a mean-field solution (which gives a Slater determinant for repulsive interactions).The pseudo-BCS wave function thus obtained reproduces the density matrix or minimizes the difference between the input and resulting density matrices.One application of the pseudo-BCS wave function is in auxiliary-field quantum Monte Carlo (AFQMC) calculations as the trial wave function to control the sign/phase problem.AFQMC is often among the most accurate general methods for correlated fermion systems.We show that the pseudo-BCS form further reduces the constraint bias and leads to improved accuracy compared to the usual Slater determinant trial wave functions, using the two-dimensional Hubbard model as an example.Furthermore, the pseudo-BCS trial wave function allows a new systematically improvable self-consistent approach, with pseudo-BCS trial wave function iteratively generated by AFQMC via the one-body density matrix.
We introduce an efficient and numerically stable technique to make use of a BCS trial wave function in the computation of correlation functions of strongly correlated quantum fermion systems. The technique is applicable to any projection approach involving paths of independent-fermion propagators, for example, in mean-field or auxiliary-field quantum Monte Carlo (AFQMC) calculations. Within AFQMC, in the absence of the sign problem, the methodology allows the use of a BCS reference state which can greatly reduce the required imaginary time of projection and improves Monte Carlo sampling efficiency and statistical accuracy for systems where pairing correlations are important. When the sign problem is present, the approach provides a powerful generalization of the constrained-path AFQMC technique which usually uses Slater determinant trial wave functions. As a demonstration of the capability of the methodology, we present benchmark results for the attractive Hubbard model, both spin balanced (no sign problem) and with a finite spin polarization (with sign problem).