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Projector Quantum Monte Carlo Methods for Linear and Non-linear Wavefunction Ansatzes

Lauretta R. Schwarz

Published Oct 1, 2017• 2 citations
Publisher

Abstract

This thesis is concerned with the development of a Projector Quantum Monte Carlo method for non-linear wavefunction ansatzes and its application to strongly correlated materials. This new approach is partially inspired by a prior application of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method to the three-band ($p-d$) Hubbard model. Through repeated stochastic application of a projector FCIQMC projects out a stochastic description of the Full Configuration Interaction (FCI) ground state wavefunction, a linear combination of Slater determinants spanning the full Hilbert space. The study of the $p-d$ Hubbard model demonstrates that the nature of this FCI expansion is profoundly affected by the choice of single-particle basis. In a counterintuitive manner, the effectiveness of a one-particle basis to produce a sparse, compact and rapidly converging FCI expansion is not necessarily paralleled by its ability to describe the physics of the system within a single determinant. The results suggest that with an appropriate basis, single-reference quantum chemical approaches may be able to describe many-body wavefunctions of strongly correlated materials. Furthermore, this thesis presents a reformulation of the projected imaginary time evolution of FCIQMC as a Lagrangian minimisation. This naturally allows for the optimisation of polynomial complex wavefunction ansatzes with a polynomial rather than exponential scaling with system size. The proposed approach blurs the line between traditional Variational and Projector Quantum Monte Carlo approaches whilst involving developments from the field of deep-learning neural networks which can be expressed as a modification of the projector. The ability of the developed approach to sample and optimise arbitrary non-linear wavefunctions is demonstrated with several classes of Tensor Network States all of which involve controlled approximations but still retain systematic improvability towards exactness. Thus, by applying the method to strongly-correlated Hubbard models, as well as $\textit{ab-initio}$ systems, including a fully periodic $\textit{ab-initio}$ graphene sheet, many-body wavefunctions and their one- and two-body static properties are obtained. The proposed approach can handle and simultaneously optimise large numbers of variational parameters, greatly exceeding those of alternative Variational Monte Carlo approaches.

DOI
10.17863/cam.13792

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This thesis is concerned with the development of a Projector Quantum Monte Carlo method for non-linear wavefunction ansatzes and its application to strongly correlated materials. This new approach is partially inspired by a prior application of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method to the three-band (p−dp-dp−d) Hubbard model. Through repeated stochastic application of a projector FCIQMC projects out a stochastic description of the Full Configuration Interaction (FCI) ground state wavefunction, a linear combination of Slater determinants spanning the full Hilbert space. The study of the p−dp-dp−d Hubbard model demonstrates that the nature of this FCI expansion is profoundly affected by the choice of single-particle basis. In a counterintuitive manner, the effectiveness of a one-particle basis to produce a sparse, compact and rapidly converging FCI expansion is not necessarily paralleled by its ability to describe the physics of the system within a single determinant. The results suggest that with an appropriate basis, single-reference quantum chemical approaches may be able to describe many-body wavefunctions of strongly correlated materials. Furthermore, this thesis presents a reformulation of the projected imaginary time evolution of FCIQMC as a Lagrangian minimisation. This naturally allows for the optimisation of polynomial complex wavefunction ansatzes with a polynomial rather than exponential scaling with system size. The proposed approach blurs the line between traditional Variational and Projector Quantum Monte Carlo approaches whilst involving developments from the field of deep-learning neural networks which can be expressed as a modification of the projector. The ability of the developed approach to sample and optimise arbitrary non-linear wavefunctions is demonstrated with several classes of Tensor Network States all of which involve controlled approximations but still retain systematic improvability towards exactness. Thus, by applying the method to strongly-correlated Hubbard models, as well as ab-initio\textit{ab-initio}ab-initio systems, including a fully periodic ab-initio\textit{ab-initio}ab-initio graphene sheet, many-body wavefunctions and their one- and two-body static properties are obtained. The proposed approach can handle and simultaneously optimise large numbers of variational parameters, greatly exceeding those of alternative Variational Monte Carlo approaches.
We reformulate the projected imaginary-time evolution of the full configuration interaction quantum Monte Carlo method in terms of a Lagrangian minimization. This naturally leads to the admission of polynomial complex wave function parametrizations, circumventing the exponential scaling of the approach. While previously these functions have traditionally inhabited the domain of variational Monte Carlo approaches, we consider recent developments for the identification of deep-learning neural networks to optimize this Lagrangian, which can be written as a modification of the propagator for the wave function dynamics. We demonstrate this approach with a form of tensor network state, and use it to find solutions to the strongly correlated Hubbard model, as well as its application to a fully periodic ab initio graphene sheet. The number of variables which can be simultaneously optimized greatly exceeds alternative formulations of variational Monte Carlo methods, allowing for systematic improvability of the wave function flexibility towards exactness for a number of different forms, while blurring the line between traditional variational and projector quantum Monte Carlo approaches.
In this work we propose a novel approach to solve the Schrödinger equation which combines projection onto the ground state with a path-filtering truncation scheme. The resulting projector configuration interaction (PCI) approach realizes a deterministic version of the full configuration interaction quantum Monte Carlo (FCIQMC) method [Booth, G. H.; Thom, A. J. W.; Alavi, A. J. Chem. Phys. 2009, 131, 054106]. To improve upon the linearized imaginary-time propagator, we develop an optimal projector scheme based on an exponential Chebyshev expansion in the limit of an infinite imaginary time step. After writing the exact projector as a path integral in determinant space, we introduce a path filtering procedure that truncates the size of the determinantal basis and approximates the Hamiltonian. The path filtering procedure is controlled by one real threshold that determines the accuracy of the PCI energy and is not biased toward any determinant. Therefore, the PCI approach can equally well describe static and dynamic electron correlation effects. This point is illustrated in benchmark computations on N2 at both equilibrium and stretched geometries. In both cases, the PCI achieves chemical accuracy with wave functions that contain less than 0.5% determinants of full CI space. We also report computations on the ground state of C2 with up to quaduple-ζ basis sets and wave functions as large as 200 million determinants, which allow a direct comparison of the PCI, FCIQMC, and density matrix renormalization group (DMRG) methods. The size of the PCI wave function grows modestly with the number of unoccupied orbitals, and its accuracy may be tuned to match that of FCIQMC and DMRG.
By performing a stochastic dynamic in a space of Slater determinants, the full configuration interaction quantum Monte Carlo (FCIQMC) method has been able to obtain energies which are essentially free from systematic error to the basis set correlation energy, within small and systematically improvable error bars. However, the weakly exponential scaling with basis size makes converging the energy with respect to basis set costly and in larger systems, impossible. To ameliorate these basis set issues, here we use perturbation theory to couple the FCIQMC wavefunction to an explicitly correlated strongly orthogonal basis of geminals, following the [2](R12) approach of Valeev et al. The required one- and two-particle density matrices are computed on-the-fly during the FCIQMC dynamic, using a sampling procedure which incurs relatively little additional computation expense. The F12 energy corrections are shown to converge rapidly as a function of sampling, both in imaginary time and number of walkers. Our pilot calculations on the binding curve for the carbon dimer, which exhibits strong correlation effects as well as substantial basis set dependence, demonstrate that the accuracy of the FCIQMC-F12 method surpasses that of all previous FCIQMC calculations, and that the F12 correction improves results equivalent to increasing the quality of the one-electron basis by two cardinal numbers.
The full configuration interaction quantum Monte Carlo (FCIQMC) method, as well as its "initiator" extension (i-FCIQMC), is used to tackle the complex electronic structure of the carbon dimer across the entire dissociation reaction coordinate, as a prototypical example of a strongly correlated molecular system. Various basis sets of increasing size up to the large cc-pVQZ are used, spanning a fully accessible N-electron basis of over 10(12) Slater determinants, and the accuracy of the method is demonstrated in each basis set. Convergence to the FCI limit is achieved in the largest basis with only O[10(7)] walkers within random errorbars of a few tenths of a millihartree across the binding curve, and extensive comparisons to FCI, CCSD(T), MRCI, and CEEIS results are made where possible. A detailed exposition of the convergence properties of the FCIQMC methods is provided, considering convergence with elapsed imaginary time, number of walkers and size of the basis. Various symmetries which can be incorporated into the stochastic dynamic, beyond the standard abelian point group symmetry and spin polarisation are also described. These can have significant benefit to the computational effort of the calculations, as well as the ability to converge to various excited states. The results presented demonstrate a new benchmark accuracy in basis-set energies for systems of this size, significantly improving on previous state of the art estimates.
Full Configuration Interaction Quantum Monte Carlo for Accurate Ab Initio\textit{Ab Initio}Ab Initio Nuclear Structure Calculations
We introduce novel full configuration interaction quantum Monte Carlo (FCIQMC) as an accurate many-body solver for ab initio\textit{ab initio}ab initio nuclear structure calculations. This stochastic approach directly samples the exact wave function in the full configuration space, enabling high-fidelity treatment of high-order many-body correlations in strongly interacting nuclear systems. Using interactions from chiral effective field theory, we have computed ground-state energies and charge radii of 4^44He, 8^88Be, 12^{12}12C and 16^{16}16O with sub-percent-level many-body uncertainties. These results establish FCIQMC as a stochastic full-configuration-space solver capable of treating systems beyond the reach of the conventional no-core shell model, and as an accurate benchmark for truncated many-body expansion methods.
Full configuration interaction quantum Monte Carlo for accurate ab initio\textit{ab initio}ab initio nuclear structure calculations: algorithms and calculation details
Full configuration interaction quantum Monte Carlo (FCIQMC) is a stochastic many-body solver that has been widely applied to electronic, molecular, and condensed-matter systems. In this work we apply FCIQMC to ab initio\textit{ab initio}ab initio nuclear structure calculations using interactions derived from chiral effective field theory. We describe the algorithm in detail, including imaginary-time propagation, excitation generation, estimator choices, the initiator approximation with adaptive shift correction, and reduced-density-matrix (RDM) sampling. Benchmark calculations in small model spaces, where deterministic full configuration interaction (FCI) results are available, validate the stochastic calculation of energies, radii, and RDM-based pure estimators. For large model spaces, we analyze the residual finite-walker bias through systematic walker-number convergence and infinite-walker extrapolations. We also demonstrate that FCIQMC can be extended beyond ground-state calculations by computing the low-lying spectrum of 6^66Li.