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Variational approach to the many-body problem with error estimation

Claes Rogius Svensson

Published Jan 1, 2018• 0 citations
OpenAlex

Abstract

In this thesis a method for doing approximate calculations of the ground state of quantum mechanical many-body systems is developed and implemented. This method uses the Hartree-Fock method as a starting point and approximates the ground state as a linear combination of non-orthogonal Slater determinants. The Slater determinants are added consecutively by minimising their associated energy while keeping the previously found Slater determinants fixed. This method was tested on a quantum harmonic oscillator filled with interacting fermions and was found to give good approximations to the ground state of this system with only a few Slater determinants. Error estimation of the approximated ground state is done by calculating the energy variance. The energy variance was used in two different methods for improving upon the approximated energy, both of which was found to work well.

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In this thesis a method for doing approximate calculations of the ground state of quantum mechanical many-body systems is developed and implemented. This method uses the Hartree-Fock method as a starting point and approximates the ground state as a linear combination of non-orthogonal Slater determinants. The Slater determinants are added consecutively by minimising their associated energy while keeping the previously found Slater determinants fixed. This method was tested on a quantum harmonic oscillator filled with interacting fermions and was found to give good approximations to the ground state of this system with only a few Slater determinants. Error estimation of the approximated ground state is done by calculating the energy variance. The energy variance was used in two different methods for improving upon the approximated energy, both of which was found to work well.
A method is implemented wherein numerical approximations to the ground and first few excited states of a quantum mechanical N -body 1D harmonic oscillator are found through variational methods, representing the states as a linear combination of normalized pseudo-states which are themselves linear combinations of non-orthogonal Slater determinants. These states are then used as a low energy basis for configuration interaction. An expression is derived for an analytical matrix derivative of the energy functional, in order to improve the speed of the variation. The speed and accuracy using the analytical derivative is compared to that of the numerical derivative, and a number of different gradient descent methods are tried and compared.
Computing many-body ground state energies and resolving electronic structure calculations are fundamental problems for fields such as quantum chemistry or condensed matter. Several quantum computing algorithms that address these problems exist, although it is often challenging to establish rigorous bounds on their performances. Here we detail three algorithms that produce approximate ground states for many-body and electronic structure problems, generalizing some previously known results for 2-local Hamiltonians. Each method comes with asymptotic bounds on the energies produced. The first one produces a separable state which improves on random product states. We test it on a spinless Hubbard model, validating numerically the theoretical result. The other two algorithms produce entangled states via shallow or deep circuits, improving on the energies of given initial states. We demonstrate their performance via numerical experiments on a 2-dimensional Hubbard model, starting from a checkerboard product state, as well as on some chemistry Hamiltonians, using the Hartree-Fock state as reference. In both cases, we show that the approximate energies produced are close to the exact ones. These algorithms provide a way to systematically improve the estimation of ground state energies and can be used stand-alone or in conjunction with existing quantum algorithms for ground states.
An approach for obtaining the ground state of Coulombic many-body systems is presented. This approach is based on the path-integral renormalization-group method with nonorthogonal Slater determinants, is free of the negative sign problem, and can handle higher dimensional systems with consideration of the correlation effect. Furthermore, it can be easily extended to the multicomponent quantum systems that contain more than two kinds of quantum particles. According to our results obtained with the present approach, it achieves the same accuracy as the variational Monte Carlo method with a few Slater determinants and enables us to study the entire ground state consisting of electrons and nuclei without the need to use the Born-Oppenheimer approximation.
We develop a new general theory for large quantum fluctuations in a Fermion many-body system that cannot be described by fluctuations around the Hartree-Fock ground state but arises from resonance of different correlation structures. We start with an exact coherent state representation of a Fermion system on a unitary group. We show that the Hamiltonian in the coherent state representation has a close connection with the Hartree-Fock energy functional. From this, we can derive a new approximation called the resonating Hartree-Fock approximation in which a state is approximated by a superposition of non-orthogonal Slater determinants with different correlation structures. We derive the variation equations to determine a resonating Hartree-Fock wavefunction. We show that the resonance between degenerate broken symmetry Slater determinants may partially recover the symmetry. We discuss how to choose trial Slater determinants in a resonating Hartree-Fock wavefunction. We suggest that resonance of Slater determinats representing localized defects, such as solitons, polarons and breathers, produced in the long range order of the HF ground state may be the most important content of large quantum fluctuations in condensed matter systems.