# Quantum Computation For Electronic Structure Calculations

This dissertation contains four projects: transforming electronic structure Hamiltonian to approximating Ising-type Hamiltonian to enable electronic structure calculations by quantum annealing, quantum-assisted restricted Boltzmann machine for electronic structure calculations, hybrid quantum classical neural network for calculating ground state energies of molecules and qubit coupled cluster single and double excitations variational quantum eigensolver for electronic structure. In chapter 1 we present a general introduction of quantum computer, including a brief introduction of two quantum computing model: gate model and quantum annealing model. We also give a general review about electronic structure calculations on quantum computer. In chapter 2, we show an approximating mapping between the electronic structure Hamiltonian and the Ising Hamiltonian. The whole mapping is enabled by first enlarging the qubits space to transform the electronic structure Hamiltonian to a diagonal Hamiltonian. Then introduce ancilla qubits to transform the diagonal Hamiltonian to an Ising-type Hamiltonian. We also design an algorithm to use the transformed Hamiltonian to obtain the approximating ground energy of the original Hamiltonian. The numerical simulation results of the transformed Hamiltonian for H

_{2}, He_{2}, HeH^{+}, and LiH molecules match the exact numerical calculations of the original Hamiltonian. This demonstrates that one can map the molecular Hamiltonian to an Ising-type Hamiltonian which could easily be implemented on currently available quantum hardware. In chapter 3, we report a hybrid quantum algorithm employing a restricted Boltzmann machine to obtain accurate molecular potential energy surfaces. By exploiting a quantum algorithm to help optimize the underlying objective function, we obtained an efficient procedure for the calculation of the electronic ground state energy for a small molecule system. Our approach achieves high accuracy for the ground state energy for H_{2}, LiH, H_{2}O at a specific location on its potential energy surface with a finite basis set. With the future availability of larger-scale quantum computers, quantum machine learning techniques are set to become powerful tools to obtain accurate values for electronic structures. In chapter 4, we present a hybrid quantum classical neural network that can be trained to perform electronic structure calculation and generate potential energy curves of simple molecules. The method is based on the combination of parameterized quantum circuit and measurements. With unsupervised training, the neural network can generate electronic potential energy curves based on training at certain bond lengths. To demonstrate the power of the proposed new method, we present results of using the quantum-classical hybrid neural network to calculate ground state potential energy curves of simple molecules such as H_{2}, LiH and BeH_{2}. The results are very accurate and the approach could potentially be used to generate complex molecular potential energy surfaces. In chapter 5, we introduce a new variational quantum eigensolver (VQE) ansatz based on the particle preserving exchange gate to achieve qubit excitations. The proposed VQE ansatz has gate complexity up-bounded to O(*n*^{4}) where*n*is the number of qubits of the Hamiltonian. Numerical results of simple molecular systems such as BeH_{2}, H_{2}O, N_{2}, H_{4}and H_{6}using the proposed VQE ansatz gives very accurate results within errors about 10^{-3}Hartree.## Funding

### Purdue Research Foundation

### Integrated Data Science Initiative Grant (IDSI F.90000303), Purdue University

### Quantum Computing Algorithms and Applications for Coherent and Strongly Correlated Chemical Systems

Basic Energy Sciences

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Directorate for Engineering

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