# Quantum Algorithm Development for Electronic Structure Calculations

This dissertation concerns the development of quantum computing algorithms for solving electronic structure problems. Three projects are contained: comparison of quantum computing methods for the water molecule, the design and implementation of Fully Controlled Variational Quantum Eigensolver(FCVQE) method, and quantum computing for atomic and molecular resonances.

Chapter 1 gives a general introduction to quantum computing and electronic structure calculations. It includes basic concepts in quantum computing, such as quantum bits (qubits), quantum gates, and an important quantum algorithm, Phase Estimation Algorithm(PEA). It also shows the procedure of molecular Hamiltonian derivation for quantum computers.

Chapter 2 discusses several published quantum algorithms and original quantum algorithms to solve molecules' electronic structures, including the Trotter-PEA method, the first- and second-order Direct-PEA methods, Direct Measurement method, and pairwise Variational Quantum Eigensolver(VQE) method. These quantum algorithms are implemented into quantum circuits simulated by classical computers to solve the ground state energy and excited state energies of the water molecule. Detailed analysis is also given for each method's error and complexity.

Chapter 3 proposes an original design for VQE, which is called Fully Controlled Variational Quantum Eigensolver(FCVQE). Based on Givens Rotation matrices, this design constructs ansatz preparation circuits exploring all possible states in the given space. This method is applied to solving the ground state energy curves for different molecules, including NaH, H

_{2}O, and N_{2}. The results from simulators turn out to be accurate compared with exact solutions. Gate complexity is discussed at the end of the chapter.Chapter 4 attempts to apply quantum simulation to atomic and molecular resonances. The original design implements the molecule's resonance Hamiltonian into the quantum circuit, and the resonance properties can be obtained from the final measurement results. It is shown that the resonance energy and width of a model system can be calculated by executing the circuit using Qiskit simulators and IBM real quantum computers as well. A proof of concept is also shown for the resonance properties of a real molecule, H

_{2}^{-}. In the future, when there are more available qubits, longer coherence time, and less noise in quantum computers, this method can be used for larger molecular systems with better accuracy.