The variational quantum eigensolver (VQE) approach is currently one of the most promising strategies for simulating chemical systems on quantum hardware. In this work, I will describe a new quantum algorithm and a new set of classical algorithms based on VQE. The quantum algorithm, ADAPT-VQE, shows promise in mitigating many of the known limitations of VQEs: Ansatz ambiguity, local minima, and barren plateaus are all addressed to varying degrees by ADAPT-VQE. The classical algorithm family, O2DX-UCCSD, draws inspiration from VQEs, but is classically solvable in polynomial time. This group of algorithms yields equations similar to those of the linearized coupled cluster theory (LCCSD) but is more systematically improvable and, for X = 3 or X = ∞, can break single bonds, which LCCSD cannot do. The overall aim of this work is to showcase the richness of the VQE algorithm and the breadth of its derivative applications. / Doctor of Philosophy / A core goal of quantum chemistry is to compute accurate ground-state energies for molecules. Quantum computers promise to simulate quantum systems in ways that classical computers cannot. It is believed that quantum computers may be able to characterize molecules that are too large for classical computers to treat accurately. One approach to this is the variational quantum eigensolver, or VQE. The idea of a VQE is to use a quantum computer to measure the molecular energy associated with a quantum state which is parametrized by some classical set of parameters. A classical computer will use a classical optimization scheme to update those parameters before the quantum computer measures the energy again. This loop is expected to minimize the quantum resources needed for a quantum computer to be useful, since much of the work is outsourced to classical computers. In this work, I describe two novel algorithms based on the VQE which solve some of its problems.

In the past two decades, myriad algorithms to elucidate the characteristics and dynamics of molecular systems have been developed for quantum computers. In this dissertation, we explore how these algorithms can be adapted to other fields, both to closely related subjects such as materials science, and more surprising subjects such as information theory. Special emphasis is placed on the Variational Quantum Eigensolver algorithm adapted to solve band structures of a periodic system; three distinct implementations are developed, each with its own advantages and disadvantages. We also see how unitary quantum circuits designed to model individual electron excitations within a molecule can be modified to prepare a quantum states strictly orthogonal to a space of known states, an important component to solve problems in thermodynamics and spectroscopy. Finally, we see how the core behavior in several quantum algorithms originally developed for quantum chemistry can be adapted to implement compressive sensing, a protocol in information theory for extrapolating large amounts of information from relatively few measurements. This body of work demonstrates that quantum algorithms developed to study molecules have immense interdisciplinary uses in fields as varied as materials science and information theory.

Increasing ubiquity of computational resources has enabled simulation of complex electronic systems and modern materials. The PAOFLOW software package is a tool designed to construct and analyze tight binding Hamiltonians from the solutions of DFT calculations. PAOFLOW leverages localized basis sets to greatly reduce computational costs of post-processing QE simulation results, enabling efficient determination of properties such as electronic density, band structures in the presence of electric or magnetic fields, magnetic or spin circular dichroism, spin-texture, Fermi surfaces, spin or anomalous Hall conductivity (SHC or AHC), electronic transport, and more. PAOFLOW's broad functionality is detailed in this work, and several independent studies where PAOFLOW's capabilities directly enabled research on promising candidates for ferroelectric and spintronic based technologies are described. Today, Quantum computers are at the forefront of computational information science. Materials scientists and quantum chemists can use quantum computers to simulate interacting systems of fermions, without having to perform the iterative methods of classical computing. This dissertation also describes a study where the band structure for silicon is simulated for the first time on quantum hardware and broadens this concept for simulating band structures of generic crystalline structures on quantum machines.