Quantum Computational Simulations for Condensed Matter Systems

In condensed matter physics, and especially in the study of strongly correlated electron systems, numerical simulation techniques are crucial to determine the properties of the system including interesting phases of matter that arise from electron-electron interactions. Many of these interesting phases of matter, including but not limited to Mott-insulating materials and possibly high-temperature superconducting systems, can be modeled by the Hubbard model. Although it is one of the simplest models to include electron-electron interactions, it cannot be solved analytically in more than one dimension and thus numerical techniques must be employed. Although there have been great strides in classical numerical simulation techniques for quantum many- body systems, all currently known simulation methods suffer from exponential resource scaling in certain parameter regimes.

Quantum computing techniques promise to alleviate these exponential scaling issues to allow simulations of larger and more complex systems. In this dissertation, I will present methods and results for simulations that leverage quantum computing for simulations of the Hubbard model. These simulations include both direct simulation of the Hubbard model along with results that solve the Hubbard model using dynamical mean-field theory (DMFT). Dynamical mean-field theory is a self-consistent mapping from the Hubbard to the Anderson impurity model, which reproduces the physics of the Hubbard model directly in the thermodynamic limit.

In terms of utilization of quantum computing techniques, here I present both results of a simulation run on real quantum hardware along with algorithms developed for future quantum hardware. Specifically, I run a small DMFT simulation which utilizes both classical computing techniques and quantum computation. I also develop multiple algorithmic techniques for preparing quantum many-body states on a quantum computer and a quantum algorithm to calculate a generic response function of the system. Finally, I will give an outlook on challenges and future opportunities for using quantum computation to simulate quantum many-body systems.