Algorithmic advancements in correlated electron and quantum information simulation

Understanding and simulating strongly correlated electron systems is a long-standing goal for condensed matter physics. The onset of high-performance computers and quantum computing devices provide two powerful playgrounds to simulate quantum systems. In this thesis, I present several new algorithmic methods aimed at simulating quantum systems, both on classical and quantum hardware, in addition to using existing methods to elucidate interesting many-body physics. In Chapter 1, I present the background for the works in this thesis. This includes a broad overview of strongly correlated electron systems, as well as the Hubbard Model and conventional/unconventional superconductivity. Then I discuss background behind gate-based quantum computing. Finally I describe entanglement transitions, where many-body phases are characterized by the entanglement scaling of quantum states. In Chapter 2, I describe the numerical techniques that provide the foundation for many of the new methodology presented. I begin with a discussion on how to evaluate numerical algorithms. Then I discuss exact diagonalization, a technique with no stochastic noise which is invaluable for small systems, but suffers from exponential scaling in system size. To access larger systems, quantum Monte Carlo (QMC) is introduced. One specific QMC algorithm, projector quantum Monte Carlo is described and used to understand the so called fermionic sign problem. Then, as an alternative to stochastic methods I introduce tensor networks and the DMRG algorithm. In Chapter 3, we introduce a new method to mitigate the QMC sign problem. This involves finding single particle basis rotations, which reduce the overall ‘badness’ of the sign problem. Any reduction of the sign problem produces an exponential speedup at long projection times β. Impressively, we benchmark this method on various Hubbard models and find sparse, local rotations that can improve the average sign over not rotating, even in the thermodynamic limit. In Chapter 4, we extend the DMRG algorithm to high-performance computing through use of distributed memory tensors. This is presented in a new library, TensorTools, which our benchmarks show is a weakly scalable code. Most state-of-the-art serial DMRG libraries have little overhead on top of underlying linear algebra routines (BLAS/LAPACK/etc) and so rather than improving node-hour costs, our code drastically reduces wall clock time by up to 99X. In Chapter 5, we present results of using (serial) DMRG to understand the ground state of the Kondo-Heisenberg model. Once thought to host topological edge modes, using high quality DMRG simulations we show the expected modes are missing. Presented is the numerical simulations that allowed us to reexamine the underlying theory of the model. In Chapter 6, we explore entanglement transitions in 2D random tensor networks. We observe a transition, and characterize it using finite size scaling of entanglement entropy. Most notably, our results show an interesting discrepancy compared to the sub-linear scaling of quantum circuit models in the volume law limit, as well as the decay of mutual information, suggesting the random tensor network model may not share the same underlying mechanisms as various quantum circuit models. In Chapter 7, we extend the use of classical shadows to quantum process tomography. Notably, we benchmark our method on up to a 4 qubit process (equivalent to 8 qubit state tomography complexity) on the IonQ trapped ion quantum device. We additionally develop a number of tools to increase performance, which attempt to mitigate stochastic and device errors. In Chapter 8, I summarize this work and present discussion of the future perspectives.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Physics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Levy, Ryan

Contributors dc:contributor

• Clark, Bryan K
• Bradlyn, Barry
• Lorenz, Virginia
• Covey, Jacob P

• condensed matter physics
• quantum many-body physics
• numerical methods
• quantum information science
• quantum Monte Carlo

Statement dc:rights

• Copyright 2022 Ryan Levy

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/117698

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F117698