Dissertation Defense

Simulating the Hamiltonian dynamics of quantum systems is one of the most promising applications of digital quantum computers. In this dissertation, we develop an understanding of quantum simulation algorithms concerning their design, analysis, implementation, and application.

The theories of optimization and machine learning answer foundational questions in computer science and lead to new algorithms for practical applications. While these topics have been extensively studied in the context of classical computing, their quantum counterparts are far from well-understood. In this thesis, we explore algorithms that bridge the gap between the fields of quantum computing and machine learning. First, we consider general optimization problems with only function evaluations.

Fault-tolerant gate sets whose generators belong to the Clifford hierarchy form the basis of many protocols for scalable quantum computing architectures. At the beginning of the decade, number-theoretic techniques were employed to analyze circuits over these gate sets on single qubits, providing the basis for a number of state-of-the-art quantum compiling algorithms. In this dissertation, I further this program by employing number-theoretic techniques for higher-dimensional gate sets on both qudit and multi-qubit circuits.
 

In this thesis, we begin by reviewing some of the most important Hamiltonian simulation algorithms that are applied in simulation of quantum field theories. Then we focus on state preparation which has been the slowest subroutine in previously known algorithms. We present two distinct methods that improve upon prior results. The first method utilizes classical computational tools such as Density  Matrix Renormalization Group to produce an efficient quantum algorithm for simulating fermionic quantum field theories in 1+1 dimensions.

This thesis describes quantum algorithms for Hamiltonian simulation,
ordinary differential equations (ODEs), and partial differential
equations (PDEs).
 
Product formulas are used to simulate Hamiltonians which can be
expressed as a sum of terms which can each be simulated individually.
By simulating each of these terms in sequence, the net effect
approximately simulates the total Hamiltonian. We find that the error

In this thesis, we focus on the questions of how quantum entanglement can be generated between two spatially separated systems and, once generated, how it can be applied in quantum metrology. First we will discuss a protocol for the generation of large entangled states using long range interactions. Next, we will turn our attention to more general questions of how the Lieb-Robinson bound and other limitations on entanglement can be used to inform the design of quantum computers.