Dissertation Defense

Superconducting circuits are at the forefront of quantum computing and quantum sensing technologies, where accurate modeling and simulation are crucial for understanding and optimizing their performance. In this dissertation, we study modeling techniques and novel device designs to advance these technologies, focusing on efficient simulations, direct velocity measurement, and nonreciprocal devices for quantum information processing.

Quantum computing leverages the quantum properties of subatomic matter to enable algorithms to run faster than those possible on a regular computer. Quantum computers have become increasingly practical in recent years, with some small-scale machines available for public use. Quantum computing applications are largely dependent on the software that manipulates computations on the hardware. These applications rely on a variety of symbolic representations including quantum programs to describe and manipulate quantum information effectively.

Recently an algorithm has been constructed that shows the binary icosahedral group 2I together with a T-like gate forms the most efficient single-qubit universal gate set. To carry out the algorithm fault tolerantly requires a code that implements 2I transversally. We fill this void by constructing a family of distance d = 3 codes that all implement 2I transversally. To do this, we introduce twisted unitary t-groups, a generalization of unitary t-groups under a twisting by an irreducible representation.

The Eastin-Knill theorem shows that the transversal gates of a quantum code, which are naturally fault-tolerant, form a finite group G. We show that G is an invariant of equivalent quantum codes and thus can be considered as a well defined symmetry. This thesis studies how the symmetry G dictates the existence and parameters of quantum codes using representation theory. We focus on qubit quantum codes that have symmetry coming from finite subgroups of SU(2). We examine two different methods of deriving quantum codes from these symmetries.

This thesis explores the quantum extension of a fundamental theme in theoretical computer science: the interplay between graph theory, computational complexity, and multiprover interactive proof systems. Specifically, we examine the connections between quantum graph properties, computability theory, and entangled nonlocal games.

Statistical learning is emerging as a new paradigm in science.

This has ignited interest within our inherently quantum world in exploring quantum machines for their advantages in learning, generating, and predicting various aspects of our universe by processing both quantum and classical data. In parallel, the pursuit of scalable science through physical simulations using both digital and analog quantum computers is rising on the horizon.

Quantum simulation stands as an important application of quantum computing, offering insights into quantum many-body systems that are beyond the reach of classical computational methods. For many quantum simulation applications, accurate initial state preparation is typically the first step for subsequent computational processes. This dissertation specifically focuses on state preparation procedures for quantum states with chiral topological order, states that are notable for their robust edge modes and topological properties.

Since the discovery of Shor’s factoring algorithm, there has been a sustained interest in finding more such examples of quantum advantage, that is, tasks where a quantum device can outperform its classical counterpart. While the universal, programmable quantum computers that can run Shor’s algorithm represent one direction in which to search for quantum advantage, they are certainly not the only one. In this dissertation, we study the theory of quantum advantage along two alternative avenues: sensing and simulation.

Quantum information science offers a remarkable promise: by thinking practically about how quantum systems can be put to work to solve computational and information processing tasks, we gain novel insights into the foundations of quantum theory and computer science. Or, conversely, by (re)considering the fundamental physical building blocks of computers and sensors, we enable new technologies, with major impacts for computational and experimental physics.

Quantum computers are likely to have a significant impact on cryptography. Many commonly used cryptosystems will be completely broken once large quantum computers are available. Since quantum computers can solve the factoring problem in polynomial time, the security of RSA would not hold against quantum computers. For symmetric-key cryptosystems, the primary quantum attack is key recovery via Grover search, which provides a quadratic speedup. One way to address this is to double the key length.