Spring 2025

Optomechanical detectors offer a highly sensitive method for measuring weak forces. By optically trapping these systems in high vacuum, one can drastically reduce environmental noise and achieve exquisite control over the detector’s center-of-mass motion, rotational degrees of freedom, and physical characteristics such as charge states. This level of isolation enables the detector’s noise to reach the quantum measurement regime, where the dominant noise source is the measurement process itself.

Whether or not a physical system will thermalize from an initial state has been a key question in modern condensed matter physics. Closely related questions are determining whether observables in these systems relax to stationary values, and what those values are. Using tools from computational complexity theory, we demonstrate that given a Hamiltonian on a finite-sized system, determining whether or not it thermalizes or relaxes to a given stationary value is computationally intractable, even for a quantum computer.

Abstract: Whether or not a physical system will thermalize from an initial state has been a key question in modern condensed matter physics. Closely related questions are determining whether observables in these systems relax to stationary values, and what those values are. Using tools from computational complexity theory, we demonstrate that given a Hamiltonian on a finite-sized system, determining whether or not it thermalizes or relaxes to a given stationary value is computationally intractable, even for a quantum computer.

Quantum query complexity is a widely studied model for understanding the capabilities and limitations of quantum computers. In this dissertation, we aim to better understand this complexity measure with respect to many natural models that are not well-studied. In particular, we are interested in the following concrete questions.

1) How powerful are quantum computers that could make multiple queries in parallel relative to analogous classical computers?

2) Can there be a simple quantum algorithmic primitive that inherently combines quantum walks and quantum search?

In 1994, the field of quantum computing had a significant breakthrough when Peter Shor introduced a quantum algorithm that factors integers in (probabilistic) polynomial time.  In these talks, I'll explain the mathematical aspects of Shor's algorithm.

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.

A set of universal fault-tolerant logical gates in quantum error correcting codes is necessary for quantum computing. Transversal operations applied independently on each qubit in a code block are naturally fault-tolerant and easy to implement, but the Eastin-Knill theorem states that the resulting discrete gate set cannot be universal. Circumventing this requires complex protocols such as magic state distillation, code switching, etc. Surface code error correction has been demonstrated on several experimental platforms.

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.

Abstract: In [1] we constructed a family of two-dimensional topological stabilizer codes on continuous variable (CV) degrees of freedom, which generalize homological rotor codes and the toric-GKP code. Our topological codes are built using the concept of boson condensation -- we start from a parent stabilizer code based on an R gauge theory and condense various bosonic excitations.

Abstract: A set of universal fault-tolerant logical gates in quantum error correcting codes is necessary for quantum computing. Transversal operations applied independently on each qubit in a code block are naturally fault-tolerant and easy to implement, but the Eastin-Knill theorem states that the resulting discrete gate set cannot be universal. Circumventing this requires complex protocols such as magic state distillation, code switching, etc. Surface code error correction has been demonstrated on several experimental platforms.