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One of the most powerful and prized concepts in physics is that of universal behaviour. Universality allows us to make predictions for entire classes of systems without requiring knowledge of the microscopics, and can be found in classical and quantum systems in both equilibrium and in their dynamics. Often in many-body systems, this universal behaviour is found in regimes where effects at macroscopic length scales dominate over microscopic fluctuations, which is particularly true at a phase transition. In this thesis, we will address universality in quantum many-body physics, and its connection to the branch of mathematics known as catastrophe theory (CT). In CT, singularities in a theory take on several universal forms, known as catastrophes, which can be shown to manifest themselves in classical mechanical trajectories. We extend the concept of catastrophes to their wave variants, known as diffraction integrals, and identify how these universal features appear in many-body wavefunctions and observables. Specifically, in Chapter 2, we examine how the wavefunction of a $\delta$-kicked Hamiltonian can be mapped exactly onto the Pearcey function, along with the effects of a phase transition on the diffraction. In Chapter 3, we examine the free-Fermion representation of the one-dimensional transverse-field Ising model in a similar vein, and identify the presence of catastrophes away from and through the critical point. / Thesis / Master of Science (MSc) / In this thesis, we investigate how certain universal structures, known as catastrophes, manifest themselves in quantum many-body systems. These catastrophes, which are associated with singularities somewhere in the theory, have been shown to be universal structures, and are expected to emerge in our study of condensed-matter systems. We will identify the presence of catastrophes, how they arise in specific classes of physical theories, and how they are affected in regimes where other universal behaviour is known to arise: near locations of quantum criticality.

It is known that almost all 2-qubit gates are universal for quantum computing (Lloyd 1995; Deutsch, Barenco, Eckert 1995). However, an explicit characterization of non-universal 2-qubit gates is not known. We consider a closely related problem of characterizing the set of non-universal 2-qubit Hamiltonians. We call a 2-qubit Hamiltonian n-universal if, when applied on different pairs of qubits, it can be used to approximate any unitary operation on n qubits. It follows directly from the results of Lloyd and Deutsch, Barenco, Eckert, that almost any 2-qubit Hamiltonian is 2-universal. Our main result is a complete characterization of 2-non-universal 2-qubit Hamiltonians. There are three cases when a 2-qubit Hamiltonian H is not universal: (1) H shares an eigenvector with the gate that swaps two qubits; (2) H acts on the two qubits independently (in any of a certain family of bases); (3) H has zero trace. The last condition rules out the Hamiltonians that generate SU(4)---it can be omitted if the global phase is not important. A Hamiltonian that is not 2-universal can still be 3-universal. We give a (possibly incomplete) list of 2-qubit Hamiltonians that are not 3-universal. If this list happens to be complete, it actually gives a classification of n-universal 2-qubit Hamiltonians for all n >= 3.

It is known that almost all 2-qubit gates are universal for quantum computing (Lloyd 1995; Deutsch, Barenco, Eckert 1995). However, an explicit characterization of non-universal 2-qubit gates is not known. We consider a closely related problem of characterizing the set of non-universal 2-qubit Hamiltonians. We call a 2-qubit Hamiltonian n-universal if, when applied on different pairs of qubits, it can be used to approximate any unitary operation on n qubits. It follows directly from the results of Lloyd and Deutsch, Barenco, Eckert, that almost any 2-qubit Hamiltonian is 2-universal. Our main result is a complete characterization of 2-non-universal 2-qubit Hamiltonians. There are three cases when a 2-qubit Hamiltonian H is not universal: (1) H shares an eigenvector with the gate that swaps two qubits; (2) H acts on the two qubits independently (in any of a certain family of bases); (3) H has zero trace. The last condition rules out the Hamiltonians that generate SU(4)---it can be omitted if the global phase is not important. A Hamiltonian that is not 2-universal can still be 3-universal. We give a (possibly incomplete) list of 2-qubit Hamiltonians that are not 3-universal. If this list happens to be complete, it actually gives a classification of n-universal 2-qubit Hamiltonians for all n >= 3.

Childhood cancer has obvious impacts on the children themselves, but also has impacts on their families and siblings. While studies have been conducted on the impacts of cancer on children and their parents, there has been limited consideration of the impacts of cancer on healthy siblings. Some studies have suggested that siblings of children with cancer are the most emotionally disregarded and distressed of all family members as a result of their sibling having cancer. Exposure to stress has a negative impact on the immune system and places siblings at risk for health problems. Thus, finding mechanisms to ameliorate stress are critical. Early intervention and treatment may serve as protective factors against risky behavior and lead to more normative child development and well-being. One method of improving the psychosocial adjustment of siblings of children with cancer has been through interventions such as camp experiences that include opportunities for campers to experience social support. There is a need to better understand the efficacy of intervention programs designed to reduce stress across a variety of settings. The focus of this study was if campers experience social support (emotional, informational, and instrumental) at a three and one-half day residential camp, and the elements of camp campers’ identified as leading to social support. Participant observations of camp and semi-structured interviews with selected campers were used to help gain insights concerning the impact of the camp experience on siblings of children with cancer. Coding of responses was undertaken, which led to the identification of themes and subthemes. Findings provided a narrative description of how campers perceived social support. In addition, social comparison was identified as a critical mechanism for meeting camper’s biological, physiological and behavioral needs. Overarching perceptions of campers were that people at camp were nice and staff was instrumental in facilitating universality and a cycle of reciprocity. Implications of this study suggest intentionally designed camps have the ability to impact a broad range of attitudes and behaviors. The study also supports theory of how structures and processes of social relationships work in relation to good health; and can be applied to a camp setting.

Childhood cancer has obvious impacts on the children themselves, but also has impacts on their families and siblings. While studies have been conducted on the impacts of cancer on children and their parents, there has been limited consideration of the impacts of cancer on healthy siblings. Some studies have suggested that siblings of children with cancer are the most emotionally disregarded and distressed of all family members as a result of their sibling having cancer. Exposure to stress has a negative impact on the immune system and places siblings at risk for health problems. Thus, finding mechanisms to ameliorate stress are critical. Early intervention and treatment may serve as protective factors against risky behavior and lead to more normative child development and well-being. One method of improving the psychosocial adjustment of siblings of children with cancer has been through interventions such as camp experiences that include opportunities for campers to experience social support. There is a need to better understand the efficacy of intervention programs designed to reduce stress across a variety of settings. The focus of this study was if campers experience social support (emotional, informational, and instrumental) at a three and one-half day residential camp, and the elements of camp campers' identified as leading to social support. Participant observations of camp and semi-structured interviews with selected campers were used to help gain insights concerning the impact of the camp experience on siblings of children with cancer. Coding of responses was undertaken, which led to the identification of themes and subthemes. Findings provided a narrative description of how campers perceived social support. In addition, social comparison was identified as a critical mechanism for meeting camper's biological, physiological and behavioral needs. Overarching perceptions of campers were that people at camp were nice and staff was instrumental in facilitating universality and a cycle of reciprocity. Implications of this study suggest intentionally designed camps have the ability to impact a broad range of attitudes and behaviors. The study also supports theory of how structures and processes of social relationships work in relation to good health; and can be applied to a camp setting.

Consider the problem of designing capacity-achieving codes for multi-terminal communication scenarios. For point-to-point communication problems, one can optimize a single code to approach capacity, but for multi-terminal problems this translates to optimizing a single code to perform well over the entire region of channel parameters. A coding scheme is called universal if it allows reliable communication over the entire achievable region promised by information theory. It was recently shown that terminated low-density parity-check convolutional codes (also known as spatially-coupled low-density parity-check ensembles) have belief-propagation thresholds that approach their maximum a-posteriori thresholds. This phenomenon, called "threshold saturation via spatial-coupling," was proven for binary erasure channels and then for binary memoryless symmetric channels. This approach provides us with a new paradigm for constructing capacity approaching codes. It was also conjectured that the principle of spatial coupling is very general and that the phenomenon of threshold saturation applies to a very broad class of graphical models. In this work, we consider a noisy Slepian-Wolf problem (with erasure and binary symmetric channel correlation models) and the binary-input Gaussian multiple access channel, which deal with correlation between sources and interference at the receiver respectively. We derive an area theorem for the joint decoder and empirically show that threshold saturation occurs for these multi-user scenarios. We also show that the outer bound derived using the area theorem is tight for the erasure Slepian-Wolf problem and that this bound is universal for regular LDPC codes with large left degrees. As a result, we demonstrate near-universal performance for these problems using spatially-coupled coding systems.

abstract: Universal biology is an important astrobiological concept, specifically for the search for life beyond Earth. Over 1.2 million species have been identified on Earth, yet all life partakes in certain processes, such as homeostasis and replication. Furthermore, several aspects of biochemistry on Earth are thought to be universal, such as the use of organic macromolecules like proteins and nucleic acids. The presence of many biochemical features in empirical data, however, has never been thoroughly investigated. Moreover, the ability to generalize universal features of Earth biology to other worlds suffers from the epistemic problem of induction. Systems biology approaches offer means to quantify abstract patterns in living systems which can more readily be extended beyond Earth’s familiar planetary context. In particular, scaling laws, which characterize how a system responds to changes in size, have met with prior success in investigating universal biology. This thesis rigorously tests the hypothesis that biochemistry is universal across life on Earth. The study collects enzyme data for annotated archaeal, bacterial, and eukaryotic genomes, in addition to metagenomes. This approach allows one to quantitatively define a biochemical system and sample across known biochemical diversity, while simultaneously exploring enzyme class scaling at both the level of both individual organisms and ecosystems. Using the Kyoto Encyclopedia of Genes and Genomes (KEGG) and the Joint Genome Institute’s Integrated Microbial Genomes and Microbiomes (JGI IMG/M) database, this thesis performs the largest comparative analysis of microbial enzyme content and biochemistry to date. In doing so, this thesis quantitatively explores the distribution of enzyme classes on Earth and adds constraints to notions of universal biochemistry on Earth. / Dissertation/Thesis / Masters Thesis Geological Sciences 2020

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