Deep metric learning has recently shown extremely promising results in the classical data domain, creating well-separated feature spaces. This idea was also adapted to quantum computers via Quantum Metric Learning (QMeL). QMeL consists of a 2 step process with a classical model to compress the data to fit into the limited number of qubits, then train a Parameterized Quantum Circuit (PQC) to create better separation in Hilbert Space. However, on Noisy Intermediate Scale Quantum (NISQ) devices, QMeL solutions result in high circuit width and depth, both of which limit scalability. The proposed Quantum Polar Metric Learning (QPMeL ), uses a classical model to learn the parameters of the polar form of a qubit. A shallow PQC with Ry and Rz gates is then utilized to create the state and a trainable layer of ZZ(θ)-gates to learn entanglement. The circuit also computes fidelity via a SWAP Test for the proposed Fidelity Triplet Loss function, used to train both classical and quantum components. When compared to QMeL approaches, QPMeL achieves 3X better multi-class separation, while using only 1/2 the number of gates and depth. QPMeL is shown to outperform classical networks with similar configurations, presentinga promising avenue for future research on fully classical models with quantum loss functions.

We studied binomial edge ideals, which are at the intersection of graph theory and abstract algebra. Our focus was the multidegrees of these ideals, which contain valuable geometric information. We proved algebraic results that allowed us to write a closed formula for the multidegree of the binomial edge ideal of a graph based on combinatorial properties of the graph. Then we discovered methods to make the process more efficient. We concluded our research by using our results to find the multidegrees of the binomial edge ideals of many families of graphs.

This dissertation is an examination of collective systems of computationally limited agents that require coordination to achieve complex ensemble behaviors or goals. The design of coordination strategies can be framed as multiagent optimization problems, which are addressed in this work from both theoretical and practical perspectives. The primary foci of this study are models where computation is distributed over the agents themselves, which are assumed to possess onboard computational capabilities. There exist many assumption variants for distributed models, including fairness and concurrency properties. In general, there is a fundamental trade-off whereby weakening model assumptions increases the applicability of proposed solutions, while also increasing the difficulty of proving theoretical guarantees. This dissertation aims to produce a deeper understanding of this trade-off with respect to multiagent optimization and scalability in distributed settings. This study considers four multiagent optimization problems. The model assumptions begin with fully centralized computation for the all-or-nothing multicommodity flow problem, then progress to synchronous distributed models through examination of the unmapped multivehicle routing problem and the distributed target localization problem. The final model is again distributed but assumes an unfair asynchronous adversary in the context of the energy distribution problem for programmable matter. For these problems, a variety of algorithms are presented, each of which is grounded in a theoretical foundation that permits formal guarantees regarding correctness, running time, and other critical properties. These guarantees are then validated with in silico simulations and (in some cases) physical experiments, demonstrating empirically that they may carry over to the real world. Hence, this dissertation bridges a portion of the predictability-practicality gap with respect to multiagent optimization problems.

In combinatorial mathematics, a Steiner system is a type of block design. A Steiner triple system is a special case of Steiner system where all blocks contain 3 elements and each pair of points occurs in exactly one block. Independent sets in Steiner triple systems is the topic which is discussed in this thesis. Some properties related to independent sets in Steiner triple system are provided. The distribution of sizes of maximum independent sets of Steiner triple systems of specific order is also discussed in this thesis. An algorithm for constructing a Steiner triple system with maximum independent set whose size is restricted with a lower bound is provided. An alternative way to construct a Steiner triple system using an affine plane is also presented. A modified greedy algorithm for finding a maximal independent set in a Steiner triple system and a post-optimization method for improving the results yielded by this algorithm are established.