Matthew Gibson-Lopez

Curves as input data naturally arise in a variety of fields including finance, seismology, medicine, spatio-temporal data mining, malicious activity detection, and more. A common way to analyze these data sets is similarity matching, for which we need metrics that measure the similarity between curves. I will explore the Continuous Dynamic Time Warping (CDTW) distance: how it improves on previous, more common metrics; existing algorithms for computing it exactly or approximatively for various cases; and finally our own novel algorithm for approximating it between piecewise algebraic curves.

The power flow equations are non-linear multivariate equations that describe the relationship between power injections and bus voltages of electric power networks. Given a network topology, we are interested in finding network parameters with many equilibrium points. This corresponds to finding instances of the power flow equations with many real solutions. Current state-of-the art algorithms in computational algebra are not capable of answering this question for networks involving more than a small number of variables. To remedy this, we design a probabilistic reward function that gives a good approximation to this root count, and a state-space that mimics the space of power flow equations. We derive the average root count for a Gaussian model, and use this as a baseline for our RL agents. The agents discover instances of the power flow equations with many more solutions than the average baseline, and remain a tractable method for doing so as the network size grows. This demonstrates the potential of RL for power-flow network design and analysis as well as the potential for RL to contribute meaningfully to problems that involve complex non-linear algebra or geometry.

This work studies the continuous-time limit of a quantum analog of the classical Stochastic Gradient Descent (SGD) algorithm by formulating the problem in the Wigner description of quantum mechanics, thereby offering novelty in the description of the corresponding Stochastic Differential Equation (SDE) versus more obscure approaches regarding intuition, such as Lindblad formulations of quantum SGD algorithms. The Wigner formulation is a natural bridge between classical and quantum probabilistic algorithms, which, to our knowledge, has not been applied yet to the study of quantum analogs of SGD. We focus on the particular case of Quantum SGD in its continuous-time limit under a harmonic potential for the optimization of a quadratic function. We present both theoretical and computational studies of its convergence for the aforementioned benchmark problem. We prove, for this benchmark, the independence of the mixing time (convergence rate) from the phase-space dimension by studying the eigenvalues of the steady state inverse covariance matrix in any dimension. This work is in collaboration with UTSA students Avery Tovar and Brandon Lippe.

Quantum optimization starts from a superposition over candidate solutions and iteratively amplifies good ones. Hard constraints are awkward in this setting and are often handled with penalty terms. When the constraint is "choose exactly k items," we can instead prepare a Dicke state, a superposition over only the feasible solutions. Standard Dicke states weight every feasible solution equally, so the search starts by treating all of them as equally promising. We propose a weighted Dicke state whose probabilities come from a classical convex relaxation of the problem, injecting classical information into the starting quantum state while keeping feasibility. We study this in Grover Adaptive Search (GAS), which repeatedly amplifies solutions that beat the best one found so far. In simulation on cardinality-constrained quadratic optimization, we show the weighted start can reduce the number of amplification rounds needed to improve on a classical heuristic, shifting work from the quantum computer to the classical one. This is joint work with UT San Antonio student Jacqueline Higareda.