We study a weighted variant of $\overline{M_{0,n}}(\mathbb{R})$ known as a Hassett space: For each of the $n$ labels, we assign a weight between 0 and 1; points can coincide if the sum of their weights does not exceed one. Our goal is to find combinatorial presentations for the fundamental groups of Hassett spaces with certain restrictions on the weights. To proceed with our goal, we use two main approaches: The first approach is to recursively compute them using blowups, Seifert Van-Kampen, and knowledge for smaller $n$. The second approach is to express the Hassett space as a blow-down of $\overline{M_{0,n}}$ and modify the cactus group directly.

This talk begins with an introduction to combinatorial circuits as symmetric monoidal string diagrams. The case of classical reversible circuits is recalled. It is then shown how quantum mechanics gives rise to a groupoid of reversible circuits subsuming the classical case. The homsets in this category form groups of circuits with identical wire counts. Presentations for these groups can answer many questions in quantum computing. As a specific example, the 3-qubit dyadic Toffoli+H circuits are considered, whose presentation emerges from the E8 lattice. This presentation, in turn, yields information about the entire sub-groupoid of dyadic Toffoli+H circuits.

This is joint work in progress with Dr. Roberto Pichardo Mendoza.