Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01j6731676x
 Title: Optimal topological generators of $$U(1)$$ and short paths in $$X^{p,q}$$ and $$\text{SU}_2(\mathbb{C})$$ Authors: Stier, Zachary Advisors: Sarnak, Peter Department: Mathematics Certificate Program: Applications of Computing Program Class Year: 2020 Abstract: In broad terms, this thesis concerns efficient generators of unitary groups with respect to natural optimality heuristics. The first part considers finding optimal topological generators for $$U(1)$$, viewed as the rotations of the plane. Sarnak's golden mean conjecture states that $$(m+1)d_\varphi(m)\le1+\frac{2}{\sqrt{5}}$$ for all integers $$m\ge1$$, where $$\varphi$$ is the golden mean and $$d_\theta(m)$$ is the discrepancy function for $$m+1$$ multiples of $$\theta$$ modulo 1. In the first part of this thesis, we characterize the set $$\mathcal{S}$$ of values $$\theta$$ that share this property, as well as the set $$\mathcal{T}$$ of those with the property for some lower bound $$m\ge M$$. Remarkably, $$\mathcal{S}\mod1$$ has only 16 elements, whereas $$\mathcal{T}$$ is the set of $$GL_2(\mathbb{Z})$$-transformations of $$\varphi$$. The purpose of the second part of this thesis is in settings where good generators are known, and one wishes to navigate to a particular element. Chapter 2 gives a unified treatment of recent work by Carvalho Pinto--Petit and Sardari on finding short paths in the Lubotzky--Phillips--Sarnak Ramanujan graphs $$X^{p,q}$$; both works make similar, fundamental improvements on the subcase of factoring diagonal matrices and use this technique to give polynomial-time algorithms for factoring almost all matrices in the group. Then, we consider an extension of this technique to $$\text{SU}_2(\mathbb{C})$$, the setting of one-qubit gates and a focus of recent work by Ross--Selinger and Parzanchevski--Sarnak. We culminate with an extension of Carvalho Pinto--Petit's factorization and an implementation of the novel methods in the algorithm. Type of Material: Princeton University Senior Theses Language: en Appears in Collections: Mathematics, 1934-2020Aeronautical Engineering, 1945-1975