Colloquium Fall 2023
Speaker: Eric Samperton (Purdue University)
Room: AC1 312
Time: November 15 from 11:30-12:30
Title: Topology and Quantum Computation in Dialogue
Abstract: Our world appears to be on the cusp of realizing the dream of quantum computation. Quantum computers are an emerging hardware technology that will exploit fundamental quantum mechanical properties of nature in order to perform certain algorithms and calculations significantly more quickly than any “classical" computer (by which I mean something like our current semi-conductor based transistor hardware). Once quantum computers are successfully deployed at scale, it is expected that they will have significant impacts throughout science and engineering. We are beginning to see glimpses of this already, but before any truly spectacular applications of quantum computers can be achieved, scientists and engineers must overcome an unavoidable fact about our reality: quantum mechanical systems are noisy and prone to errors. Thus, in order to build scalable quantum computers, we must be able to implement quantum error-correction and perform fault tolerant computational operations on an error-corrected quantum memory.
It turns out that the mathematics of topology can be quite useful for formulating and attacking these problems. This is maybe not so surprising: topology provides a rigorous language for analyzing the properties of systems that are invariant under deformations—for example, the information inside of a quantum computer being “deformed” by noise!
I’ll start this talk with an introduction to quantum mechanics, quantum computers, and the problem of quantum errors. I’ll then introduce Kitaev’s toric code, perhaps the most important example of a quantum error-correcting code. The toric code has a strong topological flavor, and in particular can be closely related to a certain topological quantum field theory (TQFT). This latter perspective allows for numerous generalizations of the toric code and leads to many interesting questions at the intersection of mathematics, physics and computer science. We’ll dig into some of these.
Speaker: Zachary Stier (UC Berkeley)
Room: AC1 312
Time: December 8, 11:30-12:20
Title: Optimal topological generators for the circle
Abstract: Abstract: Suppose we are to pick an angle and take an arbitrarily
number of multiples of that angle. Which choice best fills the circle?
We will discuss how to quantify this and the continued fractions
theory needed to answer the question, which is an extension of
Sarnak's golden mean conjecture. It turns out that there are
infinitely many choices that are eventually optimal, which we can
explicitly characterize, but only 16 that are optimal from the outset.