Omar Leon Sanchez, Department of Pure Mathematics and Natalie Mullin, Combinatorics Optimization
“Quantum Walks on Cycles and Algebraic Solutions”
In this talk we will discuss continuous-time quantum walks on cycle graphs from an algebraic perspective. Natalie will introduce the problem of detecting whether or not a continuous-time quantum walk admits a uniform probability distribution. We will see that this problem boils down to whether or not the coefficients of the transition matrix satisfy a system of polynomial equations with rational coefficients. Despite the computational complexity of solving this system, it is known that there are a finite number of solutions. Omar will give a logic-based proof of the following fact: if a system of polynomial equations with rational coefficients has a finite number of solutions, then the coordinates of each solution must be algebraic numbers. Natalie will then show that a continuous-time quantum walk on a cycle of prime length never reaches a uniform probability distribution.