Editor’s note: This article was initially published in The Daily Gazette, Swarthmore’s online, daily newspaper founded in Fall 1996. As of Fall 2018, the DG has merged with The Phoenix. See the about page to read more about the DG.
For the final talk in this semester’s weekly Math/Stat lecture series, the Department welcomed University of Pennsylvania Professor of Mathematics Robin Pemantle to campus to present his recent research findings on “Quantum Random Walks.” Professor Pemantle was primarily interested in the science of multivariable function estimation until, in 2005, he received a question in an email that raised further more interesting questions regarding quantum random walks. Pemantle speculated that he could derive the answer to the queries raised using analytical techniques he had “in his back pocket,” joking that, “any problem that might be able to be solved is a problem of interest for mathematicians.” He has spent the past summer with a team of Penn math students familiarizing with and accumulating and analyzing examples of multidimensional quantum random walks.
Before presenting the interesting data and conclusions found during his research period, Pemantle outlined basic background information on quantum random walks: what it is, why it is, and how it is analyzed. A classical, well-understood 1D random walk consists of one particle proceeding along a path with an equal probability of taking a discrete step in any of 2 directions. Mathematical researchers have deduced specific formulas that determine the length of the path and how often it will cross in a certain direction. This allows researchers to predict the random motion of, say, molecules in a liquid or glass. One interesting side-note of the theory says that the particle will always return to its origin.
This means that any wayward bacchanal returning from Pub Nite in a random fashion would eventually make it back to his/her dorm no matter what. This same concept applies to one and two dimensional random walks but does not extend to higher dimensions. Classical multidimensional walks have further, more complicated underpinnings. Quantum random walks have an added dimension of chirality and fluctuating numbers of particles and are thus all the more complex to model and predict.
Here’s where Pemantle’s work comes in. His research team wondered whether computer algorithms could predict the picture of the random walk in two dimensions and onwards and model it according to Gaussian functions. Specifically, Professor Pemantle and his students focused on analysis of predicted pictures, attempting to root out why certain peaks or other characteristics appeared.
All in all, Pemantle believes this research on quantum random walks is a good example of how “experimental mathematics can be a nice success story”: much of the presented research and observations truly understanding of what was being seen. Pemantle and the Penn team will continue research this summer in hopes of further unraveling the mystery of the quantum random walk.