The Search for a Quantum Computer through Braiding Quantum Gates

Rebecca Chen

It is difficult to say when my passion for mathematics was first kindled. I have liked doing number problems and logic puzzles for as long as I can recall: one of my earliest memories is of using toothpicks to guide a brave mouse across shark-infested waters to steal the king’s cheese (a challenge found in The Puzzle Book, which I owned when I was younger). My interest was further strengthened by participation in math competitions and a math club organized by Professor Ron Ji at IUPUI. However, soon after entering high school, I began to feel that I would like to go beyond solving contest problems and engage in the creative process of mathematical discovery . . . Solutions to the Yang-Baxter equation, an important equation in mathematics and physics, lead to matrix representations of a collection of all braids known as the braid group. Such representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum information science. In particular, representations with a special property called unitarity are desired because they generate braiding quantum gates. These quantum analogs of classical gates are actively studied in the ongoing quest to build a topological quantum computer that could be exponentially more powerful than our computers today. A generalized form of the Yang-Baxter equation was proposed a few years ago by Eric Rowell et al. By solving the generalized Yang-Baxter equation, we found new unitary braid group representations. Our representations give rise to braiding quantum gates and thus have the potential to aid in the construction of useful quantum computers.