Evaluating quantum circuits and exactly solvable models by untying knots
MATHEMATICAL PICTURE LANGUAGE
Xun Gao - Harvard University
I shall report on a new approach to study some classes of quantum circuits and exactly solvable models. Concretely, using knots gives a unified framework to characterize two famous classes of classically-simulable quantum circuits: Clifford and matchgate. We evaluate these circuits in a topological way by untying the knots. Our method is suitable for programming. The method relies on the abstraction of Ising anyons/Majorana zero modes (also known as the Z2 Quon language). It lets us partially open the black box of each small tensor in the tensor network representation. As a bonus, we find a new class of classically simulable quantum circuits. Our results have an interpretation in terms of exactly-soluble, statistical-mechanics models, and they lead to a topological extension of Kramers-Wannier duality. This point of view may also help us find new types of exactly soluble models.