A Categorical Presentation of Quantum Computation with Anyons

Andreas R. Blass, Yuri G. Gurevich

Matilde Marcolli, John C. Napp

We propose a construction of anyon systems associated to quantum tori with real multiplication and the embedding of quantum tori in AF algebras. These systems generalize the Fibonacci anyons, with weaker categorical properties, and are obtained from the basic modules and the real multiplication structure.

Seth Lloyd

Chiara Marletto, Mario Rasetti

The deep structural properties of a quantum information theoretic approach to formal languages and universal computation (the unifying tool being the Quantum Spin Network Automaton scheme of computation), as well as those of the topology problem of defining the presentation of the Mapping Class Group of a smooth, compact manifold are shown to be grounded in the common categorical features of the two problems.

Yu Zhao, Yidun Wan

A bstract We develop a comprehensive framework for realizing anyon condensation of topological orders within the string-net model by constructing a Hamiltonian that bridges the parent string-net model before and the child string-net model after anyon condensation. Our approach classifies all possible types of bosonic anyon condensation in any parent string-net model and identifies the basic degrees of freedom in the corresponding child models. The Kogut-Susskind lattice gauge theory model is a special case of our model if the full degrees of freedom of the model are truncated from SU(2) representations to quantum group SU(2) k . Compared with the traditional UMTC perspective of topological orders, our method offers a finer categorical description of anyon condensation at the microscopic level. We also explicitly represent relevant UMTC categorical entities characterizing anyon condensation through our model-based physical quantities, providing practical algorithms for calculating these categorical data.

Shawn Xingshan Cui, Seung-Moon Hong, Zhenghan Wang