Knot Logic and Topological Quantum Computing with Majorana Fermions

摘要

This paper is an introduction to relationships between quantum topology andquantum computing. We take a foundational approach, showing how knots arerelated not just to braiding and quantum operators, but to quantum settheoretical foundations and algebras of fermions. We show how the operation ofnegation in logic, seen as both a value and an operator, can generate thefusion algebra for a Majorana fermion, a particle that is its own anti-particleand interacts with itself either to annihilate itself or to produce itself. Wecall negation in this mode the mark, as it operates on itself to change frommarked to unmarked states. The mark viewed recursively as a simplest discretedynamical system naturally generates the fermion algebra, the quaternions andthe braid group representations related to Majorana fermions. The paper beginswith these fundmentals. They provide a conceptual key to many of the modelsthat appear later in the paper. In particular, the Fibonacci model fortopological quantum computing is seen to be based on the fusion rules for aMajorana fermion and these in turn are the interaction rules for the mark seenas a logical particle. It requires a shift in viewpoint to see that theoperator of negation can also be seen as a logical value. The quaternionsemerge naturally from the reentering mark. All these models have their roots inunitary representations of the Artin braid group to the quaternions.