Quantum information, the Jones polynomial and the Fibonacci model
Seminar Room 1, Newton Institute
An (abstract) quantum computer is a unitary transformation U defined on a complex vector space coupled with the preparation of states |psi> for U to act upon and a probabilistic range of measurements with probability ||^2. Designing quantum algorithms U|psi> means finding 'good' unitary transformations U for the sake of certain tasks. Remarkably, it turns out that the Jones polynomial, an invariant of knots and links discovered by Vaughan Jones in the 1980's, has an associated mathematical technology that allows the construction of a sufficient set of such unitary transformations. These occur by making a braided version of an abstract Fibonacci Particle P that can act upon itself to produce itself P,P ---> P or it can act upon itself to produce a null particle P,P ----> 1 that we will denote by 1. Writing a superposition of these two possibilities we have PP = P + 1, and iteration of this equation shows why P is called a Fibonacci particle. This talk will explain how the bracket model for the Jones polynomial gives a natural way to braid the Fibonacci particles and yields a basic structure for topological quantum computing.
There are many related avenues to the basic direction of this talk and we hope to touch on some of them: quantum computation of the Jones polynomial, the quantum Hall effect (where one expects to find physical analogs of the Fibonacci anyons), Khovanov homology, quantum knots, topological entanglement and quantum entanglement.
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