Seminar GEOTOP A Louis H Kauffman, UIC (January 25, 2019)

ABSTRACT:
Quantum computation can be regarded as a methodical study of the structure preparation, evolution, and measurement of quantum systems in a computational framework. In a quantum system, a physical state corresponds to a unit length vector in a complex space (Hilbert space or finite dimensional complex space depending on the problem). A physical process corresponds to the application of a unitary transformation to a state. A measurement of a state projects that state to one of a number of orthogonal possibilities represented by an orthonormal basis for the complex space. The probability of a measurement is the absolute square of its coefficient in the state vector. A quantum computer consists in a unitary transformation and the capability to prepare states, apply the unitary transformation and measure the results. The statistics of repeated measurement gives the information for the computation. Thus the key to quantum computing is the design of unitary transformations that can accomplish a particular task. There are a number of designs that are very interesting such as Shor’s method of factoring integers on a quantum computer. The design of actual quantum computers is very difficult because it is hard to control measurement and keep the system from decohering in its interaction with the environment. It is thought that topological physical situations, such as the behaviour of anyons in the quantum Hall effect or the behaviour of Majorana Fermions in nano wires, may hold the key to the desired fault tolerant computation. In both of these instances one can formulate the unitary transformations in terms of representations of the Artin Braid group that are related to the physics of the situation. It is thought that such topological phases of physical systems will be resistant to perturbation and will make scalable quantum computing possible. It is the purpose of this talk to describe braiding related to both the quantum Hall effect (the Fibonacci model) and unitary braiding related to Majorana Fermions (braiding from Clifford algebras). The talk will be self-contained and as pictorial as possible.

KEYWORDS:
quantum computation, Hilbert space, complex space, state vector, unitary transformation, measurement, quantum Hall effect, Fibonacci model, Majorana Fermion, Artin braid group, unitary braiding, Clifford algebra