Lecture 22. Topological Quantum Computing

0:00 Quantum Hall Effect 6:49 The idea of the topological quantum computing 8:48 Fibonacci anyons 13:47 Fusion channels 22:41 Changing the order in which anyons are fused 27:16 F-matrix 32:32 Braiding of anyons 34:27 Implementing qubits with Fibonacci anyons 37:58 The effect of braiding on qubits 47:13 Computation with braiding diagrams 48:56 Generating unitary transformations via braiding 51:00 Connections to Conformal Field Theory and Quantum Groups In this lecture we discuss the principles of Topological Quantum Computing, which were proposed by Alexei Kitaev. We begin with a review of the quantum Hall effect. It is expected that anyons emerge as quasiparticles in the quantum Hall effect at certain values of a filling number. Quantum states of anyons experience a phase shift factor when one anyon is moved around another. Braiding of anyons yields various unitary operators on the space of states of an ensemble of these quasiparticles. This may be used for the implementation of quantum gates required for quantum computing. The main advantage of quantum computing with anyons is that the outcome of braiding depends only on the homotopy type of the braiding trajectory and is not sensitive to small perturbations. As a result, we get a strong protection from errors in the computation process. As an example, we discuss Fibonacci anyons, their fusion rules and their braiding operations. This is the final lecture in a graduate course "Quantum Computing". My book "Quantum Computing for High School Students" is available here: https://qubitpublishing.com/ or from Amazon. Complete playlist for this course: https://www.youtube.com/playlist?list=PLu6jbin1VpDD87rdSDF4HRfITWF_LwgOx