In the second part of my talk, I will present results of my most recent research project on quantum algorithms, quantum complexity theory and their connection to the Jones polynomial. The probabilistic output of topological quantum computers depends on the Jones polynomial (a certain invariant) of the knots which specify their programs. I will outline our simple proof that topological quantum computers can be efficiently simulated by quantum circuits and vice versa. Our results give an efficient algorithm within the quantum circuit model for obtaining additive approximations of the Jones polynomial. Furthermore, our simple method for translating quantum circuits into knots provides a direct proof that in a certain sense this problem is the hardest problem that can be efficiently solved on a quantum computer. I will apply our techniques to characterize the complexity classes BQP (Bounded-Error Quantum Polynomial Time), Quantum-NP, #P, and PSPACE.