Algorithms and Complexity for Quantum Computing

Are quantum computers more powerful than classical computers? To answer this question one must know the classical computational complexity. What is it about the problems of quantum chemistry and quantum physics that enables us to get lower bounds on the classical complexity? We also introduce a new classification of quantum speedups. We then turn to a particular problem, the ground state of the time-independent Schroedinger equation for a system of p particles. The classical deterministic complexity of this problem is exponential in p. We provide an algorithm for solving this problem on a quantum computer whose cost is linear in p. We discuss whether this exponential separation (in our oracle model) proves that quantum computers are exponentially more powerful than classical computers. We end with a selection of research directions and where to learn more.