Abstract: Quantum computation is the marriage of quantum mechanics and computer science, applying the principles of the former to the benefit of the latter. It has developed into an exciting area at the interface of physics, computer science and mathematics. Addressing its main theoretical challenges now involves a strong mathematical component.

In this talk I will illustrate this point and - after a brisk introduction to quantum computing and some of its milestones - describe a sample of problems in quantum computing whose main theme is the symmetric group. Each of them leads to a mathematical question in group theory, representation theory or the theory of Lie algebras, whose answer resolves open problems in quantum algorithms, coding and universal computation.