Quantum Computing and the Hunt for Hidden Symmetry

In 1994, Peter Shor gave e cient quantum algorithms for factoring integers and extracting discrete logarithms [20]. If we believe that nature will permit us to faithfully implement our current model of quantum computation, then these algorithms dramatically contradict the Strong Church-Turing thesis. The e ect is heightened by the fact that these algorithms solve computational problems with long histories of attention by the computational and mathematical communities alike. In this article we discuss the branch of quantum algorithms research arising from attempts to generalize the core quantum algorithmic aspects of Shor's algorithms. Roughly, this can be viewed as the problem of generalizing algorithms of Simon [21] and Shor [20], which work over abelian groups, to general nonabelian groups. The article is meant to be self-contained, assuming no knowledge of quantum computing or the representation theory of nite groups. We begin in earnest in Section 2, describing the problem of symmetry nding : given a function f : G → S on a group G, this is the problem of determining {g ∈ G | ∀x, f(x) = f(gx)}, the set of symmetries of f . We switch gears in Section 3, giving a short introduction to the circuit model of quantum computation. The connection between these two sections is eventually established in Section 4, where we discuss the representation theory of nite groups and the quantum Fourier transform a unitary transformation speci cally tuned to the symmetries of the underlying group. Section 4.2 is devoted to Fourier