Implementing high dimensional unitary representations of SU(2) on a Quantum Computer

Abstract

In this note we consider a system with a large angular momentum l whose state we can store using some log_2(l) qubits. The problem then is how to carry out spatial rotations of the system in this representation. In other words we are looking at a unitary representation of SU(2) with dimension 2l+1 and want to implement these transformations with resources polynomial in log(l). We only give a sketch of our solution which involves ``storing'' discretised spherical harmonic functions Y_{l,m}(Theta,phi) in a quantum register. Also there are some technical gaps in the construction, but they are based on plausible assumptions. Our approach is rather cumbersome and we hope somebody will find a nicer solution. For a nice, elementary explanation of what we are trying to do (not involving physics or representation theory) see section 4.6.2.