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

TL;DR

This paper explores implementing high-dimensional SU(2) unitary representations on quantum computers, focusing on rotations of large angular momentum systems using log-scale qubits, with a sketch of a potential method.

Contribution

It proposes a preliminary approach to realize SU(2) rotations in high-dimensional quantum systems using discretized spherical harmonics, despite technical gaps.

Findings

Conceptual framework for quantum SU(2) rotations

Potential polynomial resource implementation

Identified technical challenges and assumptions

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.