Quantum phase estimation and the Aharonov-Bohm effect

We consider the time evolution of a particle on a ring with a long solenoid through and show that due to the Aharonov-Bohm effect this system naturally makes up a physical implementation of the quantum phase estimation algorithm for a $\text{U}(1)$ unitary operator. The implementation of the full quantum phase estimation algorithm with a $\text{U}(N)$ unitary operator is realized through the non-Abelian Aharonov-Bohm effect. The implementation allows for a more physically intuitive understanding of the algorithm. As an example we use the path integral formulation of the implemented quantum phase estimation algorithm to analyze the classical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$.