  The Hilbert-P\'{o}lya conjecture asserts that the imaginary parts of the
nontrivial zeros of the Riemann zeta function (the Riemann zeros) are the
eigenvalues of a self-adjoint operator (a quantum mechanical Hamiltonian, in
the physical sense), as a promising approach to prove the Riemann hypothesis
(cf.\cite{SH2011}). Instead of the eigenvalues, in this paper we consider
observable-geometric phases as the realization of the Riemann zeros in a
periodically driven quantum system, which were introduced in \cite{Chen2020}
for the study of geometric quantum computation. To this end, we further
introduce the notion of non-Abelian observable-geometric phases, involving
which we give an approach to finding a physical system to study the Riemann
zeros. Since the observable-geometric phases are connected with the geometry of
the observable space according to the evolution of the Heisenberg equation,
this sheds some light on the investigation of the Riemann hypothesis.
