  We study coherent quantum phase-slips which lift the ground state degeneracy
in a Josephson junction ring, pierced by a magnetic flux of the magnitude equal
to half of a flux quantum. The quantum phase-slip amplitude is sensitive to the
normal mode structure of superconducting phase oscillations in the ring
(Mooij-Sch\"on modes). These, in turn, are affected by spatial inhomogeneities
in the ring. We analyze the case of weak periodic modulations of the system
parameters and calculate the corresponding modification of the quantum
phase-slip amplitude.
