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

Theory of coherent quantum phase-slips in Josephson junction chains with periodic spatial modulations