  We study motion of vortices in arrays of Josephson junctions at zero
temperature where it is controlled by quantum tunneling from one plaquette to
another. The tunneling process is characterized by a finite time and can be
slow compared to the superconducting gap (so that $\tau \Delta >> 1$). The
dissipation which accompanies this process arises from rare processes when a
vortex excites a quasiparticle above the gap while tunneling through a single
junction. We find that the dissipation is significant even in the case $\tau
\Delta >> 1$, in particular it is not exponentially small in this parameter. We
use the calculated energy dissipation for the single vortex jump to estimate
the physical resistance of the whole array.
