  We study quantum phase slips (QPS) in ultrathin superconducting wires.
Starting from an effective one-dimensional microscopic model, which includes
electromagnetic fluctuations, we map the problem to a (1+1)-dimensional gas of
interacting instantons. We introduce a method to calculate the tunneling
amplitude of quantum phase slips directly from Monte Carlo simulations. This
allows us to go beyond the dilute instanton gas approximation and study the
problem without any limitations of the density of QPS. We find that the
tunneling amplitude shows a characteristic scaling behavior near the
superconductor-insulator transition. We also calculate the voltage-charge
relation of the insulating state, which is the dual of the Josephson
current-phase relation in ordinary superconducting weak links. This evolves
from a sinusoidal form in the regime of dilute QPS to more exotic shapes for
higher QPS densities, where interactions are important.
