  This work introduces a novel approach that combines the multi-index Monte
Carlo (MC) method with importance sampling (IS) to estimate rare event
quantities expressed as an expectation of a smooth observable of solutions to a
broad class of McKean-Vlasov stochastic differential equations. We extend the
double loop Monte Carlo (DLMC) estimator, previously introduced in our works
(Ben Rached et al., 2022a,b), to the multi-index setting. We formulate a new
multi-index DLMC estimator and conduct a comprehensive cost-error analysis,
leading to improved complexity results. To address rare events, an importance
sampling scheme is applied using stochastic optimal control of the single level
DLMC estimator. This combination of IS and multi-index DLMC not only reduces
computational complexity by two orders but also significantly decreases the
associated constant compared to vanilla MC. The effectiveness of the proposed
multi-index DLMC estimator is demonstrated using the Kuramoto model from
statistical physics. The results confirm a reduced complexity from
$\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$ for the single level DLMC
estimator (Ben Rached et al., 2022a) to
$\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-2} (\log
\mathrm{TOL}_{\mathrm{r}}^{-1})^2)$ for the considered example, while ensuring
accurate estimation of rare event quantities within the prescribed relative
error tolerance $\mathrm{TOL}_\mathrm{r}$.
