  In lattice QCD, the trace of the inverse of the discretized Dirac operator
appears in the disconnected fermion loop contribution to an observable. As
simulation methods get more and more precise, these contributions become
increasingly important. Hence, we consider here the problem of computing the
trace $\mathrm{tr}(D^{-1})$, with $D$ the Dirac operator. The Hutchinson
method, which is very frequently used to stochastically estimate the trace of a
function of a matrix, approximates the trace as the average over estimates of
the form $x^{H} D^{-1} x$, with the entries of the vector $x$ following a
certain probability distribution. For $N$ samples, the accuracy is
$\mathcal{O}(1/\sqrt{N})$. In recent work, we have introduced multigrid
multilevel Monte Carlo: having a multigrid hierarchy with operators $D_{\ell}$,
$P_{\ell}$ and $R_{\ell}$, for level $\ell$, we can rewrite the trace
$\mathrm{tr}(D^{-1})$ via a telescopic sum with difference-levels, written in
terms of the aforementioned operators and with a reduced variance. We have seen
significant reductions in the variance and the total work with respect to
exactly deflated Hutchinson. In this work, we explore the use of exact
deflation in combination with the multigrid multilevel Monte Carlo method, and
demonstrate how this leads to both algorithmic and computational gains.
