  The first results from the Fermilab E989 experiment have confirmed the
long-standing tension between the experimental determination of the muon
anomalous magnetic moment $a_\mu=(g_\mu-2)/2$ and its SM determination using
the dispersive approach. In order to match the expected final precision from
E989, the current uncertainty on ab initio determinations using lattice QCD
must be decreased by a factor 5-15, a goal which is hampered by the
signal-to-noise ratio problem of the electromagnetic current correlator.
Multi-level Monte Carlo integration with fermions is a method which reduces the
variance of correlators exponentially in the distance of the fields. Here we
demonstrate that the variance reduction in a realistic two-level simulation
with a pion mass of 270 MeV, linear size of 3 fm and lattice spacing around
0.065 fm is sufficient to compute the tail of the current correlator with the
statistical accuracy required for the hadronic vacuum polarization contribution
to $a_\mu$. An efficient estimator is also employed for computing the
disconnected contribution.
