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.