  The hadronic contribution to the muon anomalous magnetic moment
$a_\mu=(g_\mu-2)/2$ has to be determined at the per-mille level for the
Standard Model prediction to match the expected final uncertainty from the
ongoing E989 experiment. This is 3 times better than the current precision from
the dispersive approach, and 5-15 times smaller than the uncertainty on the
purely theoretical determinations from lattice QCD. So far the stumbling-block
is the large statistical error in the Monte Carlo evaluation of the required
correlation functions which can hardly be tamed by brute force. Here we propose
to solve this problem by multi-level Monte Carlo integration, a technique which
reduces the variance of correlators exponentially in the distance of the
fields. We test our strategy by computing the Hadronic Vacuum Polarization on a
lattice with a linear extension of 3 fm, a spacing of 0.065 fm, and a pion mass
of 270 MeV. Indeed the two-level integration makes the contribution to the
statistical error from long-distances de-facto negligible by accelerating its
inverse scaling with the cost of the simulation. These findings establish
multi-level Monte Carlo as a solid and efficient method for a precise lattice
determination of the hadronic contribution to $a_\mu$. As the approach is
applicable to other computations affected by a signal-to-noise ratio problem,
it has the potential to unlock many open problems for the nuclear and particle
physics community.
