  The path integral of a quantum system with an exact symmetry can be written
as a sum of functional integrals each giving the contribution from quantum
states with definite symmetry properties. We propose a strategy to compute each
of them, normalized to the one with vacuum quantum numbers, by a Monte Carlo
procedure whose cost increases power-like with the time extent of the lattice.
This is achieved thanks to a multi-level integration scheme, inspired by the
transfer matrix formalism, which exploits the symmetry and the locality in time
of the underlying statistical system. As a result the cost of computing the
lowest energy level in a given channel, its multiplicity and its matrix
elements is exponentially reduced with respect to the standard path-integral
Monte Carlo. We briefly illustrate the approach in the simple case of the
one-dimensional harmonic oscillator and discuss in some detail its extension to
the four-dimensional Yang Mills theories. We report on our recent new results
in the SU(3) Yang--Mills theory on the relative contribution to the partition
function of the parity-odd states.
