Exploiting symmetries for exponential error reduction in path integral Monte Carlo

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 test the strategy with a one-dimensional harmonic oscillator, by computing the ratio of the parity odd over the parity even functional integrals and the two-point correlation function. The cost of the simulations scales as expected. In particular the effort for computing the lowest energy eigenvalue in the parity odd sector grows linearly with the time extent. At a fixed CPU time, the statistical error on the two-point correlation function is exponentially reduced with respect to the standard Monte Carlo procedure, and at large time distances it is lowered by many orders of magnitude.