We propose a quantum Monte Carlo approach for the study of the thermo-equilibrium persistent current in an Aharonov - Bohm ring with interacting electrons. From topological considerations we derive a formula for the current, which leads to a global quantum Monte Carlo algorithm to evaluate the global current - phase relation at a given temperature without going into simulations point-by-point in the phase. Notable features in the present approach are: (i) the problem of complex elements in the representation of the partition function is well-handled, (ii) being robust against possible randomness in the Hamiltonian, the basic formula captures all phase-dependent physics and hence makes the simulation itself extremely efficient and reliable.