Simulation of quantum many-body systems is extremely challenging, as computational resources (time, memory) grow exponentially with system size. In this talk, I will introduce one classical and one quantum algorithm to determine the finite energy and finite temperature properties of quantum many-body systems. Both algorithms rely on the Markov chain Monte Carlo, an effective method to explore high-dimensional spaces. The classical one seeks to minimize the free energy of a variational state and avoids the error accumulation problem associated with the imaginary time evolution method. The quantum one uses the quantum computer or quantum simulator as a subroutine for the Monte Carlo sampling and avoids the so-called sign problem of the quantum Monte Carlo method.