A method for calculating the quantum canonical rate constant of chemical reactions in a many body system by means of a short-time flux autocorrelation function combined with a maximum entropy numerical analytic continuation scheme is presented. The rate constant is expressed as the time integral of the real-time flux autocorrelation function. The real-time flux autocorrelation function is evaluated for short times fully quantum mechanically by path integral Monte Carlo simulations. The maximum entropy approach is then used to extract the rate from the short real-time flux autocorrelation data. We present two numerical tests, one for proton transfer in harmonic dissipative environments in the deep tunneling regime and the other for the two-level model of primary charge separation in the photosynthetic reaction center. The results obtained using the flux autocorrelation data up to the time of no more than beta(h) over bar are in excellent agreement with the exact quantum calculation over a wide range of parameters including even the tunneling regime.