  Treating electron correlation more accurately and efficiently is at the heart
of the development of electronic structure methods. In the present work, we
explore the use of stochastic approaches to evaluate high-order electron
correlation energies, whose conventional computational scaling is unpleasantly
steep, being $O(N^{n+3})$ with respect to the system size $N$ and the
perturbation order $n$ for the M{\o}ller-Plesset (MP) series. To this end,
starting from Goldstone's time-dependent formulation of \emph{ab initio}
many-body perturbation theory (MBPT), we present a reformulation of MBPT, which
naturally leads to an Monte Carlo scheme with $O(nN^2+n^2N+f(n))$ scaling at
each step, where $f(n)$ is a function of $n$ depending on the specific
numerical scheme. Proof-of-concept calculations demonstrate that the proposed
quantum Monte Carlo algorithm successfully extends the previous Monte Carlo
approaches for MP2 and MP3 to higher orders. Therefore, for the first time the
Goldstone's time-dependent formulation is made useful numerically for electron
correlation energies, not only being purely as a theoretical tool.
