  Estimating the eigenvalue or energy gap of a Hamiltonian H is vital for
studying quantum many-body systems. Particularly, many of the problems in
quantum chemistry, condensed matter physics, and nuclear physics investigate
the energy gap between two eigenstates. Hence, how to efficiently solve the
energy gap becomes an important motive for researching new quantum algorithms.
In this work, we propose a hybrid non-variational quantum algorithm that uses
the Monte Carlo method and real-time Hamiltonian simulation to evaluate the
energy gap of a general quantum many-body system. Compared to conventional
approaches, our algorithm does not require controlled real-time evolution, thus
making its implementation much more experimental-friendly. Since our algorithm
is non-variational, it is also free from the "barren plateaus" problem. To
verify the efficiency of our algorithm, we conduct numerical simulations for
the Heisenberg model and molecule systems on a classical emulator.
