Resource-efficient quantum-classical hybrid algorithm for energy gap evaluation

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 nonvariational 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 for near-term quantum devices and early fault-tolerant quantum computers. To verify the efficiency of our algorithm, we conduct numerical simulations for the Heisenberg model and molecule systems on a classical emulator.