  Digital quantum computers provide a computational framework for solving the
Schr\"{o}dinger equation for a variety of many-particle systems. Quantum
computing algorithms for the quantum simulation of these systems have recently
witnessed remarkable growth, notwithstanding the limitations of existing
quantum hardware, especially as a tool for electronic structure computations in
molecules. In this review, we provide a self-contained introduction to emerging
algorithms for the simulation of Hamiltonian dynamics and eigenstates, with
emphasis on their applications to the electronic structure in molecular
systems. Theoretical foundations and implementation details of the method are
discussed, and their strengths, limitations, and recent advances are presented.

Quantum Physics