  As we begin to reach the limits of classical computing, quantum computing has
emerged as a technology that has captured the imagination of the scientific
world. While for many years, the ability to execute quantum algorithms was only
a theoretical possibility, recent advances in hardware mean that quantum
computing devices now exist that can carry out quantum computation on a limited
scale. Thus it is now a real possibility, and of central importance at this
time, to assess the potential impact of quantum computers on real problems of
interest. One of the earliest and most compelling applications for quantum
computers is Feynman's idea of simulating quantum systems with many degrees of
freedom. Such systems are found across chemistry, physics, and materials
science. The particular way in which quantum computing extends classical
computing means that one cannot expect arbitrary simulations to be sped up by a
quantum computer, thus one must carefully identify areas where quantum
advantage may be achieved. In this review, we briefly describe central problems
in chemistry and materials science, in areas of electronic structure, quantum
statistical mechanics, and quantum dynamics, that are of potential interest for
solution on a quantum computer. We then take a detailed snapshot of current
progress in quantum algorithms for ground-state, dynamics, and thermal state
simulation, and analyze their strengths and weaknesses for future developments.
