  Computations of chemical systems' equilibrium properties and non-equilibrium
dynamics have been suspected of being a "killer app" for quantum computers.
This review highlights the recent advancements of quantum algorithms tackling
complex sampling tasks in the key areas of computational chemistry: ground
state, thermal state properties, and quantum dynamics calculations. We review a
broad range of quantum algorithms, from hybrid quantum-classical to fully
quantum, focusing on the traditional Monte Carlo family, including Markov chain
Monte Carlo, variational Monte Carlo, projector Monte Carlo, path integral
Monte Carlo, etc. We also cover other relevant techniques involving complex
sampling tasks, such as quantum-selected configuration interaction, minimally
entangled typical thermal states, entanglement forging, and Monte
Carlo-flavored Lindbladian dynamics. We provide a comprehensive overview of
these algorithms' classical and quantum counterparts, detailing their
theoretical frameworks and discussing the potentials and challenges in
achieving quantum computational advantages.
