Simulating quantum systems on classical computers requires enormous resources due to the cost of describing quantum systems with classical components. Quantum simulation sidesteps this resource cost by emulating one quantum systems with another.

Classical algorithms sidestep the resource cost through clever ansätze and approximations. A useful quantum simulation needs to provide information about quantum systems in a regime where classical algorithms fail and provide more information than directly performing an experiment directly on the quantum system of interest. In this talk, we will consider problems where quantum simulation looks promising.  We will discuss future simulations with fault-tolerant quantum computers and near-term simulations that use all the degrees of freedom of a trapped-ion quantum simulator.