  Over the last few decades, quantum chemistry has progressed through the
development of computational methods based on modern digital computers.
However, these methods can hardly fulfill the exponentially-growing resource
requirements when applied to large quantum systems. As pointed out by Feynman,
this restriction is intrinsic to all computational models based on classical
physics. Recently, the rapid advancement of trapped-ion technologies has opened
new possibilities for quantum control and quantum simulations. Here, we present
an efficient toolkit that exploits both the internal and motional degrees of
freedom of trapped ions for solving problems in quantum chemistry, including
molecular electronic structure, molecular dynamics, and vibronic coupling. We
focus on applications that go beyond the capacity of classical computers, but
may be realizable on state-of-the-art trapped-ion systems. These results allow
us to envision a new paradigm of quantum chemistry that shifts from the current
transistor to a near-future trapped-ion-based technology.
