  We develop circuit implementations for digital-level quantum Hamiltonian
dynamics simulation algorithms suitable for implementation on a reconfigurable
quantum computer, such as trapped ions. Our focus is on the co-design of a
problem, its solution, and quantum hardware capable of executing the solution
at the minimal cost expressed in terms of the quantum computing resources used
while demonstrating the solution of an instance of a scientifically interesting
problem that is intractable classically. The choice for Hamiltonian dynamics
simulation is due to the combination of its usefulness in the study of
equilibrium in closed quantum mechanical systems, a low cost in the
implementation by quantum algorithms, and the difficulty of classical
simulation. By targeting a specific type of quantum computer and tailoring the
problem instance and solution to suit physical constraints imposed by the
hardware, we are able to reduce the resource counts by a factor of $10$ in a
physical-level implementation and a factor of $30$ to $60$ in a fault-tolerant
implementation over state of the art.
