  Trotterization-based, iterative approaches to quantum simulation are
restricted to simulation times less than the coherence time of the quantum
computer, which limits their utility in the near term. Here, we present a
hybrid quantum-classical algorithm, called Variational Fast Forwarding (VFF),
for decreasing the quantum circuit depth of quantum simulations. VFF seeks an
approximate diagonalization of a short-time simulation to enable longer-time
simulations using a constant number of gates. Our error analysis provides two
results: (1) the simulation error of VFF scales at worst linearly in the
fast-forwarded simulation time, and (2) our cost function's operational meaning
as an upper bound on average-case simulation error provides a natural
termination condition for VFF. We implement VFF for the Hubbard, Ising, and
Heisenberg models on a simulator. Additionally, we implement VFF on Rigetti's
quantum computer to demonstrate simulation beyond the coherence time. Finally,
we show how to estimate energy eigenvalues using VFF.
