  Moderate-size quantum computers are now publicly accessible over the cloud,
opening the exciting possibility of performing dynamical simulations of quantum
systems. However, while rapidly improving, these devices have short coherence
times, limiting the depth of algorithms that may be successfully implemented.
Here we demonstrate that, despite these limitations, it is possible to
implement long-time, high fidelity simulations on current hardware.
Specifically, we simulate an XY-model spin chain on the Rigetti and IBM quantum
computers, maintaining a fidelity of at least 0.9 for over 600 time steps. This
is a factor of 150 longer than is possible using the iterated Trotter method.
Our simulations are performed using a new algorithm that we call the fixed
state Variational Fast Forwarding (fsVFF) algorithm. This algorithm decreases
the circuit depth and width required for a quantum simulation by finding an
approximate diagonalization of a short time evolution unitary. Crucially, fsVFF
only requires finding a diagonalization on the subspace spanned by the initial
state, rather than on the total Hilbert space as with previous methods,
substantially reducing the required resources.
