  We demonstrate a post-quench dynamics simulation of a Heisenberg model on
present-day IBM quantum hardware that extends beyond the coherence time of the
device. This is achieved using a hybrid quantum-classical algorithm that
propagates a state using Trotter evolution and then performs a classical
optimization that effectively compresses the time-evolved state into a
variational form. When iterated, this procedure enables simulations to
arbitrary times with an error controlled by the compression fidelity and a
fixed Trotter step size. We show how to measure the required cost function, the
overlap between the time-evolved and variational states, on present-day
hardware, making use of several error mitigation methods. In addition to
carrying out simulations on real hardware, we investigate the performance and
scaling behavior of the algorithm with noiseless and noisy classical
simulations. We find the main bottleneck in going to larger system sizes to be
the difficulty of carrying out the optimization of the noisy cost function.
