Variational Fast Forwarding for NISQ Simulations

Quantum simulation has the potential to transform fields from quantum chemistry and material science to nuclear physics. However, while quantum hardware is rapidly reaching the stage where it can outperform classical supercomputers, we remain in the noisy intermediate-scale quantum (NISQ) era in which devices are prone to errors and susceptible to decoherence. The resulting short coherence times of near term quantum computers restrict the length of times that may be successfully simulated thus limiting the usefulness of quantum simulations.Here we present a hybrid quantum-classical algorithm, called Variational Fast Forwarding (VFF), to address the challenge of how to perform genuinely useful simulations with NISQ devices. The number of gates required for a VFF simulation does not scale with the length of time to be simulated and thus enables simulations of long times using a fixed number of gates. To achieve this, VFF approximately diagonalises a Trotter approximation of the short-time evolution using a hybrid quantum-classical variational method, and in turn this diagonalisation allows for arbitrary time simulations with the same circuit structure.We test the capability of VFF through extensive numerical simulations of the Hubbard, Ising, and Heisenberg models, and we implement VFF on IBM's and Riggeti's quantum computers to demonstrate simulation beyond the coherence time. In contrast to prior work on variational quantum simulations, we further provide a rigorous analysis of VFF's simulation errors. We find that the simulation error of VFF scales at worst linearly in the fast-forwarded simulation time. This enables us to derive a natural termination condition for the algorithm which is framed in terms of the required average fidelity of the simulation. We further present two refinements of the VFF algorithm.