Mohsen Bagherimehrab, Dominic W. Berry, Philipp Schleich, Abdulrahman Aldossary, Jorge A. Campos Gonzalez Angulo, Alan Aspuru-Guzik
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Faster Algorithmic Quantum and Classical Simulations by Corrected Product Formulas
Hamiltonian simulation using product formulas is arguably the most
straightforward and practical approach for algorithmic simulation of a quantum
system's dynamics on a quantum computer. Here we present corrected product
formulas (CPFs), a variation of product formulas achieved by injecting
auxiliary terms called correctors into standard product formulas. We establish
several correctors that greatly improve the accuracy of standard product
formulas for simulating Hamiltonians comprised of two partitions that can be
exactly simulated, a common feature of lattice Hamiltonians, while only adding
a small additive or multiplicative factor to the simulation cost. We show that
correctors are particularly advantageous for perturbed systems, where one
partition has a relatively small norm compared to the other, as they allow the
small norm to be utilized as an additional parameter for controlling the
simulation error. We demonstrate the performance of CPFs by numerical
simulations for several lattice Hamiltonians. Numerical results show our
theoretical error bound for CPFs matches or exceeds the empirical error of
standard product formulas for these systems. CPFs could be a valuable
algorithmic tool for early fault-tolerant quantum computers with limited
computing resources. As for standard product formulas, CPFs could also be used
for simulations on a classical computer.