Multi-product Hamiltonian simulation with explicit commutator scaling
<|reference_start|>Multi-product Hamiltonian simulation with explicit commutator scaling: The well-conditioned multi-product formula (MPF), proposed by [Low, Kliuchnikov, and Wiebe, 2019], is a simple high-order time-independent Hamiltonian simulation algorithm that implements a linear combination of standard product formulas of low order. While the MPF aims to simultaneously exploit commutator scaling among Hamiltonians and achieve near-optimal time and precision dependence, its lack of a rigorous error bound on the nested commutators renders its practical advantage ambiguous. In this work, we conduct a rigorous complexity analysis of the well-conditioned MPF, demonstrating explicit commutator scaling and near-optimal time and precision dependence at the same time. Using our improved complexity analysis, we present several applications of practical interest where the MPF based on a second-order product formula can achieve a polynomial speedup in both system size and evolution time, as well as an exponential speedup in precision, compared to second-order and even higher-order product formulas. Compared to post-Trotter methods, the MPF based on a second-order product formula can achieve polynomially better scaling in system size, with only poly-logarithmic overhead in evolution time and precision.<|reference_end|>
@article{aftab2024multi-product,
    title={Multi-product Hamiltonian simulation with explicit commutator scaling},
    author={Junaid Aftab, Dong An, Konstantina Trivisa},
    journal={arXiv preprint arXiv:2403.08922},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.08922},
    primaryClass={quant-ph cs.NA math.NA}
}