Multi-product Hamiltonian simulation with explicit commutator scaling
CoRR(2024)
摘要
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.
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