Multi-product Hamiltonian simulation with explicit commutator scaling

• URL: http://arxiv.org/abs/2403.08922v1
• Date: Wed, 13 Mar 2024 19:23:59 GMT
• Title: Multi-product Hamiltonian simulation with explicit commutator scaling
• Authors: Junaid Aftab, Dong An, Konstantina Trivisa,
• Abstract summary: The well-conditioned multi-product formula (MPF) is a simple high-order time-independent Hamiltonian simulation algorithm. We conduct a rigorous analysis of the MPF, demonstrating explicit commutator scaling and near-optimal time and precision dependence. Compared to post-Trotter methods, the MPF based on a second-order product formula can achieve a better scaling in system size, with only poly-logarithmic overhead in evolution time and precision.
• Score: 2.5677613431426978
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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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