Highly oscillatory dynamics are ubiquitous in nature and practical applications, resulting from either the large operator norm of the Hamiltonian, or the rapid change of the Hamiltonian itself when the Hamiltonian is time-dependent, or both. We develop a simple algorithm, called quantum Highly Oscillatory Protocol (qHOP) that can handle both oscillatory sources simultaneously. To our knowledge, this is the first quantum algorithm that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. Furthermore, we prove that our method achieves superconvergence for the digital simulation of the Schr\"odinger equation that has wide applications in electronic structure theory, molecular dynamics and quantum machine learning.
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