Hybrid Quantum-Classical Algorithm for Hamiltonian Simulation

We introduce a hybrid classical-quantum algorithm for simulating a Hamiltonian of the form H= sum_i=1^K H_i = sum_i=1^K H_i1 otimes H_i2 otimes cdots otimes H_iM. Given that the entries of all \{ H_i1, H_i2 , cdots , H_iM\} (for all i) are classically known, we present a procedure (with three variants) in which these operators are classically diagonalized, and then this information is fed into three possible quantum procedures to obtain the block-encoding of H. The evolution operator exp(-iHt) is then obtained using the standard block-encoding/quantum singular value transformation framework. In the case where \{H_i\}_i=1^K commute pairwise, our method can be trivially extended to the case with time-dependent coefficients. We provide a detailed discussion of the efficient regime of our hybrid framework and compare it with existing quantum simulation algorithms. Our algorithm can serve as a useful complement to existing quantum simulation algorithms, thereby expanding the reach of quantum computers for practically simulating physical systems. As a side contribution, we will show how the recent technique called randomized truncation to a quantum state developed by Harrow, Lowe, and Witteveen [arXiv preprint arXiv:2510.08518, 2025] can be applied to the context of quantum simulation and particularly quantum state preparation, for which the latter can be of independent interest.