Hybrid Quantum-Classical Algorithm for Hamiltonian Simulation

Abstract

We introduce a hybrid classical-quantum algorithm for simulating a Hamiltonian of the form $H={\sum }_{i=1}^K{H}_i={\sum }_{i=1}^K{H}_{i_1}\otimes H_{i_2}\otimes \cdots \otimes H_{i_M}$. Given that the entries of all $\{H_{i_1},H_{i_2},\cdots ,H_{i_M}\}$ (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 \textit{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.