Fourier transform-based linear combination of Hamiltonian simulation

Linear combination of Hamiltonian simulation (LCHS) connects the general linear non-unitary dynamics with unitary operators and serves as the mathematical backbone of designing near-optimal quantum linear differential equation algorithms. However, the existing LCHS formalism needs to find a kernel function subject to complicated technical conditions on a half complex plane. In this work, we establish an alternative formalism of LCHS based on the Fourier transform. Our new formalism completely removes the technical requirements beyond the real axis, providing a simple and flexible way of constructing LCHS kernel functions. Specifically, we construct a different family of the LCHS kernel function, providing a $1.81$ times reduction in the quantum differential equation algorithms based on LCHS, and an $8.27$ times reduction in its quantum circuit depth at a truncation error of $\epsilon \le 10^{-8}$. Additionally, we extend the scope of the LCHS formula to the scenario of simulating linear unstable dynamics for a short or intermediate time period.