Quantum computers are expected to excel in simulating unitary dynamics, while most applications in scientific and engineering computations involve non-unitary dynamics.

First, we propose a simple method for simulating a general class of non-unitary dynamics as a linear combination of Hamiltonian simulation (LCHS) problems [Phys. Rev. Lett. 2023]. The LCHS method can achieve optimal complexity. Second, we develop a random compilation framework (random-LCHS) with circuit efficiency in the early fault-tolerant designs. Third, we develop a variational quantum algorithm (variational LCHS) for non-Hermitian system experiments. Finally, we describe a rigorous inf-LCHS Theorem for infinite-dimensional and unbounded operators.