  We propose a simple method for simulating a general class of non-unitary
dynamics as a linear combination of Hamiltonian simulation (LCHS) problems.
LCHS does not rely on converting the problem into a dilated linear system
problem, or on the spectral mapping theorem. The latter is the mathematical
foundation of many quantum algorithms for solving a wide variety of tasks
involving non-unitary processes, such as the quantum singular value
transformation (QSVT). The LCHS method can achieve optimal cost in terms of
state preparation. We also demonstrate an application for open quantum dynamics
simulation using the complex absorbing potential method with near-optimal
dependence on all parameters.
