  We introduce a family of identities that express general linear non-unitary
evolution operators as a linear combination of unitary evolution operators,
each solving a Hamiltonian simulation problem. This formulation can
exponentially enhance the accuracy of the recently introduced linear
combination of Hamiltonian simulation (LCHS) method [An, Liu, and Lin, Physical
Review Letters, 2023]. For the first time, this approach enables quantum
algorithms to solve linear differential equations with both optimal state
preparation cost and near-optimal scaling in matrix queries on all parameters.
