Abstract 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.
An et al. (Mon,) studied this question.