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

Quantum algorithm for linear non-unitary dynamics with near-optimal dependence on all parameters