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.

翻译：我们引入一系列恒等式，将一般线性非酉演化算子表示为酉演化算子的线性组合，每个酉算子对应一个哈密顿量模拟问题。该公式可指数级提升近期提出的哈密顿量模拟线性组合（LCHS）方法[参见An、Liu与Lin，《物理评论快报》，2023]的精度。该方法首次使量子算法在矩阵查询中对所有参数实现既具备最优态制备成本又达到近最优缩放尺度，从而求解线性微分方程。