The time-marching strategy, which propagates the solution from one time step to the next, is a natural strategy for solving time-dependent differential equations on classical computers, as well as for solving the Hamiltonian simulation problem on quantum computers. For more general linear differential equations, a time-marching based quantum solver can suffer from exponentially vanishing success probability with respect to the number of time steps and is thus considered impractical. We solve this problem by repeatedly invoking a technique called the uniform singular value amplification, and the overall success probability can be lower bounded by a quantity that is independent of the number of time steps. The success probability can be further improved using a compression gadget lemma. This provides a path of designing quantum differential equation solvers that is alternative to those based on quantum linear systems algorithms (QLSA). We demonstrate the performance of the time-marching strategy with a high-order integrator based on the truncated Dyson series. The complexity of the algorithm depends linearly on the amplification ratio, which quantifies the deviation from a unitary dynamics. We prove that the linear dependence on the amplification ratio attains the query complexity lower bound and thus cannot be improved in the worst case. This algorithm also surpasses existing QLSA based solvers in three aspects: (1) the coefficient matrix $A(t)$ does not need to be diagonalizable. (2) $A(t)$ can be non-smooth, and is only of bounded variation. (3) It can use fewer queries to the initial state. Finally, we demonstrate the time-marching strategy with a first-order truncated Magnus series, while retaining the aforementioned benefits. Our analysis also raises some open questions concerning the differences between time-marching and QLSA based methods for solving differential equations.

翻译：时间推进策略将解从一个时间步传播至下一个时间步，是经典计算机求解含时微分方程的自然策略，也是量子计算机求解哈密顿量模拟问题的自然方法。对于更一般的线性微分方程，基于时间推进的量子求解器可能面临成功概率随时间步数指数衰减的问题，因此被认为不具实用性。我们通过反复调用一种称为均匀奇异值放大的技术解决了这一问题，使得总体成功概率可由一个与时间步数无关的量进行下界约束，并可通过压缩引理进一步改善成功概率。这为设计量子微分方程求解器提供了一条不同于基于量子线性系统算法（QLSA）的路径。我们展示了采用基于截断戴森级数的高阶积分器的时间推进策略的性能。算法复杂度与放大比（量化偏离幺正动力学的程度）呈线性关系。我们证明，对放大比的线性依赖达到了查询复杂度下界，因此在最坏情况下无法改进。该算法在以下三个方面超越了现有基于QLSA的求解器：（1）系数矩阵$A(t)$无需可对角化；（2）$A(t)$可以是非光滑的，仅需有界变差；（3）对初始状态的查询次数更少。最后，我们在保留上述优势的同时，展示了采用一阶截断马格努斯级数的时间推进策略。我们的分析还提出了关于时间推进法与基于QLSA方法求解微分方程差异的若干开放性问题。