We investigate the limitations of quantum computers for solving nonlinear dynamical systems. In particular, we tighten the worst-case bounds of the quantum Carleman linearisation (QCL) algorithm [Liu et al., PNAS 118, 2021] answering one of their open questions. We provide a further significant limitation for any quantum algorithm that aims to output a quantum state that approximates the normalized solution vector. Given a natural choice of coordinates for a dynamical system with one or more positive Lyapunov exponents and solutions that grow sub-exponentially, we prove that any such algorithm has complexity scaling at least exponentially in the integration time. As such, an efficient quantum algorithm for simulating chaotic systems or regimes is likely not possible.

中文翻译：

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量子算法在求解湍流和混沌系统方面的局限性


我们研究了量子计算机在求解非线性动态系统方面的局限性。特别是，我们收紧了量子 Carleman 线性化 （QCL） 算法的最坏情况界限 [Liu et al.， PNAS 118， 2021]，回答了他们的一个悬而未决的问题。我们为旨在输出近似归一化解向量的量子状态的任何量子算法提供了进一步的重大限制。给定具有一个或多个正 Lyapunov 指数和亚指数增长的解的动力学系统的自然坐标选择，我们证明任何此类算法在积分时间内都至少具有指数级扩展的复杂性。因此，用于模拟混沌系统或政权的高效量子算法可能是不可能的。