We present an efficient quantum algorithm to simulate nonlinear differential equations with polynomial vector fields of arbitrary degree on quantum platforms. Models of physical systems that are governed by ordinary differential equations (ODEs) or partial differential equation (PDEs) can be challenging to solve on classical computers due to high dimensionality, stiffness, nonlinearities, and sensitive dependence to initial conditions. For sparse $n$-dimensional linear ODEs, quantum algorithms have been developed which can produce a quantum state proportional to the solution in poly(log(nx)) time using the quantum linear systems algorithm (QLSA). Recently, this framework was extended to systems of nonlinear ODEs with quadratic polynomial vector fields by applying Carleman linearization that enables the embedding of the quadratic system into an approximate linear form. A detailed complexity analysis was conducted which showed significant computational advantage under certain conditions. We present an extension of this algorithm to deal with systems of nonlinear ODEs with $k$-th degree polynomial vector fields for arbitrary (finite) values of $k$. The steps involve: 1) mapping the $k$-th degree polynomial ODE to a higher dimensional quadratic polynomial ODE; 2) applying Carleman linearization to transform the quadratic ODE to an infinite-dimensional system of linear ODEs; 3) truncating and discretizing the linear ODE and solving using the forward Euler method and QLSA. Alternatively, one could apply Carleman linearization directly to the $k$-th degree polynomial ODE, resulting in a system of infinite-dimensional linear ODEs, and then apply step 3. This solution route can be computationally more efficient. We present detailed complexity analysis of the proposed algorithms, prove polynomial scaling of runtime on $k$ and demonstrate the framework on an example.

翻译：我们提出了一种高效量子算法，用于在量子平台上模拟具有任意阶多项式向量场的非线性微分方程。受常微分方程或偏微分方程支配的物理系统模型，由于高维度、刚性、非线性以及对初始条件的敏感依赖性，在经典计算机上求解可能具有挑战性。对于稀疏的n维线性常微分方程，已有量子算法能够通过量子线性系统算法在poly(log(nx))时间内生成与解成比例的量子态。最近，这一框架通过应用Carleman线性化扩展到了具有二次多项式向量场的非线性常微分方程组，该方法将二次系统嵌入到近似线性形式中。我们进行了详细的复杂度分析，表明在特定条件下可获得显著的计算优势。本文将该算法扩展至处理具有任意（有限）阶k次多项式向量场的非线性常微分方程组。步骤包括：1）将k次多项式常微分方程映射为更高维的二次多项式常微分方程；2）应用Carleman线性化将二次常微分方程转化为无限维线性常微分方程组；3）截断并离散化线性常微分方程，通过前向欧拉法和量子线性系统算法求解。另一种方案是直接对k次多项式常微分方程应用Carleman线性化，得到无限维线性常微分方程组后执行步骤3。这种求解路径在计算上可能更高效。我们给出了所提算法的详细复杂度分析，证明了运行时间关于k的多项式缩放，并通过实例验证了该框架的有效性。