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In this paper, we present an efficient quantum algorithm to simulate nonlinear differential equations with polynomial vector fields of arbitrary (finite) degree on quantum platforms. Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise extensively in science and engineering applications. Examples of ODE models include mechanics of rigid bodies, molecular dynamics, chemical kinetics, and epidemiology. Nonlinear PDEs arise in fluid dynamics, combustion, weather forecasting, structural mechanics, plasma dynamics, and finance to name a few. In practice, it is challenging to simulate such equations on classical computers due to high dimensionality, stiffness arising from multiple spatial/temporal scales, nonlinearities, and chaotic dynamics. Typically, high performance computing is used to mitigate computational challenges and involves approximations for tractability. For sparse n-dimensional linear ODEs, quantum algorithms have been developed which can produce a quantum state proportional to the solution in poly (log (n)) ) 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 and prove polynomial scaling of runtime on k. We demonstrate the computational framework on a numerical example. [ABSTRACT FROM AUTHOR]

• We develop a systematic framework for computing rare event probabilities in stochastic differential equations. • Stochastic Koopman eigenfunctions are used to develop importance sampling estimators. • Stochastic Koopman eigenfunctions are computed using dynamic mode decomposition. • Non-rare trajectories are used to inform how rare events occur. • Even crude approximations of eigenfunctions can yield efficient estimators. We exploit the relationship between the stochastic Koopman operator and the Kolmogorov backward equation to construct importance sampling schemes for stochastic differential equations. Specifically, we propose using eigenfunctions of the stochastic Koopman operator to approximate the Doob transform for an observable of interest (e.g., associated with a rare event) which in turn yields an approximation of the corresponding zero-variance importance sampling estimator. Our approach is broadly applicable and systematic, treating non-normal systems, non-gradient systems, and systems with oscillatory dynamics or rank-deficient noise in a common framework. In nonlinear settings where the stochastic Koopman eigenfunctions cannot be derived analytically, we use dynamic mode decomposition (DMD) methods to approximate them numerically, but the framework is agnostic to the particular numerical method employed. Numerical experiments demonstrate that even coarse approximations of a few eigenfunctions, where the latter are built from non-rare trajectories, can produce effective importance sampling schemes for rare events. [ABSTRACT FROM AUTHOR]