Quantum simulators were originally proposed for simulating one partial differential equation (PDE) in particular—Schrödinger’s equation. Can quantum simulators also efficiently simulate other PDEs? While most computational methods for PDEs—both classical and quantum—are digital (they must be discretised first), PDEs have continuous degrees of freedom. This suggests that an analog representation can be more natural. While digital quantum degrees of freedom are usually described by qubits, the analog or continuous quantum degrees of freedom can be captured by qumodes. Based on a method called Schrödingerisation, we show how to directly map D-dimensional linear PDEs onto a (D+1) -qumode quantum system where analog or continuous-variable (CV) Hamiltonian simulation on D + 1 qumodes can be used. This very simple methodology does not require one to discretise PDEs first, and it is not only applicable to linear PDEs but also to some nonlinear PDEs and systems of nonlinear ordinary differential equations. We show some examples using this method, including the Liouville equation, heat equation, Fokker–Planck equation, Black–Scholes equations, wave equation and Maxwell’s equations. We also devise new protocols for linear PDEs with random coefficients, important in uncertainty quantification, where it is clear how the analog or CV framework is most natural. This also raises the possibility that some PDEs may be simulated directly on analog quantum systems by using Hamiltonians natural for those quantum systems.

中文翻译：

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偏微分方程的模拟量子模拟


量子模拟器最初是为了模拟一个偏微分方程（PDE）——薛定谔方程而提出的。量子模拟器也能有效地模拟其他偏微分方程吗？虽然大多数偏微分方程的计算方法（无论是经典的还是量子的）都是数字化的（必须首先离散化），但偏微分方程具有连续的自由度。这表明模拟表示可以更自然。虽然数字量子自由度通常由量子位描述，但模拟或连续量子自由度可以由量子模式捕获。基于一种称为薛定谔化的方法，我们展示了如何将 D 维线性偏微分方程直接映射到 (D+1)-qumode 量子系统，其中可以使用 D + 1 qumode 上的模拟或连续变量 (CV) 哈密顿模拟。这种非常简单的方法不需要首先离散偏微分方程，它不仅适用于线性偏微分方程，还适用于一些非线性偏微分方程和非线性常微分方程组。我们展示了一些使用这种方法的例子，包括刘维尔方程、热方程、福克-普朗克方程、布莱克-斯科尔斯方程、波动方程和麦克斯韦方程。我们还为具有随机系数的线性偏微分方程设计了新协议，这对于不确定性量化很重要，因为很清楚模拟或 CV 框架是如何最自然的。这也提出了通过使用对于这些量子系统自然的哈密顿量来直接在模拟量子系统上模拟某些偏微分方程的可能性。