We propose an algorithm based on variational quantum imaginary time evolution for solving the Feynman-Kac partial differential equation resulting from a multidimensional system of stochastic differential equations. We utilize the correspondence between the Feynman-Kac partial differential equation (PDE) and the Wick-rotated Schrödinger equation for this purpose. The results for a $(2+1)$ dimensional Feynman-Kac system obtained through the variational quantum algorithm are then compared against classical ODE solvers and Monte Carlo simulation. We see a remarkable agreement between the classical methods and the quantum variational method for an illustrative example on six and eight qubits. In the non-trivial case of PDEs which are preserving probability distributions – rather than preserving the $\ell_2$-norm – we introduce a proxy norm which is efficient in keeping the solution approximately normalized throughout the evolution. The algorithmic complexity and costs associated to this methodology, in particular for the extraction of properties of the solution, are investigated. Future research topics in the areas of quantitative finance and other types of PDEs are also discussed.

中文翻译：

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Feynman-Kac 公式的变分量子算法

我们提出了一种基于变分量子虚时间演化的算法，用于求解由随机微分方程的多维系统产生的 Feynman-Kac 偏微分方程。为此，我们利用 Feynman-Kac 偏微分方程 (PDE) 和 Wick 旋转薛定谔方程之间的对应关系。然后将通过变分量子算法获得的 $(2+1)$ 维 Feynman-Kac 系统的结果与经典 ODE 求解器和蒙特卡罗模拟进行比较。对于 6 个和 8 个量子位的说明性示例，我们看到经典方法和量子变分方法之间的显着一致性。在保留概率分布的 PDE 的非平凡情况下——而不是保留 $\ell_2$-范数——我们引入了一个代理范数，该范数可以有效地在整个演化过程中保持解近似归一化。研究了与该方法相关的算法复杂性和成本，特别是对于提取解决方案的属性。还讨论了量化金融和其他类型偏微分方程领域的未来研究主题。