We present a quantum computing formulation to address a challenging problem in the development of probabilistic learning on manifolds (PLoM). It involves solving the spectral problem of the high-dimensional Fokker-Planck (FKP) operator, which remains beyond the reach of classical computing. Our ultimate goal is to develop an efficient approach for practical computations on quantum computers. For now, we focus on an adapted formulation tailored to quantum computing. The methodological aspects covered in this work include the construction of the FKP equation, where the invariant probability measure is derived from a training dataset, and the formulation of the eigenvalue problem for the FKP operator. The eigen equation is transformed into a Schr\"odinger equation with a potential V, a non-algebraic function that is neither simple nor a polynomial representation. To address this, we propose a methodology for constructing a multivariate polynomial approximation of V, leveraging polynomial chaos expansion within the Gaussian Sobolev space. This approach preserves the algebraic properties of the potential and adapts it for quantum algorithms. The quantum computing formulation employs a finite basis representation, incorporating second quantization with creation and annihilation operators. Explicit formulas for the Laplacian and potential are derived and mapped onto qubits using Pauli matrix expressions. Additionally, we outline the design of quantum circuits and the implementation of measurements to construct and observe specific quantum states. Information is extracted through quantum measurements, with eigenstates constructed and overlap measurements evaluated using universal quantum gates.

翻译：本文提出了一种量子计算表述，以解决流形概率学习（PLoM）发展中的一个具有挑战性的问题。该问题涉及求解高维Fokker-Planck（FKP）算子的谱问题，这仍然是经典计算能力所不及的。我们的最终目标是开发一种在量子计算机上进行实际计算的高效方法。目前，我们专注于一种为量子计算量身定制的适应性表述。本工作涵盖的方法论方面包括FKP方程的构建（其中不变概率测度从训练数据集导出）以及FKP算子特征值问题的表述。特征方程被转化为带有势V的薛定谔方程，该势V是一个非代数函数，既非简单形式也非多项式表示。为了解决这个问题，我们提出了一种在高斯Sobolev空间内利用多项式混沌展开来构建V的多变量多项式逼近的方法。这种方法保持了势的代数特性，并使其适用于量子算法。该量子计算表述采用了有限基表示，并结合了产生算子和湮灭算子的二次量子化。我们推导了拉普拉斯算子和势的显式公式，并使用泡利矩阵表达式将它们映射到量子比特上。此外，我们概述了量子电路的设计以及用于构建和观测特定量子态的测量实现。信息通过量子测量提取，其中特征态通过通用量子门构建，并评估重叠测量。