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 Schrödinger 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.

中文翻译：

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用于流形概率学习的 FKP 算子特征值问题的量子计算机公式

我们提出了一种量子计算公式来解决流形概率学习 （PLoM） 开发中的一个挑战性问题。它涉及解决高维福克-普朗克 （FKP） 算子的频谱问题，这仍然超出了经典计算的范围。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有效方法。目前，我们专注于为量子计算量身定制的适应公式。这项工作涵盖的方法方面包括 FKP 方程的构建，其中不变概率度量来自训练数据集，以及 FKP 运算符特征值问题的公式。特征方程被转换为具有势 V 的薛定谔方程，这是一个既不是简单也不是多项式表示的非代数函数。为了解决这个问题，我们提出了一种构建 V 的多元多项式近似的方法，利用高斯 Sobolev 空间内的多项式混沌展开。这种方法保留了势能的代数性质，并使其适用于量子算法。量子计算公式采用有限基表示，将二次量子化与创造和湮灭运算符相结合。使用 Pauli 矩阵表达式推导出 Laplacian 和 potential 的显式公式并将其映射到量子比特上。此外，我们还概述了量子电路的设计以及构建和观察特定量子态的测量实施。通过量子测量提取信息，构建特征态并使用通用量子门评估重叠测量。