Quantum computer formulation of the FKP-operator eigenvalue problem for probabilistic learning on manifolds

TL;DR

This paper develops a quantum computing approach to solve the high-dimensional Fokker-Planck operator eigenvalue problem for probabilistic learning on manifolds, enabling computations beyond classical capabilities.

Contribution

It introduces a quantum formulation of the FKP eigenvalue problem, including polynomial approximation of the potential and quantum circuit design for eigenstate measurement.

Findings

Formulated the FKP eigenvalue problem for quantum algorithms.

Developed polynomial chaos expansion for potential approximation.

Outlined quantum circuit implementation for eigenstate measurement.

Abstract

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.