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

TL;DR

The paper tackles the challenging problem of computing the spectral properties of the high-dimensional Fokker-Planck operator within probabilistic learning on manifolds (PLoM) using quantum computing. It reformulates the FKP operator as a Schrödinger-type operator with a potential ${\mathcal{V}}({\bm{y}})$ and develops a polynomial-chaos expansion in Gaussian space to enable a second-quantized, qubit-encoded quantum implementation. The methodology details the construction of the potential via a Gaussian-space PCE of $f=\log(p_{\mathbf{H}})$, the corresponding convergence framework, and explicit mappings of the Laplacian and potential to the Fock space, followed by circuit design and measurement strategies for eigenstate construction and overlap extraction. Collectively, this work provides a principled bridge between probabilistic learning on manifolds and quantum algorithms, with a concrete pathway toward high-dimensional spectral problems on quantum hardware and practical surrogate modeling from small data sets.

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 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.

Figures (6)

• Figure 1: Convergence analysis for the Gaussian reference case, for $\nu=1$ (a,b,c) and $\nu=10$ (d,e,f). Graph of functions $N \mapsto {\mathcal{E}}_f^{{\hbox{ref}}} (N)$ (a,d), $N \mapsto {\mathcal{E}}_{\bm{g}}^{{\hbox{ref}}} (N)$ (b,e), and $N \mapsto {\mathcal{E}}_h^{{\hbox{ref}}} (N)$ (c,f). The horizontal axes are $\log_{10}(N)$.
• Figure 2: Convergence analysis for the Gaussian GKDE case, for $\nu=1$ (a,b,c) and $\nu=10$ (d,e,f). For $n_d = 100$ (dashed line), $1000$ (thin line), $5000$ (thick line), and $10\,000$ (line with diamonds), graph of functions $N \mapsto {\mathcal{E}}_f (N)$ (a,d), $N \mapsto {\mathcal{E}}_{\bm{g}} (N)$ (b,e), and $N \mapsto {\mathcal{E}}_h (N)$ (c,f). The horizontal axes are $\log_{10}(N)$.
• Figure 3: General form of a circuit: the left dashed box indicates the trial state construction for $q_n$ in terms of qubits, while the right dashed box indicates the measurement of an observable $A_\beta$, where $U^m_\beta$ is a unitary transformation that diagonalizes $A_\beta$.
• Figure 4: Schematic view of a Variational Quantum Eigensolver (VQE) hybrid quantum classical circuit. The slash '/' indicates a multi-qubit line. The quantum state is constructed on the quantum circuit in two steps: i) the initial guess using $U^{init}$, and ii) the optimized unitary transformation $U^{{\hbox{VQE}}}$, which depend on a set of parameters ${\bm{\tau}}$. The classical computer drives the measurements, with unitaries $\{U^m_\alpha\}$ to reconstruct the expectation value of the Rayleigh quotient and its gradients with respect to the parameters ${\bm{\tau}}$. The classical computer then operates a feedback on the quantum computer to optimize the trial state by changing $U^{{\hbox{VQE}}}$.
• Figure 5: Circuit for the construction of one factor in \ref{['eq8.1']}, i.e. $\sum_{\alpha=1}^{m_k} \varphi^{(k)}_\alpha (\eta_k)\, | \varphi^{(k)}_\alpha \rangle$. The lines connecting circles at the nodes indicates an anti-control (control on the zero value) on the corresponding qubits.

Theorems & Definitions (13)

• definition 1: Hilbert spaces $L^2(\Theta,{\mathbb{R}})$ and ${\mathbb{H}}= L^2({\mathbb{R}}^\nu; p_{\bm{Y}}({\bm{y}})\, d{\bm{y}})$
• Proposition 1: Properties of function $f = \log(p_{\bm{H}})$
• proof
• definition 2: Normalized Hermite polynomials on ${\mathbb{R}}^\nu$ as a Hilbert basis of ${\mathbb{H}}$
• Proposition 2: Polynomial chaos expansions in ${\mathbb{H}}$ for functions $f$, ${\bm{g}}$, and $h$
• proof
• Proposition 3: Polynomial chaos expansions in Gaussian space of potential ${\mathcal{V}}$
• proof
• remark 1: Choice of the polynomial chaos expansion for ${\mathcal{V}}$
• definition 3: Truncated polynomial chaos expansions of $f$, $g_j$, and $h$