Jacobi-Anger Density Estimation for Energy Distribution of Quantum States

TL;DR

JADE provides a quantum-inspired, moment-based method to estimate energy distributions from a finite set of Hamiltonian moments. By mapping the energy problem to a characteristic function via the Jacobi–Anger expansion and applying an analytic inverse Fourier transform, it yields a closed-form PDF that is optimal in a weighted $L_2$ sense. The approach handles multimodal and non-Gaussian distributions and outperforms traditional moment-based methods across quantum and classical PDFs. Its efficiency and generality promise practical impact for pre-screening quantum computations and broader applications in PDF estimation.

Abstract

The energy distribution of a quantum state is essential for accurately estimating a molecule's ground state energy in quantum computing. Directly obtaining this distribution requires full Hamiltonian diagonalization, which is computationally prohibitive for large-scale systems. A more practical strategy is to approximate the distribution from a finite set of Hamiltonian moments. However, reconstructing an accurate distribution from only a limited number of moments remains a significant challenge. In this work, we introduce Jacobi-Anger Density Estimation (JADE), a non-parametric, quantum-inspired method designed to overcome this difficulty. JADE reconstructs the characteristic function from a finite set of moments using the Jacobi-Anger expansion and then estimates the underlying distribution via an inverse Fourier transform. We demonstrate that JADE can accurately recover the energy distribution of a quantum state for a molecular system. Beyond quantum chemistry, we also show that JADE is broadly applicable to the estimation of complicated probability density functions in various other scientific and engineering fields. Our results highlight JADE as a powerful and versatile tool for practical quantum systems, with the potential to significantly enhance ground state energy estimation and related applications.

Figures (3)

• Figure 1: Schematic overview of the proposed JADE method. The process begins with computing the distribution’s moments and the corresponding expectation values of Chebyshev polynomials. These are then incorporated into the Jacobi--Anger expansion to approximate the characteristic function. Finally, the energy distribution is obtained analytically through the inverse Fourier transform.
• Figure 2: Estimation of the energy distribution for the Hartree–Fock state of a four-atom hydrogen chain. The $x$-axis is defined as $\epsilon'=\epsilon/3$, reflect the rescaling required by the Chebyshev polynomials, which are defined on $E \in [-1,1]$. Results from the Gram--Charlier expansion and JADE are compared against the exact distribution (black solid line) obtained using the Overlapper software package. For the Gram--Charlier expansion, the number of cumulants was limited to 6 and 12, since higher orders introduce severe oscillations and divergence. Whereas the Gram--Charlier expansion becomes increasingly unstable and inaccurate with the inclusion of additional moments, JADE systematically improves as more moments are included, ultimately converging to the exact distribution. The close agreement highlights JADE’s ability to recover the key features of the energy spectrum from only a finite set of moments.
• Figure 3: (a) A bimodal polynomial function, $f_X(x)=-\tfrac{21}{8}(x-1)(x+1)(x^4 - x^3 + x^2)$; (b) A randomly generated multimodal distribution using a Gaussian kernel; (c) A asymmetric uniform distribution with discontinuities at $x=-0.6$ and $x=0.8$; (d) A sigmoid-like function, $f_X(x)=\tfrac{1}{1+\exp(-5x)}$. The black solid line shows the original PDF. The blue dashed line denotes the PDF estimated by JADE. The green and red dashed lines correspond to the Gram--Charlier expansion with 5 and 10 cumulants, respectively. The pink dashed line indicates the MEM result, with its number of parameters equal to the number of moments used by JADE, and the orange solid line shows the KDE estimate obtained from 10,000 samples.