Jacobi-anger Density Estimation Reconstructs Quantum State Energy Distributions from Finite Hamiltonian Moments

Accurately determining the energy distribution of quantum states is crucial for modelling molecular behaviour, yet calculating this distribution directly becomes overwhelmingly complex for larger systems. Kyeongan Park, Gwonhak Lee, and Minhyeok Kang from Sungkyunkwan University, along with Youngjun Park and Joonsuk Huh from Yonsei University, present a new method to address this challenge. Their work introduces Jacobi-Anger Density Estimation (JADE), a technique that reconstructs the energy distribution from a limited set of Hamiltonian moments using a mathematical expansion and an inverse Fourier transform. This innovative approach accurately recovers energy distributions for molecular systems and offers a broadly applicable solution for estimating complex probability densities across diverse scientific and engineering disciplines, promising significant improvements in ground state energy estimation and related applications.

Jacobi Expansions And Mathematical Rigour

This document presents a comprehensive explanation of the JADE method, exceeding the scope of a typical research paper by thoroughly explaining the underlying principles, mathematical derivations, and justifications for its functionality. The inclusion of detailed derivations and proofs elevates this work, demonstrating a strong mathematical foundation through the use of Jacobi-Anger expansions, Fourier transforms, and weighted L2 spaces. The document is well-organized, covering not only the core JADE algorithm but also its connection to optimal function approximation and weighted L2 loss minimization, providing a broader theoretical context. While clear and concise, potential improvements include adding visualizations, analyzing computational complexity, addressing practical considerations like parameter selection and noisy data, and providing a code snippet demonstrating implementation.

Overall, this is an exceptionally well-written and comprehensive document, a significant contribution to the field of PDF estimation, and a valuable resource for researchers and practitioners alike. Recognizing that full Hamiltonian diagonalization is computationally prohibitive for large-scale systems, the team engineered a technique that bypasses this limitation by focusing on moments, which are comparatively inexpensive to compute. JADE reconstructs the characteristic function from these moments using the Jacobi-Anger expansion, then estimates the underlying probability distribution via an inverse Fourier transform, delivering a highly efficient solution for complex systems.

This approach avoids the limitations of conventional expansion-based methods like Gram-Charlier and Edgeworth expansions, which struggle with non-Gaussian, multimodal distributions common in quantum states. Researchers implemented JADE as a classical algorithm, distinguishing it from quantum approaches like coarse-QPE, which remain constrained by the resource-intensive nature of controlled-unitary operations on current quantum hardware. This work addresses a significant challenge in quantum computing, where directly obtaining the energy distribution requires computationally expensive full Hamiltonian diagonalization. JADE avoids this prohibitive calculation by reconstructing the characteristic function from moments using the Jacobi-Anger expansion, then recovering the distribution via an inverse Fourier transform. The method delivers a highly efficient solution for estimating energy distributions without complex optimization routines.

Experiments demonstrate JADE’s ability to accurately estimate even multimodal, non-Gaussian energy distributions, unlike conventional methods such as Gram-Charlier and Edgeworth expansions which struggle with distributions deviating from Gaussianity. Unlike coarse-Quantum Phase Estimation (QPE), JADE is efficiently executable on classical hardware, bypassing the resource-intensive controlled-unitary operations that limit the practicality of quantum algorithms. The research highlights JADE as a non-parametric, quantum-inspired method for efficiently estimating probability density functions from finite moments. JADE constructs an approximation of the characteristic function using the Jacobi-Anger expansion, then applies an inverse Fourier transform to yield an estimate of the energy distribution, providing a simple, non-iterative procedure. The team demonstrates that JADE successfully reconstructs these distributions for a molecular system, offering a computationally efficient alternative to full Hamiltonian diagonalization, which becomes impractical for larger systems. Importantly, the method extends beyond quantum mechanics, proving effective in estimating complex probability density functions across various scientific and engineering fields. JADE provides a mathematically simple, closed-form expression and is proven optimal in minimising weighted loss during function approximation. The authors acknowledge that the computational cost remains tied to calculating Hamiltonian moments, but highlight opportunities for further acceleration using techniques like tensor networks and matrix product states. Future research will focus on benchmarking JADE against classically obtained ground-state energies to evaluate potential quantum advantages and comparing its resource efficiency with existing quantum algorithms.