Quantum Algorithm for Low Energy Effective Hamiltonian and Quasi-Degenerate Eigenvalue Problem

• URL: http://arxiv.org/abs/2510.08088v1
• Date: Thu, 09 Oct 2025 11:20:15 GMT
• Title: Quantum Algorithm for Low Energy Effective Hamiltonian and Quasi-Degenerate Eigenvalue Problem
• Abstract summary: Quasi-degenerate eigenvalue problems are central to quantum chemistry and condensed-matter physics.<n>We propose a quantum algorithm that diagonalizes such quasi-degenerate manifold by solving an effective-Hamiltonian eigenproblem in a low-dimensional subspace.<n>Our analysis provides provable bounds on eigenvalue accuracy and subspace fidelity, together with total query complexity.
• Score: 7.107390133525336
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Quasi-degenerate eigenvalue problems are central to quantum chemistry and condensed-matter physics, where low-energy spectra often form manifolds of nearly degenerate states that determine physical properties. Standard quantum algorithms, such as phase estimation and QSVT-based eigenvalue filtering, work well when a unique ground state is separated by a moderate spectral gap, but in the quasi-degenerate regime they require resolution finer than the intra-manifold splitting; otherwise, they return an uncontrolled superposition within the low-energy span and fail to detect or resolve degeneracies. In this work, we propose a quantum algorithm that directly diagonalizes such quasi-degenerate manifolds by solving an effective-Hamiltonian eigenproblem in a low-dimensional reference subspace. This reduced problem is exactly equivalent to the full eigenproblem, and its solutions are lifted to the full Hilbert space via a block-encoded wave operator. Our analysis provides provable bounds on eigenvalue accuracy and subspace fidelity, together with total query complexity, demonstrating that quasi-degenerate eigenvalue problems can be solved efficiently without assuming any intra-manifold splitting. We benchmark the algorithm on several systems (the Fermi-Hubbard model, LiH, and the transition-metal complex [Ru(bpy)$_3$]$^{2+}$), demonstrating robust performance and reliable resolution of (quasi-)degeneracies.

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