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
• Authors: Chun-Tse Li, Tzen Ong, Chih-Yun Lin, Yu-Cheng Chen, Hsin Lin, Min-Hsiu Hsieh,
• 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.

Related papers

• Improving Generalization and Trainability of Quantum Eigensolvers via Graph Neural Encoding [0.5013248430919223]
  Ground state of a many-body Hamiltonian is a central problem across physics, chemistry, and optimization.<n>We propose an end-to-end representation learning framework that combines a graph autoencoder with a classical neural network.<n>We demonstrate improved generalization and trainability, manifested as reduced test error and a significantly milder decay of gradient variance.
  arXiv  Detail & Related papers  (2026-02-23T12:01:14Z)
• Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra [4.081730190778995]
  We introduce QFAMES, a quantum algorithm that efficiently identifies clusters of eigenvalues and determines their multiplicities.<n>QFAMES also enables the estimation of observable expectation values within targeted energy clusters.<n>We validate the effectiveness of QFAMES through numerical demonstrations.
  arXiv  Detail & Related papers  (2025-10-08T18:37:36Z)
• Grassmann Variational Monte Carlo with neural wave functions [45.935798913942904]
  We formalize the framework introduced by Pfau et al.citepfau2024accurate in terms of Grassmann geometry of the Hilbert space.<n>We validate our approach on the Heisenberg quantum spin model on the square lattice, achieving highly accurate energies and physical observables for a large number of excited states.
  arXiv  Detail & Related papers  (2025-07-14T13:53:13Z)
• Quantum algorithm for solving generalized eigenvalue problems with application to the Schrödinger equation [0.0]
  Estimating excited-state energies is challenging for classical algorithms due to exponential scaling with system size.<n>We present a quantum algorithm for estimating eigenvalues and singular values of parameterized matrix families.
  arXiv  Detail & Related papers  (2025-06-16T14:24:30Z)
• Variational Quantum Subspace Construction via Symmetry-Preserving Cost Functions [36.94429692322632]
  We propose a variational strategy based on symmetry-preserving cost functions to iteratively construct a reduced subspace for extraction of low-lying energy states.<n>As a proof of concept, we test the proposed algorithms on H4 chain and ring, targeting both the ground-state energy and the charge gap.
  arXiv  Detail & Related papers  (2024-11-25T20:33:47Z)
• A quantum eigenvalue solver based on tensor networks [0.0]
  Electronic ground states are of central importance in chemical simulations, but have remained beyond the reach of efficient classical algorithms. We introduce a hybrid quantum-classical eigenvalue solver that constructs a wavefunction ansatz from a linear combination of matrix product states in rotated orbital bases. This study suggests a promising new avenue for scaling up simulations of strongly correlated chemical systems on near-term quantum hardware.
  arXiv  Detail & Related papers  (2024-04-16T02:04:47Z)
• Neutron-nucleus dynamics simulations for quantum computers [49.369935809497214]
  We develop a novel quantum algorithm for neutron-nucleus simulations with general potentials. It provides acceptable bound-state energies even in the presence of noise, through the noise-resilient training method. We introduce a new commutativity scheme called distance-grouped commutativity (DGC) and compare its performance with the well-known qubit-commutativity scheme.
  arXiv  Detail & Related papers  (2024-02-22T16:33:48Z)
• Quantum Worst-Case to Average-Case Reductions for All Linear Problems [66.65497337069792]
  We study the problem of designing worst-case to average-case reductions for quantum algorithms. We provide an explicit and efficient transformation of quantum algorithms that are only correct on a small fraction of their inputs into ones that are correct on all inputs.
  arXiv  Detail & Related papers  (2022-12-06T22:01:49Z)
• Spectral density reconstruction with Chebyshev polynomials [77.34726150561087]
  We show how to perform controllable reconstructions of a finite energy resolution with rigorous error estimates. This paves the way for future applications in nuclear and condensed matter physics.
  arXiv  Detail & Related papers  (2021-10-05T15:16:13Z)
• Bosonic field digitization for quantum computers [62.997667081978825]
  We address the representation of lattice bosonic fields in a discretized field amplitude basis. We develop methods to predict error scaling and present efficient qubit implementation strategies.
  arXiv  Detail & Related papers  (2021-08-24T15:30:04Z)
• Spectral Analysis of Product Formulas for Quantum Simulation [0.0]
  We show that the Trotter step size needed to estimate an energy eigenvalue within precision can be improved in scaling from $epsilon$ to $epsilon1/2$ for a large class of systems. Results partially generalize to diabatic processes, which remain in a narrow energy band separated from the rest of the spectrum by a gap.
  arXiv  Detail & Related papers  (2021-02-25T03:17:25Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.