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

TL;DR

The paper tackles the challenge of solving quasi-degenerate eigenvalue problems on quantum computers, where standard phase-estimation or filtering methods struggle to resolve degeneracies within a low-energy manifold. It introduces a subspace-based approach using a Feshbach-type effective Hamiltonian $H_{\mathrm{eff}}(\lambda)$ and a block-encoded wave operator to map reduced-space solutions to full eigenstates, enabling degeneracy detection and canonical labeling without resorting to intra-manifold energy resolution. The method provides provable bounds on eigenvalue accuracy and subspace fidelity, with explicit query-cost scaling tied to the external gap $g$, the subspace overlap $\gamma$, and the approximation tolerances of QSVT and matrix-element estimation. Numerical benchmarks on the Hubbard model, LiH, and the Ru(bpy)$_3^{2+}$ complex demonstrate robust performance in resolving (quasi-)degeneracies and achieving chemical accuracy in energies and high-fidelity eigenstates. The work generalizes downfolding techniques to a quantum-computational setting with rigorous error control, highlighting its potential for complex correlated systems in quantum chemistry and condensed matter physics.

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.

Figures (14)

• Figure 1: Algorithmic workflow for eigenvalue estimation (middle block) and eigenstate preparation (lower block). Pink blocks: Given block-encodings of $H$ (and $V$), QSVT implements a polynomial approximation to the resolvent $(H_{22}-\lambda I)^{-1}$, yielding a block-encoding of the dressed Hamiltonian term $\widetilde{H}_D(\lambda)$. Generalized Hadamard tests (optionally with QAE) produce the $d\times d$ matrix $H_{\mathrm{eff}}(\lambda)=H_{11}+\widetilde{H}_D(\lambda)$, which is diagonalized to obtain the eigenbranches $\xi_i(\lambda)$. Yellow blocks: a fixed-point iteration on $\xi_i(\lambda)-\lambda$ solves the nonlinear eigenvalue problem and returns the target energies. Blue block: a block-encoded wave operator evaluated at $\lambda$ prepares the corresponding full-space eigenstates; Löwdin orthonormalization yields an orthonormal basis for quasi-degenerate manifold.
• Figure 2: Quantum circuits for constructing the block encodings of $H_{22}$ and $H_{12}$.
• Figure 3: Quantum circuit for constructing adjustable block-encoding $(H_{22}-\lambda I_{\mathcal{H}_d^\perp})$
• Figure 4: Block-encoding circuit of the approximate dressed Hamiltonian $\widetilde{H}_{D}(\lambda)$.
• Figure 5: Block-encoding circuit for the approximate wave operator $\widetilde{\Omega}(\lambda)$. Postselecting all ancillas onto $\lvert 0\cdots 0\rangle$ implements $I - f(H_{22}-\lambda I)H_{21}$ up to a known scalar.

Theorems & Definitions (20)

• Theorem 1: Informal -- Eigenvalue estimation
• Theorem 2: Informal -- Eigenstate and subspace preparation
• Theorem 3: Spectral Schur complement
• Remark 4
• Definition 5
• Lemma 6: Polynomial approximation of $1/x$
• Lemma 7: Block-encoding of the approximated dressed Hamiltonian $\widetilde{H}_D(\lambda)$
• Corollary 8: $\mathcal{O}(\widetilde{\alpha}^2\varepsilon_{\mathrm{qsvt}})$-approximate block-encoding of $H_{D}(\lambda)$
• Lemma 9: Block-encoding of the approximate wave operator
• Lemma 10: Resource summary for matrix-element estimation