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arXiv:2510.08088

Quantum Algorithm for Low-Energy Effective Hamiltonians and Subspace Eigenvalue Problem

Chun-Tse Li, Tze Tzen Ong, Chih‐Yun Lin, Yu‐Cheng Chen, Hsin Lin, Min-Hsiu Hsieh

Published Oct 9, 2025• 0 citations
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Abstract

Subspace eigenvalue problems arise ubiquitously in quantum chemistry and condensed-matter physics, where the relevant object is often a low-energy manifold rather than a single ground-state wavefunction. In this work, we propose a fault-tolerant quantum algorithm for this subspace-level task based on the Feshbach effective-Hamiltonian formalism. Given block-encoding access to the full Hamiltonian and a chosen $d$-dimensional reference subspace, the algorithm estimates eigenvalues of states with nonzero overlap with the reference subspace through a local secant fixed-point search. It then implements the associated wave operator and prepares an orthonormal basis whose span approximates the target invariant subspace. The construction combines projected block encodings with quantum singular value transformation (QSVT), which approximates the complementary-space resolvent and thereby provides both the self-energy used for eigenvalue estimation and the wave operator used for eigenstate reconstruction. For target accuracy $\varepsilon$, a single evaluation of the effective Hamiltonian has query complexity $\widetilde{O}(d^3/(g^2\varepsilon))$, up to block-encoding normalization factors, where $g$ is the distance between the target eigenvalue and the nearest pole of the effective Hamiltonian. Under the stated local regularity conditions, the secant search requires only $O(\log\log(1/\varepsilon))$ effective-Hamiltonian evaluations to reach the working precision. Classical numerical emulations for an open $4\times2$ Fermi--Hubbard cluster, all-electron LiH bond stretching, and $[\mathrm{Ru(bpy)}_{3}]^{2+}$ demonstrate the resolution and reconstruction of low-energy states and manifolds across spin-sector crossings, near-degeneracies, and dense excited-state spectra.

DOI
10.48550/arxiv.2510.08088

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Subspace eigenvalue problems arise ubiquitously in quantum chemistry and condensed-matter physics, where the relevant object is often a low-energy manifold rather than a single ground-state wavefunction. In this work, we propose a fault-tolerant quantum algorithm for this subspace-level task based on the Feshbach effective-Hamiltonian formalism. Given block-encoding access to the full Hamiltonian and a chosen ddd-dimensional reference subspace, the algorithm estimates eigenvalues of states with nonzero overlap with the reference subspace through a local secant fixed-point search. It then implements the associated wave operator and prepares an orthonormal basis whose span approximates the target invariant subspace. The construction combines projected block encodings with quantum singular value transformation (QSVT), which approximates the complementary-space resolvent and thereby provides both the self-energy used for eigenvalue estimation and the wave operator used for eigenstate reconstruction. For target accuracy ε\varepsilonε, a single evaluation of the effective Hamiltonian has query complexity O~(d3/(g2ε))\widetilde{O}(d^3/(g^2\varepsilon))O(d3/(g2ε)), up to block-encoding normalization factors, where ggg is the distance between the target eigenvalue and the nearest pole of the effective Hamiltonian. Under the stated local regularity conditions, the secant search requires only O(log⁡log⁡(1/ε))O(\log\log(1/\varepsilon))O(loglog(1/ε)) effective-Hamiltonian evaluations to reach the working precision. Classical numerical emulations for an open 4×24\times24×2 Fermi--Hubbard cluster, all-electron LiH bond stretching, and [Ru(bpy)3]2+[\mathrm{Ru(bpy)}_{3}]^{2+}[Ru(bpy)3​]2+ demonstrate the resolution and reconstruction of low-energy states and manifolds across spin-sector crossings, near-degeneracies, and dense excited-state spectra.
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computations of the ground and excited electronic states. While substantial progress has been made in designing physically informed quantum algorithms to address the ground state under the constraints of near-term quantum hardware, relatively fewer developments exist for reliably computing excited state energetics, presumably due to the growing complexity of the quantum circuits for excited eigenroots. In this work, we exploit the inherent point-group and spin symmetries of molecular systems to construct symmetry-adapted sets of orthogonal reference states which are evolved with a common symmetry-scalar unitary, enabling the simultaneous determination of ground and excited eigenroots. By minimizing the energy functional over a restricted symmetry-adapted correlated manifold, the weighted subspace-search variational quantum eigensolver enables us to directly target multiple eigenroots within each spatial and spin symmetry sector. This physically motivated selection of references and the unitary confer substantial flexibility to the weighted energy landscape, largely allowing the optimization to avoid numerical traps and converge directly to the targeted eigenstates. More crucially, symmetry resolution also enables access to a larger number of eigenstates without substantially increasing the complexity of the cost function. Through benchmarking on moderate to strongly correlated molecular systems, we demonstrate the accuracy and advantages of our approach relative to several established excited state algorithms on quantum computers.
We consider the task of simulating time evolution under a Hamiltonian H within its low-energy subspace. Assuming access to a block-encoding of H ′ := ( H − E ) / λ , for some λ > 0 and E ∈ R , the goal is to implement an ϵ -approximation to the evolution operator e − i t H when the initial state is confined to the subspace corresponding to eigenvalues [ − 1 , − 1 + Δ / λ ] of H ′ , for Δ ≤ λ . We present a quantum algorithm that requires O ( t λ Γ + λ / Γ log ⁡ ( 1 / ϵ ) ) queries to the block-encoding for any choice of Γ such that Δ ≤ Γ ≤ λ . When the parameters satisfy log ⁡ ( 1 / ϵ ) = o ( t λ ) and Δ / λ = o ( 1 ) , this result improves over generic methods with query complexity Ω ( t λ ) . Our quantum algorithm leverages spectral gap amplification and the quantum singular value transform.For a given H , the block-encoding of its H ′ must be prepared efficiently to achieve an asymptotic speedup in simulating the low-energy subspace; we refer to these Hamiltonians as g a p − a m p l i f i a b l e . We show necessary and sufficient conditions for gap amplifiability in terms of an operationally useful decomposition of H
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