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arXiv:2609.02394

A Recursive Module-Coupling Algorithm for Computing Low-Energy Eigenstates

Dihang Sun, Nannan Ma, Ching Hua Lee, Tianqi Chen, Jiangbin Gong

quant-phSubmitted Sep 2, 2026• Announced Sep 3, 2026• 0 citations• Revised Sep 3, 2026 (v2)
v1 · Sep 2, 2026v2 · Sep 3, 2026
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Abstract

Finding the eigenstates of a many-body Hamiltonian is a fundamental challenge in physics and computational science. Since the search space grows exponentially with system size, numerous classical and quantum algorithms have been developed to address this problem. A practical strategy is to identify a physics-informed low-dimensional subspace that effectively accommodates the low-lying eigenstates, thereby reducing the computational complexity. In this paper, we propose a recursive module-coupling algorithm, which iteratively treats a system as a composition of locally-coupled smaller modules, with low-energy subspace estimated successively according to the same recursive structure. Unlike the density matrix renormalization group (DMRG) approach that optimizes a global matrix product state through repeated local sweeps and obtains excited states sequentially, our algorithm constructs a physically tailored variational basis from module eigenstates and obtains several low-energy states on an equal footing, leading to substantial speedups if targeting moderate accuracy. Our proposed method further leads naturally to a recursive quantum variational algorithm, providing a systematic and modular circuit-construction framework compatible with contemporary gate-based quantum architectures. At each recursive level, block encoders are trained to map logical basis states onto the retained physical subspace, within which a variational circuit is subsequently optimized. Such a quantum-circuit implementation provides not only a quantum multistate eigensolver, but also a systematic prescription for hierarchically constructing quantum state-preparation circuits. Classical simulations demonstrate the accuracy and efficiency of the proposed method, whereas experiments on IBM quantum processors show that eigenstate preparation with reasonable fidelities is achievable even in the current NISQ era.

Comments
23 pages,18 figures

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Finding the eigenstates of a many-body Hamiltonian is a fundamental challenge in physics and computational science. Since the search space grows exponentially with system size, numerous classical and quantum algorithms have been developed to address this problem. A practical strategy is to identify a physics-informed low-dimensional subspace that effectively accommodates the low-lying eigenstates, thereby reducing the computational complexity. In this paper, we propose a recursive module-coupling algorithm, which iteratively treats a system as a composition of locally-coupled smaller modules, with low-energy subspace estimated successively according to the same recursive structure. Unlike the density matrix renormalization group (DMRG) approach that optimizes a global matrix product state through repeated local sweeps and obtains excited states sequentially, our algorithm constructs a physically tailored variational basis from module eigenstates and obtains several low-energy states on an equal footing, leading to substantial speedups if targeting moderate accuracy. Our proposed method further leads naturally to a recursive quantum variational algorithm, providing a systematic and modular circuit-construction framework compatible with contemporary gate-based quantum architectures. At each recursive level, block encoders are trained to map logical basis states onto the retained physical subspace, within which a variational circuit is subsequently optimized. Such a quantum-circuit implementation provides not only a quantum multistate eigensolver, but also a systematic prescription for hierarchically constructing quantum state-preparation circuits. Classical simulations demonstrate the accuracy and efficiency of the proposed method, whereas experiments on IBM quantum processors show that eigenstate preparation with reasonable fidelities is achievable even in the current NISQ era.
Determining low-energy eigenstates in electronic many-body quantum systems is a key challenge in computational chemistry and condensed-matter physics. Hybrid quantum-classical approaches, such as the Variational Quantum Eigensolver and Quantum Subspace Methods, offer practical solutions but face limitations in circuit depth and measurement overhead. In this article, we propose a variational strategy based on symmetry-preserving cost functions to iteratively construct a reduced subspace for the extraction of low-lying energy states. We show that, under certain conditions, our approach leads to a tridiagonal representation similar to that obtained with the Lanczos algorithm. The iterative process allows control over the trade-off between circuit depth, the number of variational parameters, and the number of measurements required to achieve the desired accuracy, making it suitable for current quantum hardware. 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.
As demonstrated in the density matrix renormalization group (DMRG) method, approximating many-body wave function of electrons using a matrix product state (MPS) is a promising way to solve electronic structure problems. The expressibility of an MPS is determined by the size of the matrices or, in other words, the bond dimension, which unfortunately may be required to be very large in quantum chemistry simulations. In this study, we propose to calculate the ground state energies of molecular systems by variationally optimizing quantum circuit MPS (QCMPS) with a relatively small number of qubits. It is demonstrated that with carefully chosen circuit structure and orbital localization scheme, QCMPS can reach a similar accuracy as that achieved in DMRG with an exponentially large bond dimension. QCMPS simulation of a linear hydrogen molecular chain with 50 orbitals can reach the chemical accuracy using only 6 qubits at a moderate circuit depth. These results suggest that QCMPS is a promising wave function ansatz in the variational quantum eigensolver algorithm for molecular systems.
The state-of-the-art quantum computing hardware has entered the noisy intermediate-scale quantum (NISQ) era. Having been constrained by the limited number of qubits and shallow circuit depth, NISQ devices have nevertheless demonstrated the potential of applications on various subjects. One example is the variational quantum eigensolver (VQE) that was first introduced for computing ground states. Although VQE has now been extended to the study of excited states, the algorithms previously proposed involve a recursive optimization scheme which requires many extra operations with significantly deeper quantum circuits to ensure the orthogonality of different trial states. Here we propose a new algorithm to determine many low energy eigenstates simultaneously. By introducing ancillary qubits to purify the trial states so that they keep orthogonal to each other throughout the whole optimization process, our algorithm allows these states to be efficiently computed in one quantum circuit. Our algorithm reduces significantly the complexity of circuits and the readout errors, and enables flexible post-processing on the eigen-subspace from which the eigenpairs can be accurately determined. We demonstrate this algorithm by applying it to the transverse Ising model. By comparing the results obtained using this variational algorithm with the exact ones, we find that the eigenvalues of the Hamiltonian converge quickly with the increase of the circuit depth. The accuracies of the converged eigenvalues are of the same order, which implies that the difference between any two eigenvalues can be more accurately determined than the eigenvalues themselves.
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.
Simulation of quantum many-body systems is one of the most promising applications of quantum computers. It is crucial to efficiently implement the time-evolution operator as a quantum circuit to execute such simulations on near-term quantum computing devices with limited computational resources. However, standard approaches such as Trotterization sometimes require a deep quantum circuit, which is hard to implement on near-term quantum computers. Here, we propose a hybrid quantum-classical algorithm, called local subspace variational quantum compilation (LSVQC), for compiling the time-evolution operator of quantum many-body systems. The LSVQC performs a variational optimization to reproduce the action of the target time-evolution operator within a physically reasonable subspace. The optimization is performed for small local subsystems based on the Lieb-Robinson bound, which allows us to execute the cost function evaluation using small-scale quantum devices and/or classical computers. We demonstrate the validity of the LSVQC algorithm through numerical simulations of a simple spin-lattice model and an effective model of a parent compound of cuprate superconductors, Sr 2 CuO 3 , constructed by the downfolding method. It is shown that the LSVQC achieves a 95% reduction of the circuit depth for simulating quantum many-body dynamics compared to the Trotterization at best while maintaining the same computational accuracy. We also demonstrate that the restriction to a subspace leads to a substantial reduction of required resources and improved accuracy compared to the case of considering the entire Hilbert space. Furthermore, we estimate the gate count needed to execute the quantum simulations using the LSVQC on near-term quantum computing architectures in the noisy intermediate-scale or early fault-tolerant quantum computing era. Our estimation suggests that the acceptable physical gate error rate for the LSVQC can be about one order of magnitude larger than that for the Trotterization.
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