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arXiv:2606.17095

Variational Quantum Eigensolver-Based Quantum Bootstrap Embedding for Molecules

Derek Peng

physics.chem-phquant-phSubmitted Jun 13, 2026• Announced Jun 16, 2026• 0 citations• Revised Aug 15, 2026 (v2)
v1 · Jun 13, 2026v2 · Aug 15, 2026
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Abstract

Simulating strongly correlated molecular systems on near-term quantum hardware remains challenging because current hardware offers limited quantum volume and moderate-fidelity qubits. One way to address this challenge is through bootstrap embedding (BE). Bootstrap embedding partitions molecules into smaller fragments that are then embedded in the ``bath'' of other fragments iteratively. Bootstrap embedding is appealing for quantum simulation because fragmentation reduces the qubit requirements of each fragment problem. In this work, we develop a quantum bootstrap embedding (QBE) workflow that uses variational quantum eigensolver (VQE) fragment solvers and examines the algorithmic choices that determine the success of the overall VQE-QBE procedure. To improve efficiency, we introduce FastAdaptVQE, a sparse-matrix-accelerated form of the adaptive variational quantum eigensolver (ADAPT-VQE) that replaces symbolic commutator evaluation with direct statevector calculations, and MatrixFreeAdaptVQE, a matrix-free extension that removes the sparse-matrix memory bottleneck that arises for larger fragments. We also explore modifying the ADAPT-VQE operator-selection step by replacing the purely greedy choice with a lookahead strategy. Benchmarks on H$_4$, F$_2$, and selected geometries of H$_6$ reach chemical accuracy, within 1 kcal/mol of BE results obtained with a full configuration interaction (FCI) solver. For $n$-butane, density-matching residuals converged, but energy errors remained outside chemical accuracy. These results show that combining QBE with VQE can accurately compute molecular energies, while density convergence alone does not ensure energy accuracy. This work lays the foundation for extending these energy calculations to larger molecular systems and quantum materials on near-term quantum hardware.

Comments
24 pages, 4 figures, 4 tables. Code available at: https://github.com/littlebullGit/QBEMol

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Simulating strongly correlated molecular systems on near-term quantum hardware remains challenging because current hardware offers limited quantum volume and moderate-fidelity qubits. One way to address this challenge is through bootstrap embedding (BE). Bootstrap embedding partitions molecules into smaller fragments that are then embedded in the ``bath'' of other fragments iteratively. Bootstrap embedding is appealing for quantum simulation because fragmentation reduces the qubit requirements of each fragment problem. In this work, we develop a quantum bootstrap embedding (QBE) workflow that uses variational quantum eigensolver (VQE) fragment solvers and examines the algorithmic choices that determine the success of the overall VQE-QBE procedure. To improve efficiency, we introduce FastAdaptVQE, a sparse-matrix-accelerated form of the adaptive variational quantum eigensolver (ADAPT-VQE) that replaces symbolic commutator evaluation with direct statevector calculations, and MatrixFreeAdaptVQE, a matrix-free extension that removes the sparse-matrix memory bottleneck that arises for larger fragments. We also explore modifying the ADAPT-VQE operator-selection step by replacing the purely greedy choice with a lookahead strategy. Benchmarks on H4_44​, F2_22​, and selected geometries of H6_66​ reach chemical accuracy, within 1 kcal/mol of BE results obtained with a full configuration interaction (FCI) solver. For nnn-butane, density-matching residuals converged, but energy errors remained outside chemical accuracy. These results show that combining QBE with VQE can accurately compute molecular energies, while density convergence alone does not ensure energy accuracy. This work lays the foundation for extending these energy calculations to larger molecular systems and quantum materials on near-term quantum hardware.
Abstract Finding the ground-state energy of molecules is an important and challenging computational problem for which quantum computing can potentially find efficient solutions. The variational quantum eigensolver (VQE) is a quantum algorithm that tackles the molecular groundstate problem and is regarded as one of the flagships of quantum computing. Yet, to date, only very small molecules were computed via VQE, due to high noise levels in current quantum devices. Here we present an alternative variational quantum scheme that requires significantly less qubits than VQE. The reduction in the qubit number allows for shallower circuits to be sufficient, rendering the method more resistant to noise. The proposed algorithm, termed variational quantum selected-configuration-interaction (VQ-SCI), is based on: (a) representing the target groundstate as a superposition of Slater determinant configurations, encoded directly upon the quantum computational basis states; and (b) selecting a-priory only the most dominant configurations. This is demonstrated through a set of groundstate calculations of the H2, LiH, BeH2, H2O, NH3 and C2H4 molecules in the sto-3g basis set, performed on IBM quantum devices. We show that the VQ-SCI reaches the full configuration interaction energy within chemical accuracy using the lowest number of qubits reported to date. Moreover, when the SCI matrix is generated ‘on the fly’, the VQ-SCI requires exponentially less memory than classical SCI methods. This offers a potential remedy to a severe memory bottleneck problem in classical SCI calculations. Finally, the proposed scheme is general and can be straightforwardly applied for finding the groundstate of any Hermitian matrix, outside the chemical context.
Fragment embedding is one way to circumvent the high computational scaling of accurate electron correlation methods. The challenge of applying fragment embedding to molecular systems primarily lies in the strong entanglement and correlation that prevent accurate fragmentation across chemical bonds. Recently, Schmidt decomposition has been shown effective for embedding fragments that are strongly coupled to a bath in several model systems. In this work, we extend a recently developed quantum embedding scheme, bootstrap embedding (BE), to molecular systems. The resulting method utilizes the matching conditions naturally arising from using overlapping fragments to optimize the embedding. Numerical simulation suggests that the accuracy of the embedding improves rapidly with fragment size for small molecules, whereas larger fragments that include orbitals from different atoms may be needed for larger molecules. BE scales linearly with system size (apart from an integral transform) and hence can potentially be useful for large-scale calculations.
Predicting molecular electronic structure using Quantum Computing approaches is a challenge in Computational Chemistry. Traditional Variational Quantum Eigensolvers (VQEs) are typically hindered by optimization challenges. The alternative Generative Quantum Eigensolver (GQE) mitigates this by reformulating ground-state search as discrete sequence generation, but its Transformer backbone suffers from quadratic computational complexity, bottlenecking scalability for larger molecules. To address this, we introduce the Mamba Quantum Eigensolver (Mamba-QE), which replaces the Transformer with a Selective State Space Model (SSM) to process operator sequences with linear O(L)\mathcal{O}(L)O(L) complexity. Additionally, we integrate a distance-conditioned mechanism enabling the direct prediction of complete potential energy surfaces (PES). Evaluated on the H2H_2H2​ molecule, Mamba-QE reproduces Full Configuration Interaction (FCI) reference energies well within chemical accuracy, achieving an energy error below 3×10−63\times10^{-6}3×10−6 Ha across 16 geometries with zero variational violations. Mamba-QE maintains high expressive power while establishing a robust, scalable foundation for quantum simulations of complex molecular systems.
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for noisy intermediate-scale quantum (NISQ) hardware to estimate eigenvalues of many-body Hamiltonians. Unlike fully quantum approaches such as quantum phase estimation (QPE), VQE trades deep coherent circuits for repeated state preparation, measurement, and classical optimization, making it more compatible with limited qubit counts and finite coherence times. Recent developments have focused on reducing quantum resource requirements while retaining chemically meaningful wavefunction structure. In this paper, we examine recent progress in VQE methods for molecular electronic structure with an emphasis on three themes: (i) strategies for circuit and ansatz complexity reduction, including adaptive and selectively screened approaches, (ii) chemically motivated workflows that combine VQE with orbital optimization, fragmentation, and localized active-space ideas to better address strong correlation, and (iii) extensions of VQE to excited-state calculations. Throughout, we emphasize the tradeoffs among parameter count, gate depth, symmetry preservation, measurement overhead, and classical preprocessing, and discuss where these approaches may become most useful for chemically challenging active spaces. We also highlight benchmarking considerations for assessing both accuracy and resource requirements, and conclude with a perspective on regimes in which VQE may offer the greatest long-term value, particularly multireference active spaces and low-lying excited-state manifolds.
Variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for noisy intermediate-scale quantum (NISQ) computers. It is promising for quantum chemical calculations (QCC) because it can calculate the ground-state energy of molecules. Although VQE has a theoretical potential to achieve a higher accuracy than classical approximation methods, it is challenging to achieve it on current NISQ computers due to the significant impact of noise. Density matrix embedding theory (DMET) is a well-known technique that can be used to decompose a molecule into multiple fragments, which is available to mitigate the noise impact on VQE. However, our preliminary evaluation shows that the naive combination of DMET and VQE hardly mitigates it. In this work, we present two proposals to mitigate the noise impact on the DMET+VQE combination. (1) The size of quantum circuits used by VQE is decreased by reducing the number of bath orbitals which represent interactions between multiple fragments in DMET. (2) While VQE is executed on a NISQ computer, reduced density matrices (RDMs), which are used to calculate the energy of fragments in DMET, are calculated accurately by executing quantum circuits using a noise-less quantum computer simulator. The evaluation using a noisy quantum computer simulator shows that our proposals improve the accuracy of the DMET+VQE combination by an order of magnitude.
Variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for noisy intermediate-scale quantum (NISQ) computers. It is promising for quantum chemical calculations (QCC) because it can calculate the ground-state energy of a target molecule. Although VQE has a potential to achieve a higher accuracy than classical approximation methods in QCC, it is challenging to achieve it on current NISQ computers due to the significant impact of noises. Density matrix embedding theory (DMET) is a well-known technique to divide a molecule into multiple fragments, which is available to mitigate the noise impact on VQE. However, our preliminary evaluation shows that the naive combination of DMET and VQE does not outperform a gold standard classical method. In this work, we present three approaches to mitigate the noise impact for the DMET+VQE combination. (1) The size of quantum circuits used by VQE is decreased by reducing the number of bath orbitals which represent interactions between multiple fragments in DMET. (2) Reduced density matrices (RDMs), which are used to calculate a molecular energy in DMET, are calculated accurately based on expectation values obtained by executing quantum circuits using a noise-less quantum computer simulator. (3) The parameters of a quantum circuit optimized by VQE are refined with mathematical post-processing. The evaluation using a noisy quantum computer simulator shows that our approaches significantly improve the accuracy of the DMET+VQE combination. Moreover, we demonstrate that on a real NISQ device, the DMET+VQE combination applying our three approaches achieves a higher accuracy than the gold standard classical method.