We present the quantum effective fragment molecular orbital (Q-EFMO) method, a hybrid framework that evaluates selected molecular-fragment correlation energies with the variational quantum eigensolver and assembles them through the size-consistent EFMO energy expression. A Hartree-Fock EFMO calculation supplies monomer references, many-body polarization, and distant-pair EFP interactions; independent VQE/UCCSD calculations provide correlation increments for monomers and selected near-field dimers. Consequently, the maximum quantum register is determined by the largest active fragment or fragment pair rather than by total system size. For a three-layer LiH benchmark in STO-3G, a full 2.5 to 3.5 Angstrom separation scan shows that two orbital-reduction schemes approach the full CCSD reference monotonically. The more aggressive scheme remains below 1 kcal/mol for separations of 2.6 Angstrom and greater and reaches 0.10 kcal/mol at 3.5 Angstrom, using 6 qubits for monomers and 12 for the largest dimers, compared with 14 qubits for the frozen-core dimer baseline. An implementation-specific cost proxy decreases from 1000 to 40, a factor of 25. These noise-free results establish the equations, software path, and resource scaling; broader chemical validation, larger bases, and hardware tests remain necessary.
We present the quantum effective fragment molecular orbital (Q-EFMO) method, a hybrid framework that evaluates selected molecular-fragment correlation energies with the variational quantum eigensolver and assembles them through the size-consistent EFMO energy expression. A Hartree-Fock EFMO calculation supplies monomer references, many-body polarization, and distant-pair EFP interactions; independent VQE/UCCSD calculations provide correlation increments for monomers and selected near-field dimers. Consequently, the maximum quantum register is determined by the largest active fragment or fragment pair rather than by total system size. For a three-layer LiH benchmark in STO-3G, a full 2.5 to 3.5 Angstrom separation scan shows that two orbital-reduction schemes approach the full CCSD reference monotonically. The more aggressive scheme remains below 1 kcal/mol for separations of 2.6 Angstrom and greater and reaches 0.10 kcal/mol at 3.5 Angstrom, using 6 qubits for monomers and 12 for the largest dimers, compared with 14 qubits for the frozen-core dimer baseline. An implementation-specific cost proxy decreases from 1000 to 40, a factor of 25. These noise-free results establish the equations, software path, and resource scaling; broader chemical validation, larger bases, and hardware tests remain necessary.
Quantum simulation of chemistry in realistic environments is constrained by the orbital cost of explicit solvent, ions, and other surroundings. We present the quantum-classical effective fragment potential (Q-EFP) method, in which a chemically active region is treated with a quantum algorithm and the environment is represented by the effective fragment potential (EFP) in GAMESS. Coulomb and polarization potentials enter the active-region one-electron Hamiltonian, while EFP-EFP and short-range environment contributions are assembled classically. Statevector UCCSD/STO-3G benchmarks for LiH in a mixed water/methanol environment, H2O in a five-water environment, and BeH2 in an ammonium/nitrate environment differ from matched classical CCSD/EFP calculations by 0.03, 0.38, and 0.01 kcal/mol, respectively. The active calculations require 4 to 8 qubits, compared with estimated full-system counts of approximately 84 to 246 qubits, corresponding to register reductions of about 10-fold to 35-fold. These proof-of-concept results validate the embedded Hamiltonian and Q-GAMESS workflow while separating environmental size from quantum-register size. Q-EFP is complementary to real-space Q-EFMO fragmentation and virtual-orbital Q-FVO reduction, enabling a layered route to larger solvated systems.
Quantum computers offer significant potential for complex system analysis, yet their application in large systems is hindered by limitations such as qubit availability and quantum hardware noise. While the variational quantum eigensolver (VQE) was proposed to address these issues, its scalability remains limited. Many efforts, including new ansätze and Hamiltonian modifications, have been made to overcome these challenges. In this work, we introduced the novel Fragment Molecular Orbital/Variational Quantum Eigensolver (FMO/VQE) algorithm. This method combines the fragment molecular orbital (FMO) approach with VQE and efficiently utilizes qubits for quantum chemistry simulations. Employing the UCCSD ansatz, the FMO/VQE achieved an absolute error of just 0.053 mHa with 8 qubits in a [Formula: see text] system using the STO-3G basis set, and an error of 1.376 mHa with 16 qubits in a [Formula: see text] system with the 6-31G basis set. These results indicated a significant advancement in scalability over conventional VQE, maintaining accuracy with fewer qubits. Therefore, our FMO/VQE method exemplifies how integrating fragment-based quantum chemistry with quantum algorithms can enhance scalability, facilitating more complex molecular simulations and aligning with quantum computing advancements.
We introduce a radical-fragment many-body expansion at the two-body level (MBE2) for quantum chemistry of linear alkanes. Instead of heterolytic bond cleavage with hydrogen capping atoms and electrostatic embedding like in Fragment Molecular Orbital (FMO), we perform homolytic C-C bond cleavage to produce open-shell radical fragments (CH3, CH2) treated with restricted open-shell Hartree-Fock (ROHF) in isolation. The two-body MBE2 assembly formula reconstructs total alkane energies from only four unique fragment calculations regardless of chain length, reducing the maximum qubit requirement. We benchmark this framework against five energy solvers (RHF, CCSD, VQE, ADAPT-VQE, and SQD) across 11 linear alkanes from butane (C4H10) to hexacosane (C26H54). The MBE2 decomposition achieves a 12.3x qubit reduction for C26H54 (from 368 to 30 qubits) and a 12.8x reduction in unique calculations via symmetry exploitation. MBE2-VQE and MBE2-SQD (executed on IBM quantum hardware) closely track their respective classical MBE2 references, demonstrating that fragmentation-based quantum chemistry is viable for scaling quantum solvers to large molecular systems.
The Variational Quantum Eigensolver (VQE) is a quantum algorithm for estimating ground-state energies, with promising applications in material science, drug discovery, and battery research. A key challenge is the limited number of qubits available on current quantum devices, which restricts the size of molecular systems that can be studied. To address this limitation, we apply the Fragment Molecular Orbital (FMO) method in combination with VQE, referred to as FMO-VQE. This approach divides a system into smaller fragments, making the quantum calculations more tractable. While earlier studies demonstrated this method only for hydrogen clusters, we extend the application to lithium cobalt oxide, a widely used cathode material in lithium-ion batteries. Using FMO-VQE, we estimate the ground-state energy of this complex system while reducing the number of required qubits from 24 to 14, without significant loss of accuracy compared to classical methods. This reduction highlights the potential of FMO-VQE to overcome hardware limitations and make quantum simulations of larger molecules feasible. The results suggest a practical path for applying near-term quantum computers to real-world challenges, opening opportunities for advancements in the battery industry and drug design.
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, F2, and selected geometries of H6 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.
In this work, we utilize the framework of many-body expansion (MBE) to decompose electronic structures into fragments by incrementing virtual orbitals, aiming to accurately solve the ground and excited state energies of each fragment using the variational quantum eigensolver and deflation algorithms. While our approach is primarily based on unitary coupled cluster singles and doubles (UCCSD) and its generalization, we also introduce modifications and approximations to conserve quantum resources in MBE by partially generalizing the UCCSD operator and neglecting the relaxation of the reference states. As a proof of concept, we investigate the potential energy surfaces for the bond-breaking processes of the ground state of two molecules (H 2 O and N 2 ) and calculate the ground and excited state energies of three molecules (LiH, CH +, and H 2 O). The results demonstrate that our approach can, in principle, provide reliable descriptions in all the tests including strongly correlated systems when appropriate approximations are chosen. Additionally, we perform model simulations to investigate the impact of shot noise on the total MBE energy and show that precise energy estimation is crucial for lower-order MBE fragments.