Abstract Variational quantum eigensolver (VQE) has emerged as a promising method for investigating ground state properties in quantum chemistry, materials science, and condensed matter physics. However, the conventional VQE method generally lacks systematic improvement and convergence guarantees, particularly when dealing with strongly correlated systems. In light of these challenges, we present a novel framework called orbital expansion VQE (OE-VQE) to address these limitations. The key idea is to devise an efficient convergence path by utilizing shallower quantum circuits, starting from a highly compact active space and gradually expanding it until convergence to the ground state is achieved. To validate the effectiveness of the OE-VQE framework, we conducted benchmark simulations on several small yet representative molecules, including the H 6 chain, H 10 ring and N 2 . The simulation results demonstrate that our proposed convergence paths significantly enhance the performance of conventional VQE. Overall, our work sheds valuable insight into the simulation of molecules based on shallow quantum circuits, offering a promising avenue for advancing the efficiency and accuracy of VQE approaches in tackling complex molecular systems.
Abstract Variational quantum eigensolver (VQE) has emerged as a promising method for investigating ground state properties in quantum chemistry, materials science, and condensed matter physics. However, the conventional VQE method generally lacks systematic improvement and convergence guarantees, particularly when dealing with strongly correlated systems. In light of these challenges, we present a novel framework called orbital expansion VQE (OE-VQE) to address these limitations. The key idea is to devise an efficient convergence path by utilizing shallower quantum circuits, starting from a highly compact active space and gradually expanding it until convergence to the ground state is achieved. To validate the effectiveness of the OE-VQE framework, we conducted benchmark simulations on several small yet representative molecules, including the H 6 chain, H 10 ring and N 2 . The simulation results demonstrate that our proposed convergence paths significantly enhance the performance of conventional VQE. Overall, our work sheds valuable insight into the simulation of molecules based on shallow quantum circuits, offering a promising avenue for advancing the efficiency and accuracy of VQE approaches in tackling complex molecular 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.
The variational quantum eigensolver (VQE) has emerged as one of the leading quantum algorithms for solving electronic structure problems on near-term noisy intermediate-scale quantum devices. However, its practical application to quantum chemistry remains challenging due to the limited coherence time, imperfect quantum gate fidelity, and the large number of measurements required, which together confine current electronic structure simulations to relatively small active spaces. In this work, we present an orbital-optimized VQE framework based on orbital compression, designed to improve the accuracy of electronic structure calculations while maintaining relatively small active spaces. Frozen natural orbitals (FNO) and split virtual orbitals (SVO) are first employed to construct compact active spaces for VQE simulations, leading to the FNO/SVO-VQE approach. Orbital optimization is then incorporated to further recover electron correlation effects, resulting in the FNO/SVO-OO-VQE methods. We apply the proposed method to simulate potential energy surfaces for molecular dissociation and the activation energy of formaldehyde decomposition. Numerical results demonstrate that both FNO-OO-VQE and SVO-OO-VQE improve the variational accuracy while substantially reducing measurement cost.
The use of the variational quantum eigensolver (VQE) for quantum chemistry is one of the most promising applications for noisy intermediate-scale quantum (NISQ) devices. A major limitation is represented by the need to build compact and shallow circuit ansatzes having the variational flexibility to catch the complexity of the electronic structure problem. To alleviate this drawback, we introduce a modified VQE scheme in which the form of the molecular Hamiltonian is adapted to the circuit ansatz through an optimization procedure. Exploiting the invariance of the Hamiltonian by molecular orbital rotations, we can optimize it using gradients that can be calculated without significant computational overload. The proposed method, named Wavefunction Adapted Hamiltonian Through Orbital Rotation (WAHTOR), has been applied to small molecules in numerical state vector simulations. The results demonstrate that, at variance with standard VQE, the method is less dependent on circuit topology and less prone to be trapped into high-energy local minima. It is able to recover a significant amount of electron correlation even with only empirical ansatzes with shallow circuit depth. Noisy calculations demonstrate the robustness and feasibility of the proposed methodology and indicate the hardware requirements to effectively apply the procedure using forthcoming NISQ devices.
Abstract Near-term quantum devices offer promising avenues for addressing the electronic structure problem in quantum chemistry, yet their limited qubits and susceptibility to noise constrain algorithmic scalability. Although the variational quantum eigensolver (VQE) has shown potential for small-scale systems, further improvements are necessary to handle large basis sets and large many-electron molecules efficiently. In this work, we introduce RO-VQE, an improved approach derived from the earlier optimized orbital algorithm that employs a randomized procedure for selecting and optimizing orbitals. We evaluate RO-VQE on hydrogen chains systems (H2 and H4)—a well-established benchmark for quantum chemistry methods—using minimal, split-valence, and correlation-consistent basis sets. For these systems, RO-VQE improves ground-state energy estimations compared to conventional VQE methods, matching the accuracy of systematic strategies in these test cases. These proof-of-concept results suggest that randomized active-space selection may offer a practical compromise in NISQ-era quantum computations, offering a flexible alternative to more deterministic methods, particularly for systems constrained by limited quantum resources.
We present a self-consistent field (SCF) approach within the adaptive derivative-assembled problem-tailored ansatz variational quantum eigensolver (ADAPT-VQE) framework for efficient quantum simulations of chemical systems on near-term quantum computers. To this end, our ADAPT-VQE-SCF approach combines the idea of generating an ansatz with a small number of parameters, resulting in shallow-depth quantum circuits with a direct minimization of an energy expression that is correct to second order with respect to changes in the molecular orbital basis. Our numerical analysis, including calculations for the transition-metal complex ferrocene [Fe (C 5 H 5 ) 2 ], indicates that convergence in the self-consistent orbital optimization loop can be reached without a considerable increase in the number of two-qubit gates in the quantum circuit by comparison to a VQE optimization in the initial molecular orbital basis. Moreover, the orbital optimization can be carried out simultaneously within each iteration of the ADAPT-VQE cycle. ADAPT-VQE-SCF thus allows us to implement a routine analogous to the complete active space SCF, a cornerstone of state-of-the-art computational chemistry, in a hardware-efficient manner on near-term quantum computers. Hence, ADAPT-VQE-SCF paves the way toward a paradigm shift for quantitative quantum-chemistry simulations on quantum computers by requiring fewer qubits and opening up for the use of large and flexible atomic orbital basis sets in contrast to earlier methods that are predominantly based on the idea of full active spaces with minimal basis sets.
Abstract Molecular simulations with the variational quantum eigensolver (VQE) are a promising application for emerging noisy intermediate-scale quantum computers. Constructing accurate molecular ansatze that are easy to optimize and implemented by shallow quantum circuits is crucial for the successful implementation of such simulations. Ansatze are, generally, constructed as series of fermionic-excitation evolutions. Instead, we demonstrate the usefulness of constructing ansatze with ``qubit-excitation evolutions', which, contrary to fermionic excitation evolutions, obey ``qubit commutation relations'. We show that qubit excitation evolutions, despite the lack of some of the physical features of fermionic excitation evolutions, accurately construct ansatze, while requiring asymptotically fewer gates. Utilizing qubit excitation evolutions, we introduce the iterative qubit excitation based VQE (IQEB-VQE) algorithm. The IQEB-VQE performs molecular simulations using a problem-tailored ansatz, grown iteratively by appending evolutions of single and double qubit excitation operators. By performing numerical simulations for small molecules, we benchmark the IQEB-VQE, and compare it against other competitive VQE algorithms. In terms of circuit efficiency and time complexity, we find that the IQEB-VQE systematically outperforms the previously most circuit-efficient, practically scalable VQE algorithms.