Abstract The use of the variational quantum eigensolver (VQE) on near-term quantum computers for solving Kohn–Sham Hamiltonian is still underexplored. In this study, we apply the weighted subspace search variational quantum eigensolver (weighted SSVQE) to obtain the occupied KS orbitals, their corresponding energies, and the total energy, exploring the effects of varying entanglement unitary blocks, circuit depths, and optimization strategies. While the choice of optimization algorithm has minimal impact on the total energy, it significantly influences convergence behavior, stability, and computational time. Our results further reveal the existence of an optimal weight configuration that achieves both faster convergence and improved energy accuracy. Furthermore, our findings highlight the critical role of ansatz depth and entanglement unitary blocks on total energy calculations across molecular geometries. The most effective design begins with a low-expressibility, low-entangling ansatz, such as linear entanglement unitary blocks, and repeats it until a sufficient number of layers is achieved. Using homoatomic and heteroatomic molecule as test sets, the approach remains valid and effective across systems exhibiting different types of electron correlation. This approach proves highly efficient, enabling shallow circuits that avoid convergence issues and closely match standard density functional calculations. This work thus offers a design strategy for shallow, convergence-friendly VQE circuits that match standard density functional energies.
Abstract The use of the variational quantum eigensolver (VQE) on near-term quantum computers for solving Kohn–Sham Hamiltonian is still underexplored. In this study, we apply the weighted subspace search variational quantum eigensolver (weighted SSVQE) to obtain the occupied KS orbitals, their corresponding energies, and the total energy, exploring the effects of varying entanglement unitary blocks, circuit depths, and optimization strategies. While the choice of optimization algorithm has minimal impact on the total energy, it significantly influences convergence behavior, stability, and computational time. Our results further reveal the existence of an optimal weight configuration that achieves both faster convergence and improved energy accuracy. Furthermore, our findings highlight the critical role of ansatz depth and entanglement unitary blocks on total energy calculations across molecular geometries. The most effective design begins with a low-expressibility, low-entangling ansatz, such as linear entanglement unitary blocks, and repeats it until a sufficient number of layers is achieved. Using homoatomic and heteroatomic molecule as test sets, the approach remains valid and effective across systems exhibiting different types of electron correlation. This approach proves highly efficient, enabling shallow circuits that avoid convergence issues and closely match standard density functional calculations. This work thus offers a design strategy for shallow, convergence-friendly VQE circuits that match standard density functional energies.
The variational quantum eigensolver (VQE) is a method of choice to solve the electronic structure problem for molecules on near-term gate-based quantum computers.However, the circuit depth is expected to grow significantly with the problem size.Increased depth can both degrade the accuracy of the results and reduce trainability.In this work, we propose an approach to reduce ansatz circuit depth.Our approach, called "PermVQE," adds an additional optimization loop to the VQE that permutes qubits in order to solve for the qubit Hamiltonian that maximally localizes correlations in the ground state.The choice of permutations is based on mutual information, which is a measure of interaction between electrons and/or holes in spin-orbitals.Encoding strongly entangled spin-orbitals into proximal qubits on a quantum chip naturally reduces the circuit depth needed to prepare the ground state.For representative molecular systems, LiH, H 2 , (H 2 ) 2 , H = 4 , H + 3 , and N 2 , we demonstrate that placing entangled qubits in close proximity leads to shallower depth circuits required to reach a given eigenvalue-eigenvector accuracy.The approach is designed for hardware-efficient ansatz of any qubit connectivity, and examples are demonstrated for linear and twodimensional grid architectures.The main ideas can also be applied to simulate molecules with other ansatz as well as variational quantum algorithms beyond the VQE.In particular, we demonstrate the beneficial effect of qubit permutations to build fermionic-adaptive derivative assembled pseudo-Trotter ansatz on a linear qubit connectivity architecture with nearly a twofold reduction of the number of controlled NOT gates.
Development of resource-friendly quantum algorithms remains highly desirable for noisy intermediate-scale quantum computing. Based on the variational quantum eigensolver (VQE) with unitary coupled-cluster Ansatz, we demonstrate that partitioning of the Hilbert space made possible by the point-group symmetry of the molecular systems greatly reduces the number of variational operators by confining the variational search within a subspace. In addition, we found that instead of including all subterms for each excitation operator, a single-term representation suffices to reach required accuracy for various molecules tested, resulting in an additional shortening of the quantum circuit by a factor of 4–8. With these strategies, VQE calculations on a noise-free quantum simulator achieve energies within a few meVs of those obtained with the full unitary coupled-cluster Ansatz with single and double excitations for the H4-square, H4-chain, and H6-hexagon molecules, while the number of cnot gates, a measure of the quantum-circuit depth, is reduced by a factor of as large as 35. Furthermore, we introduced an efficient “score” parameter to rank the excitation operators, so that the operators causing larger energy reduction can be applied first. Using the H4 square and H4 chain as examples, We demonstrated on noisy quantum simulators that the first few variational operators can bring the energy within the chemical accuracy, while additional operators do not improve the energy since the accumulative noise outweighs the gain from the expansion of the variational Ansatz.
The weighted subspace-search variational quantum eigensolver (SSVQE) is a prominent algorithm for calculating excited-state properties of molecular quantum systems. In this work, we elaborate on some of its fundamental features with the aim of improving its practical realization. First, we demonstrate that the initial ansätze for various excited states could be prepared into a single pure state through a minimal number of ancilla qubits, followed by the optimization of a subsequent global unitary rotation in the targeted subspace. Since the ancillas' sole purpose is to purify an underlying ensemble ρw state with spectral weights w, their measurement would just collapse ρw with probabilities wj to one of its eigenstates ∣Ψj⟩. We thus observe that our realization of SSVQE is equivalent to the original SSVQE improved by importance sampling. Then, we elaborate by numerical means on the potential influence of the auxiliary weights w on the accuracy of the sought-after eigenstates and eigenenergies. Clear trends are discovered which are contrasted with some recent mathematical results concerning the ensemble variational principle that underlies SSVQE.
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.
Hardware-efficient empirical variational ansätze for Variational Quantum Eigensolver (VQE) simulations of quantum chemistry often lack a direct connection to classical quantum chemistry methods. In this work, we propose a method to bridge this gap by introducing a novel approach to constructing a starting point for variational quantum circuits, leveraging quantum mutual information from classical quantum chemistry states to design simple yet effective heuristic ansätze with a topology reflecting the molecular system’s correlations. As a first step, we make use of quantum chemistry calculations, such as Mo̷ller–Plesset (MP2) perturbation theory, to initially provide approximate Natural Orbitals, which have been shown to be the best candidate one-electron basis for developing compact empirical wave functions. 1 Second, we evaluate the quantum mutual information matrix, which provides insights about the main correlations between qubits of the quantum circuit, and enables a direct design of entangling blocks for the circuit. The resulting ansatz is then used with a VQE to obtain a short-depth variational ground state of the electronic Hamiltonian. To validate our approach, we perform a comprehensive statistical analysis through simulations of various molecular systems ( H 2, LiH, H 2 O ) and apply it to the more complex NH 3 molecule. The reported results demonstrate that the proposed methodology gives rise to highly effective ansätze, outperforming the standard empirical ladder-entangler ansatz. Overall, our approach can be used as an effective state preparation, providing a promising route for designing efficient variational quantum circuits for large molecular systems.
Development of resource-friendly quantum algorithms remains highly desirable for noisy intermediate-scale quantum computing.Based on the variational quantum eigensolver (VQE) with unitary coupled-cluster Ansatz, we demonstrate that partitioning of the Hilbert space made possible by the point-group symmetry of the molecular systems greatly reduces the number of variational operators by confining the variational search within a subspace.In addition, we found that instead of including all subterms for each excitation operator, a single-term representation suffices to reach required accuracy for various molecules tested, resulting in an additional shortening of the quantum circuit by a factor of 4-8.With these strategies, VQE calculations on a noise-free quantum simulator achieve energies within a few meVs of those obtained with the full unitary coupled-cluster Ansatz with single and double excitations for the H 4 -square, H 4 -chain, and H 6 -hexagon molecules, while the number of CNOT gates, a measure of the quantum-circuit depth, is reduced by a factor of as large as 35.Furthermore, we introduced an efficient "score" parameter to rank the excitation operators, so that the operators causing larger energy reduction can be applied first.Using the H 4 square and H 4 chain as examples, We demonstrated on noisy quantum simulators that the first few variational operators can bring the energy within the chemical accuracy, while additional operators do not improve the energy since the accumulative noise outweighs the gain from the expansion of the variational Ansatz.