Subspace-Search Variational Quantum Eigen solver (SSVQE) is a searching method of multiple states and relies on the unitarity of transformations to ensure the orthogonality of output states for multiple states. Therefore, this method is thought to be a promising method for quantum chemistry because ordinary Variational Quantum Eigen solver (VQE) can only calculate the excited states step by step from the ground state based on Variational Quantum deflation (VQD). We compare the advantage of VQE, SSVQE with/without the constraint term and/or Tabu search term, which is added by the Lagrange multiplier method so as to calculate the desired energy levels. We evaluated the advantage by calculating each level of H2 and HeH, respectively. As there simulation results, the accuracy calculated by constrained VQE with Tabu search indicates higher accuracy than that of our other algorithm, for analysis on H2. The accuracy calculated by constrained SSVQE indicates higher that of the constrained VQE with Tabu search. We found it is beneficial for enhancing the accuracy to use constraint terms decreasing convergence times to use Tabu search terms according to the nature of molecules. We demonstrate that constraint and Tabu search terms contribute to the accuracy and convergence time on quantum chemical calculating.
Within the evolving domain of quantum computational chemistry, the Variational Quantum Eigensolver (VQE) has been developed to explore not only the ground state but also the excited states of molecules. In this study, we compare the performance of Variational Quantum Deflation (VQD) and Subspace-Search Variational Quantum Eigensolver (SSVQE) methods in determining the low-lying excited states of
LiH. Our investigation reveals that while VQD exhibits a slight advantage in accuracy, SSVQE stands out for its efficiency, allowing the determination of all low-lying excited states through a single parameter optimization procedure. We further evaluate the effectiveness of optimizers, including Gradient Descent (GD), Quantum Natural Gradient (QNG), and Adam optimizer, in obtaining
LiH's first excited state, with the Adam optimizer demonstrating superior efficiency in requiring the fewest iterations. Moreover, we propose a novel approach combining Folded Spectrum VQE (FS-VQE) with either VQD or SSVQE, enabling the exploration of highly excited states. We test the new approaches for finding all three
H4's excited states. Folded Spectrum SSVQE (FS-SSVQE) can find all three highly excited states near
−1.0 Ha with only one optimizing procedure, but the procedure converges slowly. In contrast, although Folded spectrum VQD (FS-VQD) gets highly excited states with individual optimizing procedures, the optimizing procedure converges faster.
Variational quantum eigensolver (VQE) is an efficient computational method promising chemical accuracy in electronic structure calculations on a universal-gate quantum computer. However, such a simple task as computing the electronic energy of a hydrogen molecular cation, H 2 +, is not possible for a general VQE protocol because the calculation will invariably collapse to a lower energy of the corresponding neutral form, H 2 . The origin of the problem is that VQE effectively performs an unconstrained energy optimization in the Fock space of the original electronic problem. We show how this can be avoided by introducing necessary constraints directing VQE toward the electronic state of interest. The proposed constrained VQE can find an electronic state with a certain number of electrons, a certain spin, or any other property. Moreover, the new algorithm naturally removes unphysical kinks in potential energy surfaces (PESs), which frequently appeared in the regular VQE and required significant additional quantum resources for their removal. We demonstrate the performance of the constrained VQE by simulating PESs of various states of H 2 and H 2 O on Rigetti Computing Inc.’s 19Q-Acorn quantum processor.
We propose a new approach to improve the accuracy of ground state estimates in Variational Quantum Eigensolver (VQE) algorithms by employing subspace representations with soft-coded orthogonality constraints. As in other subspace-based VQE methods, such as the Subspace-Search VQE (SSVQE) and Multistate Contracted VQE (MCVQE), once the parameters are optimized to maximize the subspace overlap with the low-energy sector of the Hamiltonian, one diagonalizes the Hamiltonian restricted to the subspace. Unlike these methods, where \emph{hard-coded} orthogonality constraints are enforced at the circuit level among the states spanning the subspace, we consider a subspace representation where orthogonality is \emph{soft-coded} via penalty terms in the cost function. We show that this representation allows for shallower quantum circuits while maintaining high fidelity when compared to single-state (standard VQE) and multi-state (SSVQE or MCVQE) representations, on two benchmark cases: a
3×3 transverse-field Ising model and random realizations of the Edwards--Anderson spin-glass model on a
4×4 lattice.
Variational quantum eigensolver (VQE) is demonstrated to be the promising methodology for quantum chemistry based on near-term quantum devices. However, many problems are yet to be investigated for this methodology, such as the influences of optimization algorithm and basis size on the accuracy and efficiency for quantum computing. To address these issues, five molecules (H2, LiH, HF, N2 and F2) are studied in this work based on the VQE method using unitary coupled cluster (UCC) ansatz. The performance of the gradient optimization L-BFGS-B is compared with that of the direct search method COBYLA. The former converges more quickly, but the accuracy of energy surface is a little lower. The basis set shows a vital influence on the accuracy and efficiency. A large basis set generally provides an accurate energy surface, but induces a significant increase in computing time. The 631g basis is generally required from the energy surface of the simplest H2 molecule. For practical applications of VQE, complete active space (CAS) is suggested based on limited quantum resources. With the same number of qubits, more occupied orbitals included in CAS gives a better accuracy for the energy surface and a smaller evaluation number in the VQE optimization. Additionally, the electronic structure, such as filling fraction of orbitals, the bond strength of a molecule and the maximum nuclear charge also influences the performance of optimization, where half occupation of orbitals generally requires a large computation cost.
The development of Fault-Tolerant Quantum Computer (FTQC) gradually raises a possibility to implement the Quantum Phase Estimation (QPE) algorithm. However, QPE works only for normalized systems. This requires the minimum and maximum of eigenvalues of a Hamiltonian. Variational Quantum Eigensolver (VQE) well developed in the Noisy Intermediate Scale Quantum (NISQ) era is necessary for preparing the initial eigenvectors that close to the exact states. In this paper, we propose a method to derive all of the states with high accuracy by using the Variational Quantum State Eigensolver (VQSE) and Subspace-Search VQE (SSVQE) methods. We show that by using the VQSE and the SSVQE prescreening methods, we can derive all of the hydrogen molecules states correctly.