Abstract An exciting application of quantum computers is the simulation of physical systems. The goal of the variational quantum eigensolver (VQE) is to approximate the ground state of a quantum system. This is done by applying the variational principle, finding the expectation value of the Hamiltonian in an ansatz quantum state, then minimizing this expectation value with respect to the parameters of the ansatz. In the VQE, the calculation of the expectation value in the ansatz state is done on a quantum computer, while the adjustment of the parameters of the ansatz to find the minimum energy is done classically. In this article we study the spin–spin coupling H = A S → 1 · S → 2 . This coupling appears in the hyperfine interaction of atomic systems, resulting from the interaction of the electronic spin with the nuclear spin. The form of the interaction satisfies the requirements of the VQE, that the Hamiltonian must be expressed as sums of products of Pauli matrices, making it an interesting trial system for a quantum computer. In our calculation technique, the expectation value is implemented using two qubits on a quantum computer. The ansatz state is simplified to a single free parameter, which is then optimized classically. The algorithm converges rapidly, and the correct ground state is deduced. This serves as a useful example of implementing the VQE to solve a physically realizable problem in quantum mechanics.
Abstract An exciting application of quantum computers is the simulation of physical systems. The goal of the variational quantum eigensolver (VQE) is to approximate the ground state of a quantum system. This is done by applying the variational principle, finding the expectation value of the Hamiltonian in an ansatz quantum state, then minimizing this expectation value with respect to the parameters of the ansatz. In the VQE, the calculation of the expectation value in the ansatz state is done on a quantum computer, while the adjustment of the parameters of the ansatz to find the minimum energy is done classically. In this article we study the spin–spin coupling H = A S → 1 · S → 2 . This coupling appears in the hyperfine interaction of atomic systems, resulting from the interaction of the electronic spin with the nuclear spin. The form of the interaction satisfies the requirements of the VQE, that the Hamiltonian must be expressed as sums of products of Pauli matrices, making it an interesting trial system for a quantum computer. In our calculation technique, the expectation value is implemented using two qubits on a quantum computer. The ansatz state is simplified to a single free parameter, which is then optimized classically. The algorithm converges rapidly, and the correct ground state is deduced. This serves as a useful example of implementing the VQE to solve a physically realizable problem in quantum mechanics.
Exponential scaling presents a significant challenge in electronic structure calculations performed on classical computers. This paper explores how quantum computer algorithms can accurately represent quantum systems. Variational Quantum Eigensolver (VQE) algorithm is used to compute the ground state energy of hydrogen and helium sequences by implementing variational principle and quantum gates as trial wavefunction. This technique combines classical optimization with quantum computing calculations to simulate quantum systems on noisy and resource-limited computers. The resulting calculated energy is highly consistent to the corresponding exact values and Hartree-Fock calculations with a trend of when the number of atoms increases the calculated energy becomes more negative, leading to a decrease in the percentage error. Moreover, the convergence of the ground state energy of hydrogen and helium atoms was effectively optimized. The desired energy was reached, proven by adjusting the expectation value, and gradually achieving unity in state overlap. These findings demonstrate the VQE method's accuracy in calculating simple quantum systems and its scalability for larger atomic and molecular system, such as those in quantum chemistry and material science. However, challenges in quantum computer simulations, such as limited in qubit numbers and the presence of noise, require further advancements. Therefore, implementing a larger basis sets, advanced qubit mapping, specific chemistry ansatz, and flexible optimization techniques is one way to improve overall calculation.
The Variational Quantum Algorithms (VQAs) are hybrid quantum-classical algorithms and they can be used in the Nosiy Intermadiate Scale Quantum (NISQ) devises. The Variational Quantum Eigensolver (VQE) was suggested as a first VQA. VQE is based on the variational method of quantum mechanics and it is used to find the ground state energy of a quantum system. In this study, VQE is used for the analysis of NMR spectra for the AB and AB2 spin systems. The frequencies and the spin coupling values are obtained from the sample spectra for these spin systems. Then the Hamiltonians are written in terms of pauli spin operators and transformed into a suitable forms for quantum computer. By employing VQE the ground state energies are obtained for the related spin systems. They are found to be in good agreement with the results obtained from the known variation method.
Abstract To understand the properties and interactions of materials, and determining the ground state energies is one of the important challenges in quantum chemistry, materials science, and quantum mechanics, where quantum computing can play an important role for studying the properties of materials. In this study, we have explored the quantum processor application to compute the He molecule ground state energy which utilizes the Variational Quantum Eigensolver (VQE) algorithm. In here, we have implemented VQE on a state-of-the-art quantum processor, optimizing a parameterized quantum circuit to minimize the energy expectation value of the He molecule's Hamiltonian on the four qubits processor. The obtained results of this work show a significant improvement in accuracy compared to classical computational methods, such as Hartree-Fock and density functional theory, which demonstrate the compute potential of quantum algorithms in quantum many-body problems. Thus, these results demonstrate the advantages of quantum computing in achieving high accuracy in simulations of molecular and material properties, and pave the way for future applications in more complex systems. This work highlights the potential of quantum processors in the fields of quantum chemistry, computational physics, and data science.
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm for finding the minimum eigenvalue of a Hamiltonian that involves the optimization of a parameterized quantum circuit. Since the resulting optimization problem is in general nonconvex, the method can converge to suboptimal parameter values that do not yield the minimum eigenvalue. In this work, we address this shortcoming by adopting the concept of variational adiabatic quantum computing (VAQC) as a procedure to improve VQE. In VAQC, the ground state of a continuously parameterized Hamiltonian is approximated via a parameterized quantum circuit. We discuss some basic theory of VAQC to motivate the development of a hybrid quantum-classical homotopy continuation method. The proposed method has parallels with a predictor-corrector method for numerical integration of differential equations. While there are theoretical limitations to the procedure, we see in practice that VAQC can successfully find good initial circuit parameters to initialize VQE. We demonstrate this with two examples from quantum chemistry. Through these examples, we provide empirical evidence that VAQC, combined with other techniques (an adaptive termination criteria for the classical optimizer and a variance-based resampling method for the expectation evaluation), can provide more accurate solutions than “plain” VQE, for the same amount of effort.
Abstract Quantum computing is a technology that utilizes the principles of quantum mechanics to perform complex computational processes. In this work, we use Qiskit Module from IBM to do quantum computational calculation using Variational Quantum Eigensolver (VQE) algorithm. VQE is a hybrid quantum-classical method that combines a quantum computer to measure energies and a classical computer to process the measurement results and update the parameters of the quantum computer. The purpose of VQE is to find the ground state energy of a chemical system. In the previous study, many of the VQE calculations have been done on simple molecules. So, in this study, we would like to use Hydrazine molecule as our object of VQE calculation. Furthermore, these results will be compared with the results from the classical calculation (MP2, CCSD(T), QCISD(T), and CASSCF) methods for testing the effectiveness of VQE using Unitary Coupled-Cluster Single and Double excitations (UCCSD) Ansatz. The quantum algorithm based on the UCCSD Ansatz led to a simplification of the algorithm by reducing the circuit depth. Then, the possibility to use active space approximation, can be used to reduce the quantum gates while trying to keep a good level of accuracy. In this study, we chose (2,2) and (4,4) active spaces. Based on the results, as we increase the size of the active space during the evaluation of the single-point energy, the estimated ground states obtained from the VQE algorithm yield nearly identical values. Conversely, in CASSCF calculations, expanding the active space introduces more energy corrections, thus making it more sensitive. Additionally, when examining potential energy surfaces, VQE demonstrates results that gradually align with CCSD(T) and QCISD(T) methods.
Abstract Material discovery is a phenomenon practiced since the evolution of the world. The discovery of materials has led to significant development in varied fields such as Science, Engineering and Technology. Computationally simulating molecules has been an area of interest in the industry as well as academia. However, simulating large molecules can be computationally expensive in terms of computing power and complexity. Quantum computing is a recent development that can improve the efficiency in predicting properties of atoms and molecules which will be useful for material design. The Variational Quantum Eigensolver (VQE) is one such quantum algorithm used to calculate the ground state energy of molecules or ions. In this study, we have done a comparative analysis of the parameters that constitute the VQE algorithm. This includes components such as basis, qubit mapping, ansatz, and optimizers used. We have also developed a database consisting of 79 single atoms and their variations of oxidation states and 33 molecules with the data of their Hamiltonian and ground state energy and dipole moment.