In the study of solving the ground state of quantum systems by variational principle, variational quantum algorithm is widely adopted to solve the ground state. As one kind of quantum classical hybrid algorithm, variational quantum eigensolver plays an important role. In this paper, we calculate ground state of quantum Rabi model in the ultra-strong coupling regime by invoking the variational quantum eigensolver in the open-source quantum computing simulator Qiskit. In the 3-qubit computing basis, we use the standard binary encoding method to encode the operators and quantum states in the quantum Rabi model, and measure the minimum average value (ground state energy) on the quantum circuit constructed by the Hamiltonian variational ansatz. Finally, comparing the calculation results of the variational quantum eigensolver with the exact values obtained by classical numerical simulations, we discuss the dependence of computational accuracy of the variational quantum eigensolver on the consumption of the number of the qubits and the coupling strength in quantum Rabi model.
In the study of solving the ground state of quantum systems by variational principle, variational quantum algorithm is widely adopted to solve the ground state. As one kind of quantum classical hybrid algorithm, variational quantum eigensolver plays an important role. In this paper, we calculate ground state of quantum Rabi model in the ultra-strong coupling regime by invoking the variational quantum eigensolver in the open-source quantum computing simulator Qiskit. In the 3-qubit computing basis, we use the standard binary encoding method to encode the operators and quantum states in the quantum Rabi model, and measure the minimum average value (ground state energy) on the quantum circuit constructed by the Hamiltonian variational ansatz. Finally, comparing the calculation results of the variational quantum eigensolver with the exact values obtained by classical numerical simulations, we discuss the dependence of computational accuracy of the variational quantum eigensolver on the consumption of the number of the qubits and the coupling strength in quantum Rabi model.
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.
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.
We propose a divide-and-conquer method for the quantum-classical hybrid algorithm to solve larger problems with small-scale quantum computers.Specifically, we concatenate a variational quantum eigensolver (VQE) with a reduction in the system dimension, where the interactions between divided subsystems are taken as an effective Hamiltonian expanded by the reduced basis.Then the effective Hamiltonian is further solved by the VQE, which we call deep VQE.Deep VQE allows us to apply quantum-classical hybrid algorithms on small-scale quantum computers to large systems with strong intrasubsystem interactions and weak intersubsystem interactions, or strongly correlated spin models on large regular lattices.As proof-of-principle numerical demonstrations, we use the proposed method for quasi-one-dimensional models, including one-dimensionally coupled 12-qubit Heisenberg antiferromagnetic models on kagome lattices as well as two-dimensional Heisenberg antiferromagnetic models on square lattices.The largest problem size of 64 qubits is solved by simulating 20-qubit quantum computers with a reasonably good accuracy approximately a few %.The proposed scheme enables us to handle the problems of > 1000 qubits by concatenating VQEs with a few tens of qubits.While it is unclear how accurate ground-state energy can be obtained for such a large system, our numerical results on a 64-qubit system suggest that deep VQE provides a good approximation (discrepancy within a few percent) and has room for further improvement.Therefore, deep VQE provides us a promising pathway to solve practically important problems on noisy intermediate-scale quantum computers.
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.
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.
An analytical variational method for the ground state of the biased quantum Rabi model in the ultra-strong coupling regime is presented. This analytical variational method can be obtained by a unitary transformation or alternatively by assuming the form of the ground state wave function. The key of the method is to introduce a variational parameter λ , which can be determined by minimizing the energy functional. Using this method, we calculate the physical observables with high accuracy in comparison with the numerical exact ones. Our method evidently improves over the widely used general rotating-wave approximation (GRWA) in both qualitative and quantitative aspects.