In the circumstances of quantum chemistry, these n states represent spin orbitals, so one-electron energy eigenfunctions and often molecular orbitals found by the Hartree–Fock (HF) method. We describe the fermionic systems using the formalism of second quantization, where n single-particle states can be either empty ∣ 0 ⟩ or occupied ∣ 1 ⟩ by a spinless fermionic particle. The estimation of the ground-state energy is iteratively optimized by a classical controller changing the parameters ( θ 1 and θ 0) we just control two parameters ( θ 0 and θ 1). Thus, the final shape of qubit Hamiltonian of the hydrogen molecule will be written as follows: , it was witnessed that two-qubits could be detached after the binary tree transformation and hence we end up with two-qubits. The final qubit Hamiltonian contains four terms, each term contains a tensor product of two Pauli operators acted on two-qubits for the hydrogen molecule. When using the binary-tree transformation, this creates a qubit Hamiltonian diagonal in the second and fourth qubit, which has the total particle and spin Z 2 symmetries encoded in those qubits, we recommend the readers to go through Appendix A for more details of removing those two electrons. We display the L spin orbitals by registering the first L / 2 spin-up ones and then the L / 2 spin-down ones. The hydrogen molecule Hamiltonian is mapped first onto four qubits using a binary-tree transformation. The two shared electrons in the hydrogen molecule will have opposite spins.įinally, the second quantized molecular Hamiltonian must be converted to qubit Hamiltonian. Two s orbitals on hydrogen atoms, each orbital filled with either spin-up or spin-down electrons and thus the total number of spin orbitals is 4. However, VQE uses a classic optimizer ( i.e., COBYLA and SPSA ) to find the updated ansatz parameters ( θ i) that reduce the expectation value of H as follows: This is often the same as for the VQE algorithm. Our algorithm contains the following major phases: (1) we convert the fermion system to qubit system, furthermore, we also convert the molecular Hamiltonian into qubit Hamiltonian using the known binary-tree transformation and this also includes the fermionic–qubit operators transformation (2) we prepare the quantum state ∣ Ψ ( θ ) ⟩, normally called ansatz (we describe all elements of ∣ Ψ ( θ i ) ⟩ in Section 3) (3) we measure the expectation value ⟨ Ψ ( θ i ) ∣ H ∣ Ψ ( θ i ) ⟩ and (4) we search for θ i that makes ⟨ Ψ ( θ i ) ∣ H ∣ Ψ ( θ i ) ⟩ the minimum. VQE is a quantum/classical hybrid algorithm that can be used to find eigenvalues of a (often large) matrix H. For more details on VQE, we recommend the reader here to read Refs. To find the ground-state energies of the hydrogen molecule, we follow the VQE to simulate the hydrogen molecule H 2, but we do not implement our simulation on the noisy quantum processor and also we do not use the classical optimizer to update the quantum circuit parameters. We provide full details of this algorithm by, first, describing the fermionics-qubit transformation, and then, the molecular Hamiltonian–qubit Hamiltonian transformation as well. In this article, we calculate the hydrogen molecular ground-state energies using our algorithm based on quantum varaitional principle (QVP). developed the variational quantum eigensolver (VQE) to calculate the ground-state energies of a molecular system using quantum circuits with less overhead resources. Some of these algorithms require many quantum overhead resources and exceed the limits of the available noisy intermediate-scale quantum (NISQ) such as the quantum phase estimation (QPE) algorithm. Therefore, several algorithms have been developed to calculate ground-state energies for chemical systems. The dominant objective in the computational chemistry is to find the ground-state energies of many body-interacting fermionic Hamiltonians. Accomplishing such a high level of accuracy depends on the effectiveness of the quantum algorithms. A quantum algorithm and a quantum computer could possibly beat all computational quantum chemistry approximations. Computational quantum chemistry relies on approximate methods that often succeed in predicting chemicals characteristics of larger systems, and each approximation has different levels of accuracy according to the system complexity.
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