Fig. 1
Schematic of the variational quantum eigensolver (VQE) method that minimizes the energy of the Hamiltonian \(\langle {\psi (\vec {\theta })}| \hat {H}_{el} |{\psi (\vec {\theta })}\rangle \) by adjusting variational parameters θ. It uses classical computing resources denoted by green color and quantum computing resources denoted by blue. A simulation starts by constructing a fermionic Hamiltonian and finding mean-field solution |ψHF〉. Next, the fermionic Hamiltonian is mapped into qubit Hamiltonian, represented as a sum of Pauli strings \(H=\sum _{j}\alpha _{j}\prod _{i}\sigma _{i}^{j}\). Then the ansatz to represent the wave function is chosen and initialized with the initial set of parameters \(\vec {\theta }_{0}\). The trial state is prepared on a quantum computer as a quantum circuit consisting of parametrized gates. The rest of the procedure is performed repeatedly until the convergence criterion is met. At iteration k the energy of the Hamiltonian is computed by measuring every Hamiltonian term \(\langle {\psi (\vec {\theta _{k}})}|P_{j}|{\psi (\vec {\theta _{k}})}\rangle \) on a quantum computer and adding them on a classical computer. The energy \(E(\vec {\theta _{k}})\) is fed into the classical algorithm that updates parameters for the next step of optimization \(\vec {\theta }_{k+1}\) according to the chosen optimization algorithm