Excited states calculations with the quantum approximate optimization algorithm

The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical algorithm introduced to solve complex combinatorial optimization problems, such as the Max Cut. It exploits a parameterized quantum circuit to estimate the ground state of a cost Hamiltonian Hc, that encodes the solution of the combinatorial problem. The best values of the parameters are determined via classical optimization techniques. The QAOA is also used to obtain the ground state of molecules. In this work, we will extend its applicability by introducing a procedure that allows us to calculate also the excited states in an iterative way: once the ground state is known, one can obtain a new Hamiltonian whose ground state is the first excited state of the original Hamiltonian. Now the QAOA can be applied again. The proposed method has been tested using the H2 and the LiH molecule. For the quantum part of the algorithm, the STO-3G basis has been used for the one-body and two-body integral calculation, considering only 2 molecular orbitals and 2 electrons with opposite spins, a necessary step to build the second quantized Hamiltonian. The creation and annihilation operators have been mapped to qubits using the Jordan Wigner transformation. For the classical optimizer, the Basin-Hopping method with the BFGS algorithm has been used. We first calculate the ground state energy and wave function. For both the molecules, the ground states can be successfully estimated for small inter-nuclear distances. At larger values the results are not good, due to the degeneration between the ground state and the first excited state. Then we apply our procedure for the calculation of first excited state for both molecules. Again, we show convergence to the right results for short inter-nuclear distances. The degeneration problem is no longer present, but the errors related to the calculation of the ground states are transferred to the calculation of the first excited states.