  In this educational paper, we will discuss calculations on the hydrogen
molecule both on classical and quantum computers. In the former case, we will
discuss the calculation of molecular integrals that can then be used to
calculate potential energy curves at the Hartree--Fock level and to correct
them by obtaining the exact results for all states in the minimal basis. Some
aspects of spin-symmetry will also be discussed. In the case of quantum
computing, we will start out from the second-quantized Hamiltonian and qubit
mappings. Using quantum phase estimation, we then provide the circuits for two
different algorithms: Trotteization and qubitization. Finally, the significance
of quantum error correction will be briefly discussed.
