The Electronic Structure of the Hydrogen Molecule: A Tutorial Exercise in Classical and Quantum Computation

In this educational paper, we will discuss calculations
on the
hydrogen molecule on both 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.
We then provide circuits for quantum phase estimation using two different
algorithms: Trotterization and qubitization. Finally, the significance
of quantum error correction will be briefly discussed.