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

TL;DR

This paper provides an educational overview of calculating the electronic structure of the hydrogen molecule using classical methods like Hartree-Fock and quantum algorithms such as quantum phase estimation, highlighting both approaches' procedures and significance.

Contribution

It offers a comprehensive tutorial on classical and quantum computational techniques for hydrogen molecule calculations, including detailed circuit constructions and discussions on quantum error correction.

Findings

Classical calculations yield potential energy curves and exact states.

Quantum algorithms like Trotteization and qubitization are detailed.

Quantum error correction's role is briefly discussed.

Abstract

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.