Computational Chemistry on Quantum Computers

TL;DR

This paper explores quantum computational techniques for simulating molecular Hamiltonians, focusing on the creation, measurement, and analysis of small molecules like H2 and He2 using algorithms such as LCAO, LCU, and PEA.

Contribution

It demonstrates the application of quantum algorithms to molecular Hamiltonians, highlighting the challenges with Hamiltonian size and the effectiveness of low-order expansions.

Findings

Hamiltonian size increases with molecule complexity

Low-order expansions suffice for qualitative analysis

Measurement resolution depends on the number of quantum registers

Abstract

The purpose of this experiment was to use the known analytical techniques to study the creation, simulation, and measurements of molecular Hamiltonians. The techniques used consisted of the Linear Combination of Atomic Orbitals (LCAO), the Linear Combination of Unitaries (LCU), and the Phase Estimation Algorithm (PEA). The molecules studied were $H_2$ with and without spin, as well as $He_2$ without spin. Hamiltonians were created under the LCAO basis, and reconstructed using the Jordan-Winger transform in order to create a linear combination of Pauli spin operators. The lengths of each molecular Hamiltonian greatly increased from the $H_2$ without spin, to $He_2$. This resulted in a reduced ability to simulate the Hamiltonians under ideal conditions. Thus, only low orders of l = 1 and l = 2 were used when expanding the Hamiltonian in accordance to the LCU method of simulation. The resulting Hamiltonians were measured using PEA, and plotted against function of $\frac{2\pi(K)}{N}$ and the probability distribution of each register. The resolution of the graph was dependent on the amount of registers, N, being used. However, the reduction of order hardly changed the image of the $H_2$ graphs. Qualitative comparisons between the three molecules were drawn.