Quantum computing simulation of the hydrogen molecular ground-state energies with limited resources

Ali Abu-Nada

doi:10.1515/phys-2021-0071

Abstract

Abstract In this article, the hydrogen molecular ground-state energies using our algorithm based on quantum variational principle are calculated. They are calculated through a simulator since the system of the present study (i.e., the hydrogen molecule) is relatively small and hence the ground-state energies for this molecule are efficiently classically simulable using a simulator. Complete details of this algorithm are elucidated. For this, a full description on the fermions–qubits and the molecular Hamiltonian–qubit Hamiltonian transformations, is given. The authors search for qubit system parameters ( θ 0 {\theta }_{0} and θ 1 {\theta }_{1} ) that yield the minimum energies for the system and also study the ground state energies as a function of the molecular bond length. Proposed circuit is humble and does not include many parameters compared with that of Kandala et al., the authors control only two parameters ( θ 0 {\theta }_{0} and θ 1 {\theta }_{1} ).

Highlights

Feynman emphasized that simulating quantum systems on a computer that uses quantum bits is very efficient [1]
Computational quantum chemistry relies on approximate methods that often succeed in predicting chemicals characteristics of larger systems, and each approximation has different levels of accuracy according to the system complexity [6,7]
Our algorithm contains the following major phases: (1) we convert the fermion system to qubit system, we convert the molecular Hamiltonian into qubit Hamiltonian using the known binary-tree transformation and this includes the fermionic–qubit operators transformation [11,12]; (2) we prepare the quantum state ∣Ψ(θ)⟩, normally called ansatz (we describe all elements of ∣Ψ(θi)⟩ in Section 3); (3) we measure the expectation value ⟨Ψ(θi)∣H∣Ψ(θi)⟩; and (4) we search for θi that makes ⟨Ψ(θi)∣H∣Ψ(θi)⟩ the minimum

Summary

Introduction

Feynman emphasized that simulating quantum systems on a computer that uses quantum bits (qubits) is very efficient [1]. Our algorithm contains the following major phases: (1) we convert the fermion system to qubit system, we convert the molecular Hamiltonian into qubit Hamiltonian using the known binary-tree transformation and this includes the fermionic–qubit operators transformation [11,12]; (2) we prepare the quantum state ∣Ψ(θ)⟩, normally called ansatz (we describe all elements of ∣Ψ(θi)⟩ in Section 3); (3) we measure the expectation value ⟨Ψ(θi)∣H∣Ψ(θi)⟩; and (4) we search for θi that makes ⟨Ψ(θi)∣H∣Ψ(θi)⟩ the minimum This is often the same as for the VQE algorithm.

Fermion–qubit transformation

Overview of the algorithm and simulation details

Results

Conclusion