Ground State Energy Estimation on Current Quantum Hardware Through The Variational Quantum Eigensolver: A Comprehensive Study

TL;DR

This paper provides a comprehensive guide for implementing the Variational Quantum Eigensolver (VQE) on NISQ devices to estimate molecular ground state energies, comparing performance across noise models and real hardware.

Contribution

It offers detailed methodological insights, clarifies under-documented aspects, and evaluates VQE performance on actual quantum hardware and simulators.

Findings

VQE can estimate ground state energies with varying accuracy across noise models.

Implementation details improve reproducibility and understanding of VQE on current hardware.

Real hardware experiments reveal the impact of noise on VQE performance.

Abstract

While numerical simulations are presented in most papers introducing new methods to enhance the VQE performance, comprehensive, comparative, and applied studies remain relatively rare. We present a comprehensive, yet concise guide for the implementation of the VQE for molecular problems on NISQ devices, specifically applied to estimate the ground state energy of the BeH2 molecule using hardware-efficient and chemically informed ans\"atze. This work clarifies several under-documented aspects in the literature, such as the construction of the electronic Hamiltonian, the transformation of fermionic operators into qubit operators via second quantization, and the mathematical framework's details for the unitary coupled cluster single and double (UCCSD) ansatz. Our methodology, implemented using Qiskit 1.2, the latest release as of the date of this writing, is demonstrated on a noiseless simulator and further tested with noisy quantum circuits. The resilience of the VQE to quantum noise remains an open question. This study compares the computational accuracy of ground state energy estimations for molecules using the VQE across three different current quantum hardware noise models. Furthermore, our experiment on IBM's 156-qubit actual quantum computer revealed valuable insights on the real performance of the VQE on current quantum hardware.