Summary

In this chapter, we have introduced classical and hybrid classical-quantum variational methods to find the lowest energy eigenvalue for a quantum system and their implementation with a classical PyQMC variational Monte Carlo Python package, which interoperates with the PySCF, and Qiskit Nature using the STO-3G basis with the Python-based PySCF driver.

We have illustrated these methods, solving for the ground state and plotting the BOPES of the hydrogen molecule, the lithium hydride molecule, and the macro molecule.

The results we obtained with Qiskit Nature VQE and QPE are in good agreement with those obtained with the PyQMC and PySCF RHF packages for several combinations of fermionic-to-qubit Hamiltonian mappers and classical gradient descent solvers and by reducing the quantum workload to the outermost two electrons of the formamide molecule. We hope these results will encourage the reader to replay these experiments with different choices of solvers and with other...