oo-VQE

Schematic representation of an orbitally-optimized VQE workflow. \(\boldsymbol \theta\) and \(\boldsymbol \kappa\) represent the set of circuit and orbital rotation parameters, respectively. The expectation values of \(H\) are minimized via a classical two-step process; NR stands for Newton-Raphson.

The oo-VQE algorithm (refer to the figure) is implemented in two other dedicated classes: qc2.algorithms.qiskit.oo_vqe.oo_VQE and qc2.algorithms.pennylane.oo_vqe.oo_VQE. It is important to note that, in addition to handling circuit parameters, oo-VQE also optimizes the initial Hartree-Fock molecular orbital coefficients through additional classical Newton-Raphson steps (\(C'_{\rm MO} \rightarrow e^{-\boldsymbol \kappa} C_{\rm MO}\)) . This is done by resorting to analytic first and second derivatives of the energy with respect to \(\boldsymbol \kappa\) as implemented in the OrbitalOptimization class; for details, see [MMN+20, SAHR81, YSG+21, ZGS+23].

Similarly to VQE, the oo-VQE algorithm class in qc2 is primarily designed to be instantiated through the qc2Data.algorithm attribute; this is further explained in Running quantum-classical algorithms via qc2Data class. However, it can also be instantiated and operated independently, as long as an instance of qc2Data is available. An illustrative example is shown below:

from ase.build import molecule

from qiskit_algorithms.optimizers import SLSQP
from qiskit.primitives import Estimator

from qc2.data import qc2Data
from qc2.algorithms.qiskit import oo_VQE
from qc2.algorithms.utils import ActiveSpace

# set ASE Atoms object
mol = molecule('H2O')

# instantiate qc2Data class
qc2data = qc2Data(
    molecule=mol,
    ...
)

# ... run qchem ab initio HF calculation via qc2-ASE

# set up oo-VQE class
oo_vqe = oo_VQE(
    qc2data=qc2data,
    active_space=ActiveSpace(
        num_active_electrons=(2, 2),
        num_active_spatial_orbitals=4
    ),
    mapper='jw',
    optimizer=SLSQP(),
    estimator=Estimator(),
)

# run algorithm
oo_vqe.run()