VQE

Schematic representation of a VQE workflow. \(\boldsymbol \theta\) represents the set of circuit parameters which are variationally optimized via an external classical routine. Expectation values of the molecular Hamiltonian \(H\) (cost function) are evaluated at circuit level.

The VQE algorithm can be invoked through the qc2.algorithms.qiskit.vqe.VQE and qc2.algorithms.pennylane.vqe.VQE classes. Despite requiring a few SDK-specific details, in general the algorithm and its instantiation entails the following input information:

• Reference state & occupation numbers: a reference quantum circuit and initial qubit states; this is by default Hartree-Fock.

• Molecule active space: this determines the total number of qubits and is input as an instance of ActiveSpace.

• Ansatz: type of variational form used; by default, this is set to UCCSD.

• Initial circuit parameters: Initial guesses for the ansatz variational parameters \(\theta_{i}\).

• Fermionic-to-qubit mapper & qubit Hamiltonian: Strategy to transform the initial fermionic Hamiltonian (constructed from classical qchem data) to qubit space; by default, this set to Jordan-Wigner.

• Classical optimizer: the optimization routine for the circuit variational parameters. Default routines are: SLSQP for Qiskit Nature and GradientDescentOptimizer for PennyLane .

Like in the case of qc2-ASE calculators, the algorithm VQE class is naturally abstracted within qc2Data. So, in actual runs, users only need to instantiate it as part of its algorithm method (to be discussed in Running quantum-classical algorithms via qc2Data class section). For illustrative purposes only, the following is a pseudo-code example demonstrating how VQE can be instantiated independently:

from ase.build import molecule

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

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

# set ASE Atoms object
mol = molecule('H2')

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

# set up VQE class
vqe = VQE(
    qc2data=qc2data,
    active_space=ActiveSpace(
        num_active_electrons=(1, 1),
        num_active_spatial_orbitals=2
    ),
    mapper='bk',
    optimizer=COBYLA(),
    estimator=Estimator(),
)

where the active_space argument is set to an instance of the ActiveSpace class. The 'bk' string in mapper invokes the Bravyi-Kitaev fermionic-to-qubit transformation (qiskit_nature.BravyiKitaevMapper) as implemented in FermionicToQubitMapper

A corresponding example using qc2.algorithms.pennylane.vqe.VQE is:

from ase.build import molecule

import pennylane as qml

from qc2.data import qc2Data
from qc2.algorithms.pennylane import VQE
from qc2.algorithms.utils import ActiveSpace

# set ASE Atoms object
mol = molecule('H2')

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

# set up VQE class
vqe = VQE(
    qc2data=qc2data,
    active_space=ActiveSpace(
        num_active_electrons=(1, 1),
        num_active_spatial_orbitals=2
    ),
    mapper='jw',
    optimizer=qml.GradientDescentOptimizer(stepsize=0.5),
    device='default.qubit'
)