Quantum Chemistry

PennyLane provides the qchem module to perform quantum chemistry simulations. It contains a differentiable Hartree-Fock solver and the functionality to construct a fully-differentiable molecular Hamiltonian that can be used as input to quantum algorithms such as the variational quantum eigensolver (VQE) algorithm. The qchem module also provides tools for building other observables such as molecular dipole moment, spin and particle number observables. The theoretical foundation of the quantum chemistry functionality in PennyLane is explained in our white paper.

Building the electronic Hamiltonian

The qchem module provides access to a driver function molecular_hamiltonian() to generate the electronic Hamiltonian in a single call. For example,

import pennylane as qml
from pennylane import numpy as np

symbols = ["H", "H"]
geometry = np.array([[0., 0., -0.66140414], [0., 0., 0.66140414]])
hamiltonian, qubits = qml.qchem.molecular_hamiltonian(symbols, geometry)

where:

  • hamiltonian is the qubit Hamiltonian of the molecule represented as a PennyLane Hamiltonian and

  • qubits is the number of qubits needed to perform the quantum simulation.

The molecular_hamiltonian() function can also be used to construct the molecular Hamiltonian with an external backend that uses the OpenFermion-PySCF plugin interfaced with the electronic structure package PySCF, which requires separate installation. This backend is non-differentiable and can be selected by setting method='pyscf' in molecular_hamiltonian(). Additionally, if the electronic Hamiltonian is built independently using OpenFermion tools, it can be readily converted to a PennyLane observable using the import_operator() function.

Furthermore, the net charge, the spin multiplicity, the atomic basis functions and the active space can also be specified for each backend.

hamiltonian, qubits = qml.qchem.molecular_hamiltonian(
    symbols,
    geometry,
    charge=0,
    mult=1,
    basis='sto-3g',
    method='pyscf',
    active_electrons=2,
    active_orbitals=2
)

Importing molecular structure data

The atomic structure of a molecule can be either defined as an array or imported from an external file using the read_structure() function:

symbols, geometry = qml.qchem.read_structure('h2.xyz')

VQE simulations

The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical computational scheme, where a quantum computer is used to prepare the trial wave function of a molecule and to measure the expectation value of the electronic Hamiltonian, while a classical optimizer is used to find its ground state.

PennyLane supports treating Hamiltonians just like any other observable, and the expectation value of a Hamiltonian can be calculated using qml.expval:

dev = qml.device('default.qubit', wires=4)

symbols = ["H", "H"]
geometry = np.array([[0., 0., -0.66140414], [0., 0., 0.66140414]])
hamiltonian, qubits = qml.qchem.molecular_hamiltonian(symbols, geometry)

@qml.qnode(dev)
def circuit(params):
    qml.BasisState(np.array([1, 1, 0, 0]), wires=[0, 1, 2, 3])
    qml.DoubleExcitation(params, wires=[0, 1, 2, 3])
    return qml.expval(hamiltonian)

params = np.array(0.20885146442480412, requires_grad=True)
circuit(params)
tensor(-1.13618912, requires_grad=True)

The circuit parameter can be optimized using the interface of choice.

Note

For more details on VQE and the quantum chemistry functionality available in qchem, check out the PennyLane quantum chemistry tutorials.

Quantum chemistry functions and classes

PennyLane supports the following quantum chemistry functions and classes.

Molecular integrals and matrices

attraction_integral

Return a function that computes the nuclear attraction integral for two contracted Gaussian functions.

attraction_matrix

Return a function that computes the electron-nuclear attraction matrix for a given set of basis functions.

contracted_norm

Compute the normalization constant for a contracted Gaussian function.

core_matrix

Return a function that computes the core matrix for a given set of basis functions.

dipole_integrals

Return a function that computes the dipole moment integrals over the molecular orbitals.

electron_integrals

Return a function that computes the one- and two-electron integrals in the molecular orbital basis.

electron_repulsion

Compute the electron-electron repulsion integral between four primitive Gaussian functions.

expansion

Compute Hermite Gaussian expansion coefficients recursively for two Gaussian functions.

gaussian_kinetic

Compute the kinetic integral for two primitive Gaussian functions.

gaussian_moment

Compute the one-dimensional multipole moment integral for two primitive Gaussian functions.

gaussian_overlap

Compute overlap integral for two primitive Gaussian functions.

hermite_moment

Compute the Hermite moment integral recursively.

kinetic_integral

Return a function that computes the kinetic integral for two contracted Gaussian functions.

kinetic_matrix

Return a function that computes the kinetic matrix for a given set of basis functions.

mol_density_matrix

Compute the molecular density matrix.

moment_integral

Return a function that computes the multipole moment integral for two contracted Gaussians.

moment_matrix

Return a function that computes the multipole moment matrix for a set of basis functions.

nuclear_attraction

Compute nuclear attraction integral between primitive Gaussian functions.

overlap_integral

Return a function that computes the overlap integral for two contracted Gaussian functions.

overlap_matrix

Return a function that computes the overlap matrix for a given set of basis functions.

primitive_norm

Compute the normalization constant for a primitive Gaussian function.

repulsion_integral

Return a function that computes the electron-electron repulsion integral for four contracted Gaussian functions.

repulsion_tensor

Return a function that computes the electron repulsion tensor for a given set of basis functions.

Differentiable Hartree-Fock

hf_energy

Return a function that computes the Hartree-Fock energy.

nuclear_energy

Return a function that computes the nuclear-repulsion energy.

scf

Return a function that performs the self-consistent-field calculations.

Hartree-Fock with external packages

decompose

Decomposes the molecular Hamiltonian into a linear combination of Pauli operators using OpenFermion tools.

meanfield

Generates a file from which the mean field electronic structure of the molecule can be retrieved.

one_particle

Generates the FermionOperator representing a given one-particle operator required to build many-body qubit observables.

two_particle

Generates the FermionOperator representing a given two-particle operator required to build many-body qubit observables.

Differentiable observables

diff_hamiltonian

Return a function that computes the qubit Hamiltonian.

dipole_moment

Return a function that computes the qubit dipole moment observable.

fermionic_dipole

Return a function that builds the fermionic dipole moment observable.

fermionic_hamiltonian

Return a function that computes the fermionic Hamiltonian.

fermionic_observable

Create a fermionic observable from molecular orbital integrals.

jordan_wigner

Convert a fermionic operator to a qubit operator using the Jordan-Wigner mapping.

molecular_hamiltonian

Generate the qubit Hamiltonian of a molecule.

qubit_observable

Convert a fermionic observable to a PennyLane qubit observable.

simplify

Add together identical terms in the Hamiltonian.

Other observables

dipole_of

Computes the electric dipole moment operator in the Pauli basis.

observable

Builds the fermionic many-body observable whose expectation value can be measured in PennyLane.

particle_number

Compute the particle number observable \(\hat{N}=\sum_\alpha \hat{n}_\alpha\) in the Pauli basis.

spin2

Compute the total spin observable \(\hat{S}^2\).

spinz

Computes the total spin projection observable \(\hat{S}_z\).

Qubit tapering

clifford

Compute a Clifford operator from a set of generators and Pauli-X operators.

optimal_sector

Get the optimal sector which contains the ground state.

paulix_ops

Generate the single qubit Pauli-X operators \(\sigma^{x}_{i}\) for each symmetry \(\tau_j\), such that it anti-commutes with \(\tau_j\) and commutes with all others symmetries \(\tau_{k\neq j}\).

symmetry_generators

Compute the generators \(\{\tau_1, \ldots, \tau_k\}\) for a Hamiltonian over the binary field \(\mathbb{Z}_2\).

taper

Transform a Hamiltonian with a Clifford operator and then taper qubits.

taper_hf

Transform a Hartree-Fock state with a Clifford operator and then taper qubits.

taper_operation

Transform a gate operation with a Clifford operator and then taper qubits.

Utility functions

active_space

Build the active space for a given number of active electrons and active orbitals.

excitations

Generate single and double excitations from a Hartree-Fock reference state.

excitations_to_wires

Map the indices representing the single and double excitations generated with the function excitations() to the wires that the Unitary Coupled-Cluster (UCCSD) template will act on.

factorize

Return the double-factorized form of a two-electron integral tensor.

hf_state

Generate the occupation-number vector representing the Hartree-Fock state.

import_operator

Convert an external operator to a PennyLane operator.

read_structure

Read the structure of the polyatomic system from a file and returns a list with the symbols of the atoms in the molecule and a 1D array with their positions \([x_1, y_1, z_1, x_2, y_2, z_2, \dots]\) in atomic units (Bohr radius = 1).

Molecule class and basis functions

atom_basis_data

Generate default basis set parameters for an atom.

BasisFunction

Create a basis function object.

Molecule

Create a molecule object that stores molecular information and default basis set parameters.

mol_basis_data

Generates default basis set parameters for a molecule.