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Using PennyLane with PySCF and OpenFermion
Using PennyLane with PySCF and OpenFermion
Published: January 02, 2023. Last updated: September 09, 2025.
The quantum chemistry module in PennyLane, qchem
, provides built-in
methods to compute molecular integrals, solve Hartree-Fock equations, and construct
fully-differentiable molecular
Hamiltonians. PennyLane also lets you take advantage of various
external resources and libraries to build upon existing tools. In this demo we will show you how
to integrate PennyLane with PySCF and
OpenFermion to compute molecular integrals,
construct molecular Hamiltonians, and import initial states.
Building molecular Hamiltonians
In PennyLane, Hamiltonians for quantum chemistry are built with the
molecular_hamiltonian()
function by specifying a backend for solving the
Hartree–Fock equations. The default backend is the differentiable Hartree–Fock solver of the
qchem
module. A molecular Hamiltonian can also be constructed with
non-differentiable backends that use the electronic structure package
PySCF or the
OpenFermion-PySCF plugin. These
backends can be selected by setting the keyword argument method='pyscf'
or
method='openfermion'
in molecular_hamiltonian()
. This requires
PySCF
or OpenFermion-PySCF
to be installed by the user depending on the desired backend:
pip install pyscf # for method='pyscf`
pip install openfermionpyscf # for method='openfermion`
For example, the molecular Hamiltonian for a water molecule can be constructed with the pyscf
backend as:
import pennylane as qml
import numpy as np
symbols = ["H", "O", "H"]
geometry = np.array([[-0.0399, -0.0038, 0.0000],
[ 1.5780, 0.8540, 0.0000],
[ 2.7909, -0.5159, 0.0000]])
molecule = qml.qchem.Molecule(symbols, geometry)
H, qubits = qml.qchem.molecular_hamiltonian(molecule, method="pyscf")
print(H)
-46.46418132900196 * I([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]) + 12.412625796786344 * Z(0) + 0.12507005020261677 * (Y(0) @ Z(1) @ Y(2)) + 0.12507005020261677 * (X(0) @ Z(1) @ X(2)) + -6.264577773654327e-05 * (Y(0) @ Z(1) @ Z(2) @ Z(3) @ Y(4)) + -6.264577773654327e-05 * (X(0) @ Z(1) @ Z(2) @ Z(3) @ X(4)) + 0.04274503289428947 * (Y(0) @ Z(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Y(6)) + 0.04274503289428947 * (X(0) @ Z(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ X(6)) + 0.07164199409271013 * (Y(0) @ Z(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 0.07164199409271013 * (X(0) @ Z(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + 0.00027347991087344435 * (Y(0) @ Z(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + 0.00027347991087344435 * (X(0) @ Z(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + 1.653877900171149 * Z(2) + 0.23671148640587403 * (Z(0) @ Z(2)) + 7.60454623320554e-05 * (Y(2) @ Z(3) @ Y(4)) + 7.60454623320554e-05 * (X(2) @ Z(3) @ X(4)) + 1.631270619277116e-05 * (Z(0) @ Y(2) @ Z(3) @ Y(4)) + 1.631270619277116e-05 * (Z(0) @ X(2) @ Z(3) @ X(4)) + -0.121349356204751 * (Y(2) @ Z(3) @ Z(4) @ Z(5) @ Y(6)) + -0.121349356204751 * (X(2) @ Z(3) @ Z(4) @ Z(5) @ X(6)) + -0.02712001981070407 * (Z(0) @ Y(2) @ Z(3) @ Z(4) @ Z(5) @ Y(6)) + -0.02712001981070407 * (Z(0) @ X(2) @ Z(3) @ Z(4) @ Z(5) @ X(6)) + -0.2816231687777095 * (Y(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + -0.2816231687777095 * (X(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + -0.06751923559371835 * (Z(0) @ Y(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + -0.06751923559371835 * (Z(0) @ X(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + -0.0011293550744284084 * (Y(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + -0.0011293550744284084 * (X(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + -0.0002740718912897058 * (Z(0) @ Y(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + -0.0002740718912897058 * (Z(0) @ X(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + 1.203393471851158 * Z(4) + 0.19661468635456458 * (Z(0) @ Z(4)) + 0.00023971851063759317 * (Y(4) @ Z(5) @ Y(6)) + 0.00023971851063759317 * (X(4) @ Z(5) @ X(6)) + 6.0599465976718034e-05 * (Z(0) @ Y(4) @ Z(5) @ Y(6)) + 6.0599465976718034e-05 * (Z(0) @ X(4) @ Z(5) @ X(6)) + 0.001296621110521866 * (Y(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 0.001296621110521866 * (X(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + 0.00029949049119751216 * (Z(0) @ Y(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 0.00029949049119751216 * (Z(0) @ X(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + -0.3693680852999779 * (Y(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + -0.3693680852999779 * (X(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + -0.08684679664918639 * (Z(0) @ Y(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + -0.08684679664918639 * (Z(0) @ X(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + 1.309644597901104 * Z(6) + 0.2416354662230169 * (Z(0) @ Z(6)) + 0.22851845083218353 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 0.22851845083218353 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + 0.05609324457867964 * (Z(0) @ Y(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 0.05609324457867964 * (Z(0) @ X(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + 0.0005075743650135072 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + 0.0005075743650135072 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + 0.0001301053559041618 * (Z(0) @ Y(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + 0.0001301053559041618 * (Z(0) @ X(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + 1.3693425800377725 * Z(8) + 0.27232519926700777 * (Z(0) @ Z(8)) + 0.7829997663616644 * Z(10) + 0.19300118524938328 * (Z(0) @ Z(10)) + 0.00020954442991638893 * (Y(10) @ Z(11) @ Y(12)) + 0.00020954442991638893 * (X(10) @ Z(11) @ X(12)) + 4.255432627101938e-05 * (Z(0) @ Y(10) @ Z(11) @ Y(12)) + 4.255432627101938e-05 * (Z(0) @ X(10) @ Z(11) @ X(12)) + 0.8084689671660954 * Z(12) + 0.21101855800623628 * (Z(0) @ Z(12)) + 12.412625796786344 * Z(1) + 1.1861767319420167 * (Z(0) @ Z(1)) + 0.104331558813001 * (Y(0) @ Y(2)) + 0.104331558813001 * (X(0) @ X(2)) + -6.0952559254996176e-05 * (Y(0) @ Z(2) @ Z(3) @ Y(4)) + -6.0952559254996176e-05 * (X(0) @ Z(2) @ Z(3) @ X(4)) + 0.04588079149822992 * (Y(0) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Y(6)) + 0.04588079149822992 * (X(0) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ X(6)) + 0.05858441547836735 * (Y(0) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 0.05858441547836735 * (X(0) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + 0.00022412880040975609 * (Y(0) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + 0.00022412880040975609 * (X(0) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + 0.12507005020261683 * (Y(1) @ Z(2) @ Y(3)) + 0.104331558813001 * (Z(0) @ Y(1) @ Z(2) @ Y(3)) + 0.12507005020261683 * (X(1) @ Z(2) @ X(3)) + 0.104331558813001 * (Z(0) @ X(1) @ Z(2) @ X(3)) + 0.01458387061862001 * (Y(0) @ X(1) @ X(2) @ Y(3)) + -0.01458387061862001 * (Y(0) @ Y(1) @ X(2) @ X(3)) + -0.01458387061862001 * (X(0) @ X(1) @ Y(2) @ Y(3)) + 0.01458387061862001 * (X(0) @ Y(1) @ Y(2) @ X(3)) + 6.981972855069865e-06 * (Y(0) @ X(1) @ X(3) @ Y(4)) + 6.981972855069865e-06 * (Y(0) @ Y(1) @ Y(3) @ Y(4)) + 6.981972855069865e-06 * (X(0) @ X(1) @ X(3) @ X(4)) + 6.981972855069865e-06 * (X(0) @ Y(1) @ Y(3) @ X(4)) + -0.005652977934002365 * (Y(0) @ X(1) @ X(3) @ Z(4) @ Z(5) @ Y(6)) + -0.005652977934002365 * (Y(0) @ Y(1) @ Y(3) @ Z(4) @ Z(5) @ Y(6)) + -0.005652977934002365 * (X(0) @ X(1) @ X(3) @ Z(4) @ Z(5) @ X(6)) + -0.005652977934002365 * (X(0) @ Y(1) @ Y(3) @ Z(4) @ Z(5) @ X(6)) + -0.00882536009140516 * (Y(0) @ X(1) @ X(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + -0.00882536009140516 * (Y(0) @ Y(1) @ Y(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + -0.00882536009140516 * (X(0) @ X(1) @ X(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + -0.00882536009140516 * (X(0) @ Y(1) @ Y(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + -3.504711775214738e-05 * (Y(0) @ X(1) @ X(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + -3.504711775214738e-05 * (Y(0) @ Y(1) @ Y(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + -3.504711775214738e-05 * (X(0) @ X(1) @ X(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + -3.504711775214738e-05 * (X(0) @ Y(1) @ Y(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + -6.264577773654327e-05 * (Y(1) @ Z(2) @ Z(3) @ Z(4) @ Y(5)) + -6.0952559254996176e-05 * (Z(0) @ Y(1) @ Z(2) @ Z(3) @ Z(4) @ Y(5)) + -6.264577773654327e-05 * (X(1) @ Z(2) @ Z(3) @ Z(4) @ X(5)) + -6.0952559254996176e-05 * (Z(0) @ X(1) @ Z(2) @ Z(3) @ Z(4) @ X(5)) + -6.981972855069865e-06 * (Y(0) @ X(1) @ X(2) @ Z(3) @ Z(4) @ Y(5)) + 6.981972855069865e-06 * (Y(0) @ Y(1) @ X(2) @ Z(3) @ Z(4) @ X(5)) + 6.981972855069865e-06 * (X(0) @ X(1) @ Y(2) @ Z(3) @ Z(4) @ Y(5)) + -6.981972855069865e-06 * (X(0) @ Y(1) @ Y(2) @ Z(3) @ Z(4) @ X(5)) + 0.002745882857363149 * (Y(0) @ X(1) @ X(4) @ Y(5)) + -0.002745882857363149 * (Y(0) @ Y(1) @ X(4) @ X(5)) + -0.002745882857363149 * (X(0) @ X(1) @ Y(4) @ Y(5)) + 0.002745882857363149 * (X(0) @ Y(1) @ Y(4) @ X(5)) + 4.7851295571635575e-06 * (Y(0) @ X(1) @ X(5) @ Y(6)) + 4.7851295571635575e-06 * (Y(0) @ Y(1) @ Y(5) @ Y(6)) + 4.7851295571635575e-06 * (X(0) @ X(1) @ X(5) @ X(6)) + 4.7851295571635575e-06 * (X(0) @ Y(1) @ Y(5) @ X(6)) + 1.5383169868573355e-05 * (Y(0) @ X(1) @ X(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 1.5383169868573355e-05 * (Y(0) @ Y(1) @ Y(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 1.5383169868573355e-05 * (X(0) @ X(1) @ X(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + 1.5383169868573355e-05 * (X(0) @ Y(1) @ Y(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + -0.0038038519664684725 * (Y(0) @ X(1) @ X(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + -0.0038038519664684725 * (Y(0) @ Y(1) @ Y(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + -0.0038038519664684725 * (X(0) @ X(1) @ X(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + -0.0038038519664684725 * (X(0) @ Y(1) @ Y(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + 0.04274503289428947 * (Y(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Y(7)) + 0.04588079149822992 * (Z(0) @ Y(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Y(7)) + 0.04274503289428947 * (X(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ X(7)) + 0.04588079149822992 * (Z(0) @ X(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ X(7)) + 0.005652977934002365 * (Y(0) @ X(1) @ X(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Y(7)) + -0.005652977934002365 * (Y(0) @ Y(1) @ X(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ X(7)) + -0.005652977934002365 * (X(0) @ X(1) @ Y(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Y(7)) + 0.005652977934002365 * (X(0) @ Y(1) @ Y(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ X(7)) + -4.7851295571635575e-06 * (Y(0) @ X(1) @ X(4) @ Z(5) @ Z(6) @ Y(7)) + 4.7851295571635575e-06 * (Y(0) @ Y(1) @ X(4) @ Z(5) @ Z(6) @ X(7)) + 4.7851295571635575e-06 * (X(0) @ X(1) @ Y(4) @ Z(5) @ Z(6) @ Y(7)) + -4.7851295571635575e-06 * (X(0) @ Y(1) @ Y(4) @ Z(5) @ Z(6) @ X(7)) + 0.006887843637461323 * (Y(0) @ X(1) @ X(6) @ Y(7)) + -0.006887843637461323 * (Y(0) @ Y(1) @ X(6) @ X(7)) + -0.006887843637461323 * (X(0) @ X(1) @ Y(6) @ Y(7)) + 0.006887843637461323 * (X(0) @ Y(1) @ Y(6) @ X(7)) + -7.689944472291374e-05 * (Y(0) @ X(1) @ X(7) @ Z(8) @ Z(9) @ Y(10)) + -7.689944472291374e-05 * (Y(0) @ Y(1) @ Y(7) @ Z(8) @ Z(9) @ Y(10)) + -7.689944472291374e-05 * (X(0) @ X(1) @ X(7) @ Z(8) @ Z(9) @ X(10)) + -7.689944472291374e-05 * (X(0) @ Y(1) @ Y(7) @ Z(8) @ Z(9) @ X(10)) + -3.330119571970294e-06 * (Y(0) @ X(1) @ X(7) @ Z(8) @ Z(9) @ 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Y(11) @ X(12)) + 7.373115091407838e-05 * (Z(6) @ Y(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Z(12) @ Y(13)) + 7.373115091407838e-05 * (Z(6) @ X(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Z(12) @ X(13)) + -2.0248930237264964e-05 * (Y(6) @ X(7) @ X(10) @ Z(11) @ Z(12) @ Y(13)) + 2.0248930237264964e-05 * (Y(6) @ Y(7) @ X(10) @ Z(11) @ Z(12) @ X(13)) + 2.0248930237264964e-05 * (X(6) @ X(7) @ Y(10) @ Z(11) @ Z(12) @ Y(13)) + -2.0248930237264964e-05 * (X(6) @ Y(7) @ Y(10) @ Z(11) @ Z(12) @ X(13)) + 0.017367312313811206 * (Y(6) @ X(7) @ X(12) @ Y(13)) + -0.017367312313811206 * (Y(6) @ Y(7) @ X(12) @ X(13)) + -0.017367312313811206 * (X(6) @ X(7) @ Y(12) @ Y(13)) + 0.017367312313811206 * (X(6) @ Y(7) @ Y(12) @ X(13)) + 0.03010926069804785 * (Y(6) @ Z(7) @ Z(9) @ Y(10)) + 0.03010926069804785 * (X(6) @ Z(7) @ Z(9) @ X(10)) + 7.061174268039775e-05 * (Y(6) @ Z(7) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + 7.061174268039775e-05 * (X(6) @ Z(7) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + 0.1814336529902883 * (Z(6) @ Z(9)) + 0.02981808659561775 * (Y(6) @ Z(7) @ Z(8) @ Y(10)) + 0.02981808659561775 * (X(6) @ Z(7) @ Z(8) @ X(10)) + 6.410687728281299e-05 * (Y(6) @ Z(7) @ Z(8) @ Z(10) @ Z(11) @ Y(12)) + 6.410687728281299e-05 * (X(6) @ Z(7) @ Z(8) @ Z(10) @ Z(11) @ X(12)) + 0.00029117410243010373 * (Y(6) @ Y(9) @ Z(10) @ Y(11) @ Z(7) @ Y(8)) + 0.00029117410243010373 * (Y(6) @ X(9) @ Z(10) @ X(11) @ Z(7) @ Y(8)) + 0.00029117410243010373 * (X(6) @ Y(9) @ Z(10) @ Y(11) @ Z(7) @ X(8)) + 0.00029117410243010373 * (X(6) @ X(9) @ Z(10) @ X(11) @ Z(7) @ X(8)) + 6.504865397584347e-06 * (Y(6) @ Y(9) @ Z(10) @ Z(11) @ Z(12) @ Y(13) @ Z(7) @ Y(8)) + 6.504865397584347e-06 * (Y(6) @ X(9) @ Z(10) @ Z(11) @ Z(12) @ X(13) @ Z(7) @ Y(8)) + 6.504865397584347e-06 * (X(6) @ Y(9) @ Z(10) @ Z(11) @ Z(12) @ Y(13) @ Z(7) @ X(8)) + 6.504865397584347e-06 * (X(6) @ X(9) @ Z(10) @ Z(11) @ Z(12) @ X(13) @ Z(7) @ X(8)) + 6.280080983592223e-05 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Z(11) @ Y(12)) + 6.280080983592223e-05 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ Z(11) @ X(12)) + 0.025106395160108385 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10) @ Z(12)) + 0.025106395160108385 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ X(10) @ Z(12)) + 0.1373512666361355 * (Z(6) @ Z(11)) + 0.011308061483328212 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10) @ Z(11)) + 0.011308061483328212 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ X(10) @ Z(11)) + 0.00012759083494219466 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Y(12)) + 0.00012759083494219466 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ X(12)) + 5.6517814404880896e-05 * (Z(6) @ Y(11) @ Z(12) @ Y(13)) + 5.6517814404880896e-05 * (Z(6) @ X(11) @ Z(12) @ X(13)) + -6.479002510627102e-05 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10) @ Y(11) @ Z(12) @ Y(13)) + -6.479002510627102e-05 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10) @ X(11) @ Z(12) @ X(13)) + -6.479002510627102e-05 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ X(10) @ Y(11) @ Z(12) @ Y(13)) + -6.479002510627102e-05 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ X(10) @ X(11) @ Z(12) @ X(13)) + 0.014564757847849817 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ X(11) @ X(12) @ Y(13)) + -0.014564757847849817 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Y(11) @ X(12) @ X(13)) + -0.014564757847849817 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ X(11) @ Y(12) @ Y(13)) + 0.014564757847849817 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Y(11) @ Y(12) @ X(13)) + 0.1513775403533093 * (Z(6) @ Z(13)) + 0.010541637312258564 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10) @ Z(13)) + 0.010541637312258564 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ X(10) @ Z(13)) + -8.181904529692714e-05 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12) @ Z(13)) + -8.181904529692714e-05 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12) @ Z(13)) + 0.1675608365822894 * (Z(7) @ Z(9)) + 0.1195206938218008 * (Z(7) @ Z(11)) + 3.626888416761579e-05 * (Z(7) @ Y(11) @ Z(12) @ Y(13)) + 3.626888416761579e-05 * (Z(7) @ X(11) @ Z(12) @ X(13)) + 0.1340102280394981 * (Z(7) @ Z(13)) + 0.1814336529902883 * (Z(7) @ Z(8)) + 0.02981808659561775 * (Y(7) @ Z(9) @ Z(10) @ Y(11)) + 0.02981808659561775 * (X(7) @ Z(9) @ Z(10) @ X(11)) + 6.410687728281299e-05 * (Y(7) @ Z(9) @ Z(10) @ Z(11) @ Z(12) @ Y(13)) + 6.410687728281299e-05 * (X(7) @ Z(9) @ Z(10) @ Z(11) @ Z(12) @ X(13)) + 0.00029117410243010373 * (Y(7) @ X(8) @ X(9) @ Y(10)) + -0.00029117410243010373 * (Y(7) @ Y(8) @ X(9) @ X(10)) + -0.00029117410243010373 * (X(7) @ X(8) @ Y(9) @ Y(10)) + 0.00029117410243010373 * (X(7) @ Y(8) @ Y(9) @ X(10)) + 6.504865397584347e-06 * (Y(7) @ X(8) @ X(9) @ Z(10) @ Z(11) @ Y(12)) + -6.504865397584347e-06 * (Y(7) @ Y(8) @ X(9) @ Z(10) @ Z(11) @ X(12)) + -6.504865397584347e-06 * (X(7) @ X(8) @ Y(9) @ Z(10) @ Z(11) @ Y(12)) + 6.504865397584347e-06 * (X(7) @ Y(8) @ Y(9) @ Z(10) @ Z(11) @ X(12)) + 0.03010926069804785 * (Y(7) @ Z(8) @ Z(10) @ Y(11)) + 0.03010926069804785 * (X(7) @ Z(8) @ Z(10) @ X(11)) + 7.061174268039775e-05 * (Y(7) @ Z(8) @ Z(10) @ Z(11) @ Z(12) @ Y(13)) + 7.061174268039775e-05 * (X(7) @ Z(8) @ Z(10) @ Z(11) @ Z(12) @ X(13)) + 0.1373512666361355 * (Z(7) @ Z(10)) + 0.011308061483328214 * (Y(7) @ Z(8) @ Z(9) @ Y(11)) + 0.011308061483328214 * (X(7) @ Z(8) @ Z(9) @ X(11)) + 0.00012759083494219466 * (Y(7) @ Z(8) @ Z(9) @ Z(11) @ Z(12) @ Y(13)) + 0.00012759083494219466 * (X(7) @ Z(8) @ Z(9) @ Z(11) @ Z(12) @ X(13)) + 5.6517814404880896e-05 * (Z(7) @ Y(10) @ Z(11) @ Y(12)) + 5.6517814404880896e-05 * (Z(7) @ X(10) @ Z(11) @ X(12)) + -6.479002510627102e-05 * (Y(7) @ Z(8) @ Z(9) @ X(10) @ X(11) @ Y(12)) + 6.479002510627102e-05 * (Y(7) @ Z(8) @ Z(9) @ Y(10) @ X(11) @ X(12)) + 6.479002510627102e-05 * (X(7) @ Z(8) @ Z(9) @ X(10) @ Y(11) @ Y(12)) + -6.479002510627102e-05 * (X(7) @ Z(8) @ Z(9) @ Y(10) @ Y(11) @ X(12)) + -0.014564757847849817 * (Y(7) @ Z(8) @ Z(9) @ X(10) @ X(12) @ Y(13)) + -0.014564757847849817 * (Y(7) @ Z(8) @ Z(9) @ Y(10) @ Y(12) @ Y(13)) + -0.014564757847849817 * (X(7) @ Z(8) @ Z(9) @ X(10) @ X(12) @ X(13)) + -0.014564757847849817 * (X(7) @ Z(8) @ Z(9) @ Y(10) @ Y(12) @ X(13)) + 6.280080983592223e-05 * (Y(7) @ Z(8) @ Z(9) @ Z(10) @ Z(12) @ Y(13)) + 6.280080983592223e-05 * (X(7) @ Z(8) @ Z(9) @ Z(10) @ Z(12) @ X(13)) + 0.025106395160108385 * (Y(7) @ Z(8) @ Z(9) @ Z(10) @ Y(11) @ Z(13)) + 0.025106395160108385 * (X(7) @ Z(8) @ Z(9) @ Z(10) @ X(11) @ Z(13)) + 0.1513775403533093 * (Z(7) @ Z(12)) + 0.010541637312258564 * (Y(7) @ Z(8) @ Z(9) @ Z(10) @ Y(11) @ Z(12)) + 0.010541637312258564 * (X(7) @ Z(8) @ Z(9) @ Z(10) @ X(11) @ Z(12)) + -8.181904529692714e-05 * (Y(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(13)) + -8.181904529692714e-05 * (X(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(13)) + 0.13767060913151413 * (Z(8) @ Z(10)) + 3.1221831943040884e-05 * (Z(8) @ Y(10) @ Z(11) @ Y(12)) + 3.1221831943040884e-05 * (Z(8) @ X(10) @ Z(11) @ X(12)) + 0.14973113556794843 * (Z(8) @ Z(12)) + 0.2200397733437613 * (Z(8) @ Z(9)) + 0.009560047991368666 * (Y(8) @ X(9) @ X(10) @ Y(11)) + -0.009560047991368666 * (Y(8) @ Y(9) @ X(10) @ X(11)) + -0.009560047991368666 * (X(8) @ X(9) @ Y(10) @ Y(11)) + 0.009560047991368666 * (X(8) @ Y(9) @ Y(10) @ X(11)) + -1.2959889312165584e-05 * (Y(8) @ X(9) @ X(11) @ Y(12)) + -1.2959889312165584e-05 * (Y(8) @ Y(9) @ Y(11) @ Y(12)) + -1.2959889312165584e-05 * (X(8) @ X(9) @ X(11) @ X(12)) + -1.2959889312165584e-05 * (X(8) @ Y(9) @ Y(11) @ X(12)) + 1.2959889312165584e-05 * (Y(8) @ X(9) @ X(10) @ Z(11) @ Z(12) @ Y(13)) + -1.2959889312165584e-05 * (Y(8) @ Y(9) @ X(10) @ Z(11) @ Z(12) @ X(13)) + -1.2959889312165584e-05 * (X(8) @ X(9) @ Y(10) @ Z(11) @ Z(12) @ Y(13)) + 1.2959889312165584e-05 * (X(8) @ Y(9) @ Y(10) @ Z(11) @ Z(12) @ X(13)) + 0.006087763625218351 * (Y(8) @ X(9) @ X(12) @ Y(13)) + -0.006087763625218351 * (Y(8) @ Y(9) @ X(12) @ X(13)) + -0.006087763625218351 * (X(8) @ X(9) @ Y(12) @ Y(13)) + 0.006087763625218351 * (X(8) @ Y(9) @ Y(12) @ X(13)) + 0.1472306571228828 * (Z(8) @ Z(11)) + 1.8261942630874717e-05 * (Z(8) @ Y(11) @ Z(12) @ Y(13)) + 1.8261942630874717e-05 * (Z(8) @ X(11) @ Z(12) @ X(13)) + 0.15581889919316677 * (Z(8) @ Z(13)) + 0.13767060913151413 * (Z(9) @ Z(11)) + 3.1221831943040884e-05 * (Z(9) @ Y(11) @ Z(12) @ Y(13)) + 3.1221831943040884e-05 * (Z(9) @ X(11) @ Z(12) @ X(13)) + 0.14973113556794843 * (Z(9) @ Z(13)) + 0.1472306571228828 * (Z(9) @ Z(10)) + 1.8261942630874717e-05 * (Z(9) @ Y(10) @ Z(11) @ Y(12)) + 1.8261942630874717e-05 * (Z(9) @ X(10) @ Z(11) @ X(12)) + 0.15581889919316677 * (Z(9) @ Z(12)) + 0.11270485037291608 * (Z(10) @ Z(12)) + 0.14926575904805806 * (Z(10) @ Z(11)) + 0.0001602192782908234 * (Y(10) @ Y(12)) + 0.0001602192782908234 * (X(10) @ X(12)) + 0.0001602192782908234 * (Z(10) @ Y(11) @ Z(12) @ Y(13)) + 0.0001602192782908234 * (Z(10) @ X(11) @ Z(12) @ X(13)) + 0.028682469959293605 * (Y(10) @ X(11) @ X(12) @ Y(13)) + -0.028682469959293605 * (Y(10) @ Y(11) @ X(12) @ X(13)) + -0.028682469959293605 * (X(10) @ X(11) @ Y(12) @ Y(13)) + 0.028682469959293605 * (X(10) @ Y(11) @ Y(12) @ X(13)) + 0.1413873203322097 * (Z(10) @ Z(13)) + -0.00015554494039127738 * (Y(10) @ Z(11) @ Y(12) @ Z(13)) + -0.00015554494039127738 * (X(10) @ Z(11) @ X(12) @ Z(13)) + 0.11270485037291608 * (Z(11) @ Z(13)) + 0.1413873203322097 * (Z(11) @ Z(12)) + -0.00015554494039127738 * (Y(11) @ Y(13)) + -0.00015554494039127738 * (X(11) @ X(13)) + 0.1543544392544665 * (Z(12) @ Z(13))
This generates a PennyLane Hamiltonian
that can be used in a VQE workflow or
converted to a
sparse matrix
in the computational basis.
Additionally, if you have built your electronic Hamiltonian independently using
OpenFermion tools, it can
be readily converted to a PennyLane observable using the
import_operator()
function. Here is an example:
from openfermion.ops import QubitOperator
H = 0.1 * QubitOperator('X0 X1') + 0.2 * QubitOperator('Z0')
H = qml.qchem.import_operator(H)
print(f'Type: \n {type(H)} \n')
print(f'Hamiltonian: \n {H}')
Type:
<class 'pennylane.ops.op_math.sum.Sum'>
Hamiltonian:
0.1 * (X(0) @ X(1)) + 0.2 * Z(0)
Computing molecular integrals
In order to build a
molecular Hamiltonian, we need
one- and two-electron integrals in the molecular orbital basis. These integrals are used to
construct a fermionic Hamiltonian which is then mapped onto the qubit basis. These molecular
integrals can be computed with the
electron_integrals()
function of PennyLane. Alternatively, the integrals
can be computed with the PySCF package and used in PennyLane
workflows such as building a
fermionic Hamiltonian or
quantum resource estimation.
Let’s use water as an example.
First, we define the PySCF molecule object and run a restricted Hartree-Fock calculation:
from pyscf import gto, ao2mo, scf
mol_pyscf = gto.M(atom = '''H -0.02111417 -0.00201087 0.;
O 0.83504162 0.45191733 0.;
H 1.47688065 -0.27300252 0.''')
rhf = scf.RHF(mol_pyscf)
energy = rhf.kernel()
converged SCF energy = -74.9645256918537
We obtain the molecular integrals one_ao
and two_ao
in the basis of atomic orbitals
by following the example here:
one_ao = mol_pyscf.intor_symmetric('int1e_kin') + mol_pyscf.intor_symmetric('int1e_nuc')
two_ao = mol_pyscf.intor('int2e_sph')
These integrals are then mapped to the basis of molecular orbitals:
one_mo = np.einsum('pi,pq,qj->ij', rhf.mo_coeff, one_ao, rhf.mo_coeff)
two_mo = ao2mo.incore.full(two_ao, rhf.mo_coeff)
Note that the two-electron integral tensor is represented in chemists’ notation. To use it in PennyLane, we need to convert it into the so-called physicists’ notation:
two_mo = np.swapaxes(two_mo, 1, 3)
Let’s now look at an example where these molecular integrals are used to build the fermionic Hamiltonian of water. To do that we also need to compute the nuclear energy contribution:
core_constant = np.array([rhf.energy_nuc()])
We now use the integrals to construct a fermionic Hamiltonian with PennyLane’s powerful tools for creating and manipulating fermionic operators:
H_fermionic = qml.qchem.fermionic_observable(core_constant, one_mo, two_mo)
The Hamiltonian can be mapped to the qubit basis with the jordan_wigner()
function:
H = qml.jordan_wigner(H_fermionic)
Importing initial states
Simulating molecules with quantum algorithms requires defining an initial state that should have non-zero overlap with the molecular ground state. A trivial choice for the initial state is the Hartree-Fock state which is obtained by putting the electrons in the lowest-energy molecular orbitals. For molecules with a complicated electronic structure, the Hartree-Fock state has only a small overlap with the ground state, which makes executing quantum algorithms inefficient.
Initial states obtained from affordable post-Hartree-Fock calculations can be used to make the
quantum workflow more performant. For instance, configuration interaction (CI) and coupled cluster
(CC) calculations with single and double (SD) excitations can be performed using PySCF and the
resulting wave function can be used as the initial state in the quantum algorithm. PennyLane
provides the import_state()
function that takes a PySCF solver object,
extracts the wave function and returns a state vector in the computational basis that can be used
in a quantum circuit. Let’s look at an example.
First, we run CCSD calculations for the hydrogen molecule to obtain the solver object.
from pyscf import gto, scf, cc
mol = gto.M(atom=[['H', (0, 0, 0)], ['H', (0, 0, 0.7)]])
myhf = scf.RHF(mol).run()
mycc = cc.CCSD(myhf).run()
converged SCF energy = -1.11734903499028
E(CCSD) = -1.136189572437634 E_corr = -0.01884053744735375
Then, we use the import_state()
function to obtain the
state vector.
state = qml.qchem.import_state(mycc)
print(state)
[ 0. +0.j 0. +0.j 0. +0.j -0.10467584+0.j
0. +0.j 0. +0.j 0. +0.j 0. +0.j
0. +0.j 0. +0.j 0. +0.j 0. +0.j
0.99450639+0.j 0. +0.j 0. +0.j 0. +0.j]
You can verify that this state is a superposition of the Hartree-Fock state and a doubly-excited state.
Converting fermionic operators
Fermionic operators are commonly used to construct observables for molecules and spin systems.
You can easily convert between fermionic operators created with PennyLane and OpenFermion by using
the from_openfermion()
and to_openfermion()
functions. Let’s
look at some examples. First, we create a fermionic operator with OpenFermion and convert it to a
PennyLane fermionic operator.
from openfermion import FermionOperator
openfermion_op = 0.5 * FermionOperator('0^ 2') + FermionOperator('0 2^')
pennylane_op = qml.from_openfermion(openfermion_op)
print(pennylane_op)
0.5 * a⁺(0) a(2)
+ 1.0 * a(0) a⁺(2)
The resulting operator can be used in PennyLane like any other fermionic object. We now take this PennyLane fermionic operator and convert it back to an OpenFermion operator.
openfermion_op = qml.to_openfermion(pennylane_op)
print(openfermion_op)
1.0 [0 2^] +
0.5 [0^ 2]
The from_openfermion()
and to_openfermion()
functions support
converting several operator types. You can look at the function documentations for more details
and examples.
Conclusions
This tutorial demonstrates how to use PennyLane with external quantum chemistry libraries such as PySCF and OpenFermion.
To summarize:
We can construct molecular Hamiltonians in PennyLane by using a user-installed version of PySCF by passing the argument
method=pyscf
to themolecular_hamiltonian()
function.We can directly use one- and two-electron integrals from PySCF, but we need to convert the tensor containing the two-electron integrals from chemists’ notation to physicists’ notation.
We can easily convert between OpenFermion operators and PennyLane operators using the
from_openfermion()
andto_openfermion()
functions.Finally, we can convert PySCF wave functions to PennyLane state vectors using the
import_state()
function.
About the author
Soran Jahangiri
I am a quantum scientist and software developer working at Xanadu. My work is focused on developing and implementing quantum algorithms in PennyLane.
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