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  • Related contentA brief overview of VQE
  • Related contentGivens rotations for quantum chemistry

Contents

  1. Building molecular Hamiltonians
  2. Computing molecular integrals
  3. Importing initial states
  4. Converting fermionic operators
  5. Conclusions
  6. About the author

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  1. Demos/
  2. Quantum Chemistry/

  3. Using PennyLane with PySCF and OpenFermion

Using PennyLane with PySCF and OpenFermion

Soran Jahangiri

Soran Jahangiri

Published: January 02, 2023. Last updated: January 14, 2026.

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)

-46.46418132900201 * I([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]) + 12.41262579678634 * Z(0) + 0.12507005020262402 * (Y(0) @ Z(1) @ Y(2)) + 0.12507005020262402 * (X(0) @ Z(1) @ X(2)) + -6.26457777372341e-05 * (Y(0) @ Z(1) @ Z(2) @ Z(3) @ Y(4)) + -6.26457777372341e-05 * (X(0) @ Z(1) @ Z(2) @ Z(3) @ X(4)) + 0.04274503289429084 * (Y(0) @ Z(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Y(6)) + 0.04274503289429084 * (X(0) @ Z(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ X(6)) + 0.07164199409272264 * (Y(0) @ Z(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 0.07164199409272264 * (X(0) @ Z(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + 0.00027347991087183724 * (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)) + 0.00027347991087183724 * (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)) + 1.6538779001711268 * Z(2) + 0.23671148640586961 * (Z(0) @ Z(2)) + 7.604546233294205e-05 * (Y(2) @ Z(3) @ Y(4)) + 7.604546233294205e-05 * (X(2) @ Z(3) @ X(4)) + 1.63127061929395e-05 * (Z(0) @ Y(2) @ Z(3) @ Y(4)) + 1.63127061929395e-05 * (Z(0) @ X(2) @ Z(3) @ X(4)) + -0.12134935620473168 * (Y(2) @ Z(3) @ Z(4) @ Z(5) @ Y(6)) + -0.12134935620473168 * (X(2) @ Z(3) @ Z(4) @ Z(5) @ X(6)) + -0.027120019810694736 * (Z(0) @ Y(2) @ Z(3) @ Z(4) @ Z(5) @ Y(6)) + -0.027120019810694736 * (Z(0) @ X(2) @ Z(3) @ Z(4) @ Z(5) @ X(6)) + -0.28162316877774796 * (Y(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + -0.28162316877774796 * (X(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + -0.06751923559372566 * (Z(0) @ Y(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + -0.06751923559372566 * (Z(0) @ X(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + -0.001129355074429036 * (Y(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + -0.001129355074429036 * (X(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + -0.00027407189128962536 * (Z(0) @ Y(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + -0.00027407189128962536 * (Z(0) @ X(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + 1.203393471851102 * Z(4) + 0.1966146863545511 * (Z(0) @ Z(4)) + 0.000239718510638908 * (Y(4) @ Z(5) @ Y(6)) + 0.000239718510638908 * (X(4) @ Z(5) @ X(6)) + 6.059946597708718e-05 * (Z(0) @ Y(4) @ Z(5) @ Y(6)) + 6.059946597708718e-05 * (Z(0) @ X(4) @ Z(5) @ X(6)) + 0.0012966211105193875 * (Y(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 0.0012966211105193875 * (X(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + 0.0002994904911967585 * (Z(0) @ Y(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 0.0002994904911967585 * (Z(0) @ X(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + -0.36936808530000886 * (Y(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + -0.36936808530000886 * (X(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + -0.08684679664918522 * (Z(0) @ Y(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + -0.08684679664918522 * (Z(0) @ X(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + 1.309644597901184 * Z(6) + 0.24163546622303628 * (Z(0) @ Z(6)) + 0.2285184508320912 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 0.2285184508320912 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + 0.056093244578668724 * (Z(0) @ Y(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 0.056093244578668724 * (Z(0) @ X(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + 0.0005075743650109076 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + 0.0005075743650109076 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + 0.00013010535590340672 * (Z(0) @ Y(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + 0.00013010535590340672 * (Z(0) @ X(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + 1.3693425800377712 * Z(8) + 0.27232519926700777 * (Z(0) @ Z(8)) + 0.7829997663616036 * Z(10) + 0.19300118524936874 * (Z(0) @ Z(10)) + 0.00020954442991563105 * (Y(10) @ Z(11) @ Y(12)) + 0.00020954442991563105 * (X(10) @ Z(11) @ X(12)) + 4.255432627081463e-05 * (Z(0) @ Y(10) @ Z(11) @ Y(12)) + 4.255432627081463e-05 * (Z(0) @ X(10) @ Z(11) @ X(12)) + 0.808468967166152 * Z(12) + 0.2110185580062499 * (Z(0) @ Z(12)) + 12.412625796786338 * Z(1) + 1.186176731942017 * (Z(0) @ Z(1)) + 0.10433155881299869 * (Y(0) @ Y(2)) + 0.10433155881299869 * (X(0) @ X(2)) + -6.09525592552795e-05 * (Y(0) @ Z(2) @ Z(3) @ Y(4)) + -6.09525592552795e-05 * (X(0) @ Z(2) @ Z(3) @ X(4)) + 0.045880791498227685 * (Y(0) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Y(6)) + 0.045880791498227685 * (X(0) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ X(6)) + 0.058584415478375396 * (Y(0) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 0.058584415478375396 * (X(0) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + 0.00022412880040952474 * (Y(0) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + 0.00022412880040952474 * (X(0) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + 0.12507005020262396 * (Y(1) @ Z(2) @ Y(3)) + 0.10433155881299869 * (Z(0) @ Y(1) @ Z(2) @ Y(3)) + 0.12507005020262396 * (X(1) @ Z(2) @ X(3)) + 0.10433155881299869 * (Z(0) @ X(1) @ Z(2) @ X(3)) + 0.014583870618619372 * (Y(0) @ X(1) @ X(2) @ Y(3)) + -0.014583870618619372 * (Y(0) @ Y(1) @ X(2) @ X(3)) + -0.014583870618619372 * (X(0) @ X(1) @ Y(2) @ Y(3)) + 0.014583870618619372 * (X(0) @ Y(1) @ Y(2) @ X(3)) + 6.981972855109131e-06 * (Y(0) @ X(1) @ X(3) @ Y(4)) + 6.981972855109131e-06 * (Y(0) @ Y(1) @ Y(3) @ Y(4)) + 6.981972855109131e-06 * (X(0) @ X(1) @ X(3) @ X(4)) + 6.981972855109131e-06 * (X(0) @ Y(1) @ Y(3) @ X(4)) + -0.005652977934001602 * (Y(0) @ X(1) @ X(3) @ Z(4) @ Z(5) @ Y(6)) + -0.005652977934001602 * (Y(0) @ Y(1) @ Y(3) @ Z(4) @ Z(5) @ Y(6)) + -0.005652977934001602 * (X(0) @ X(1) @ X(3) @ Z(4) @ Z(5) @ X(6)) + -0.005652977934001602 * (X(0) @ Y(1) @ Y(3) @ Z(4) @ Z(5) @ X(6)) + -0.00882536009140613 * (Y(0) @ X(1) @ X(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + -0.00882536009140613 * (Y(0) @ Y(1) @ Y(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + -0.00882536009140613 * (X(0) @ X(1) @ X(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + -0.00882536009140613 * (X(0) @ Y(1) @ Y(3) @ Z(4) @ Z(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + -3.5047117752109114e-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)) + -3.5047117752109114e-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)) + -3.5047117752109114e-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)) + -3.5047117752109114e-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)) + -6.264577773723409e-05 * (Y(1) @ Z(2) @ Z(3) @ Z(4) @ Y(5)) + -6.09525592552795e-05 * (Z(0) @ Y(1) @ Z(2) @ Z(3) @ Z(4) @ Y(5)) + -6.264577773723409e-05 * (X(1) @ Z(2) @ Z(3) @ Z(4) @ X(5)) + -6.09525592552795e-05 * (Z(0) @ X(1) @ Z(2) @ Z(3) @ Z(4) @ X(5)) + -6.981972855109131e-06 * (Y(0) @ X(1) @ X(2) @ Z(3) @ Z(4) @ Y(5)) + 6.981972855109131e-06 * (Y(0) @ Y(1) @ X(2) @ Z(3) @ Z(4) @ X(5)) + 6.981972855109131e-06 * (X(0) @ X(1) @ Y(2) @ Z(3) @ Z(4) @ Y(5)) + -6.981972855109131e-06 * (X(0) @ Y(1) @ Y(2) @ Z(3) @ Z(4) @ X(5)) + 0.0027458828573625614 * (Y(0) @ X(1) @ X(4) @ Y(5)) + -0.0027458828573625614 * (Y(0) @ Y(1) @ X(4) @ X(5)) + -0.0027458828573625614 * (X(0) @ X(1) @ Y(4) @ Y(5)) + 0.0027458828573625614 * (X(0) @ Y(1) @ Y(4) @ X(5)) + 4.785129557192126e-06 * (Y(0) @ X(1) @ X(5) @ Y(6)) + 4.785129557192126e-06 * (Y(0) @ Y(1) @ Y(5) @ Y(6)) + 4.785129557192126e-06 * (X(0) @ X(1) @ X(5) @ X(6)) + 4.785129557192126e-06 * (X(0) @ Y(1) @ Y(5) @ X(6)) + 1.538316986855677e-05 * (Y(0) @ X(1) @ X(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 1.538316986855677e-05 * (Y(0) @ Y(1) @ Y(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10)) + 1.538316986855677e-05 * (X(0) @ X(1) @ X(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + 1.538316986855677e-05 * (X(0) @ Y(1) @ Y(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ X(10)) + -0.0038038519664682835 * (Y(0) @ X(1) @ X(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + -0.0038038519664682835 * (Y(0) @ Y(1) @ Y(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12)) + -0.0038038519664682835 * (X(0) @ X(1) @ X(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + -0.0038038519664682835 * (X(0) @ Y(1) @ Y(5) @ Z(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + 0.04274503289429085 * (Y(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Y(7)) + 0.045880791498227685 * (Z(0) @ Y(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Y(7)) + 0.04274503289429085 * (X(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ X(7)) + 0.045880791498227685 * (Z(0) @ X(1) @ Z(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ X(7)) + 0.005652977934001602 * (Y(0) @ X(1) @ X(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Y(7)) + -0.005652977934001602 * (Y(0) @ Y(1) @ X(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ X(7)) + -0.005652977934001602 * (X(0) @ X(1) @ Y(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ Y(7)) + 0.005652977934001602 * (X(0) @ Y(1) @ Y(2) @ Z(3) @ Z(4) @ Z(5) @ Z(6) @ X(7)) + -4.785129557192126e-06 * (Y(0) @ X(1) @ X(4) @ Z(5) @ Z(6) @ Y(7)) + 4.785129557192126e-06 * (Y(0) @ Y(1) @ X(4) @ Z(5) @ Z(6) @ X(7)) + 4.785129557192126e-06 * (X(0) @ X(1) @ Y(4) @ Z(5) @ Z(6) @ Y(7)) + -4.785129557192126e-06 * (X(0) @ Y(1) @ Y(4) @ Z(5) @ Z(6) @ X(7)) + 0.006887843637461973 * (Y(0) @ X(1) @ X(6) @ Y(7)) + -0.006887843637461973 * (Y(0) @ Y(1) @ X(6) @ X(7)) + -0.006887843637461973 * (X(0) @ X(1) @ Y(6) @ Y(7)) + 0.006887843637461973 * (X(0) @ Y(1) @ Y(6) @ X(7)) + -7.689944472336908e-05 * (Y(0) @ X(1) @ X(7) @ Z(8) @ Z(9) @ Y(10)) + -7.689944472336908e-05 * (Y(0) @ Y(1) @ Y(7) @ Z(8) @ Z(9) @ Y(10)) + -7.689944472336908e-05 * (X(0) @ X(1) @ X(7) @ Z(8) @ Z(9) @ X(10)) + -7.689944472336908e-05 * (X(0) @ Y(1) @ Y(7) @ Z(8) @ Z(9) @ X(10)) + -3.3301195719931796e-06 * (Y(0) @ X(1) @ X(7) @ Z(8) @ Z(9) @ Z(10) @ 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@ Y(12)) + 7.061174268006357e-05 * (X(6) @ Z(7) @ Z(9) @ Z(10) @ Z(11) @ X(12)) + 0.18143365299029895 * (Z(6) @ Z(9)) + 0.029818086595610534 * (Y(6) @ Z(7) @ Z(8) @ Y(10)) + 0.029818086595610534 * (X(6) @ Z(7) @ Z(8) @ X(10)) + 6.41068772824359e-05 * (Y(6) @ Z(7) @ Z(8) @ Z(10) @ Z(11) @ Y(12)) + 6.41068772824359e-05 * (X(6) @ Z(7) @ Z(8) @ Z(10) @ Z(11) @ X(12)) + 0.0002911741024316397 * (Y(6) @ Y(9) @ Z(10) @ Y(11) @ Z(7) @ Y(8)) + 0.0002911741024316397 * (Y(6) @ X(9) @ Z(10) @ X(11) @ Z(7) @ Y(8)) + 0.0002911741024316397 * (X(6) @ Y(9) @ Z(10) @ Y(11) @ Z(7) @ X(8)) + 0.0002911741024316397 * (X(6) @ X(9) @ Z(10) @ X(11) @ Z(7) @ X(8)) + 6.504865397627281e-06 * (Y(6) @ Y(9) @ Z(10) @ Z(11) @ Z(12) @ Y(13) @ Z(7) @ Y(8)) + 6.504865397627281e-06 * (Y(6) @ X(9) @ Z(10) @ Z(11) @ Z(12) @ X(13) @ Z(7) @ Y(8)) + 6.504865397627281e-06 * (X(6) @ Y(9) @ Z(10) @ Z(11) @ Z(12) @ Y(13) @ Z(7) @ X(8)) + 6.504865397627281e-06 * (X(6) @ X(9) @ Z(10) @ Z(11) @ Z(12) @ X(13) @ Z(7) @ X(8)) + 6.28008098356742e-05 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Z(11) @ Y(12)) + 6.28008098356742e-05 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ Z(11) @ X(12)) + 0.025106395160105592 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10) @ Z(12)) + 0.025106395160105592 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ X(10) @ Z(12)) + 0.13735126663612907 * (Z(6) @ Z(11)) + 0.011308061483323941 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10) @ Z(11)) + 0.011308061483323941 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ X(10) @ Z(11)) + 0.0001275908349419338 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Y(12)) + 0.0001275908349419338 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ X(12)) + 5.6517814404867344e-05 * (Z(6) @ Y(11) @ Z(12) @ Y(13)) + 5.6517814404867344e-05 * (Z(6) @ X(11) @ Z(12) @ X(13)) + -6.479002510625793e-05 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10) @ Y(11) @ Z(12) @ Y(13)) + -6.479002510625793e-05 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10) @ X(11) @ Z(12) @ X(13)) + -6.479002510625793e-05 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ X(10) @ Y(11) @ Z(12) @ Y(13)) + -6.479002510625793e-05 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ X(10) @ X(11) @ Z(12) @ X(13)) + 0.014564757847846286 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ X(11) @ X(12) @ Y(13)) + -0.014564757847846286 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Y(11) @ X(12) @ X(13)) + -0.014564757847846286 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ X(11) @ Y(12) @ Y(13)) + 0.014564757847846286 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Y(11) @ Y(12) @ X(13)) + 0.15137754035331907 * (Z(6) @ Z(13)) + 0.010541637312259301 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Y(10) @ Z(13)) + 0.010541637312259301 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ X(10) @ Z(13)) + -8.18190452969234e-05 * (Y(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(12) @ Z(13)) + -8.18190452969234e-05 * (X(6) @ Z(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(12) @ Z(13)) + 0.16756083658229884 * (Z(7) @ Z(9)) + 0.11952069382180176 * (Z(7) @ Z(11)) + 3.6268884167454e-05 * (Z(7) @ Y(11) @ Z(12) @ Y(13)) + 3.6268884167454e-05 * (Z(7) @ X(11) @ Z(12) @ X(13)) + 0.1340102280395115 * (Z(7) @ Z(13)) + 0.18143365299029895 * (Z(7) @ Z(8)) + 0.029818086595610534 * (Y(7) @ Z(9) @ Z(10) @ Y(11)) + 0.029818086595610534 * (X(7) @ Z(9) @ Z(10) @ X(11)) + 6.41068772824359e-05 * (Y(7) @ Z(9) @ Z(10) @ Z(11) @ Z(12) @ Y(13)) + 6.41068772824359e-05 * (X(7) @ Z(9) @ Z(10) @ Z(11) @ Z(12) @ X(13)) + 0.0002911741024316397 * (Y(7) @ X(8) @ X(9) @ Y(10)) + -0.0002911741024316397 * (Y(7) @ Y(8) @ X(9) @ X(10)) + -0.0002911741024316397 * (X(7) @ X(8) @ Y(9) @ Y(10)) + 0.0002911741024316397 * (X(7) @ Y(8) @ Y(9) @ X(10)) + 6.504865397627281e-06 * (Y(7) @ X(8) @ X(9) @ Z(10) @ Z(11) @ Y(12)) + -6.504865397627281e-06 * (Y(7) @ Y(8) @ X(9) @ Z(10) @ Z(11) @ X(12)) + -6.504865397627281e-06 * (X(7) @ X(8) @ Y(9) @ Z(10) @ Z(11) @ Y(12)) + 6.504865397627281e-06 * (X(7) @ Y(8) @ Y(9) @ Z(10) @ Z(11) @ X(12)) + 0.03010926069804217 * (Y(7) @ Z(8) @ Z(10) @ Y(11)) + 0.03010926069804217 * (X(7) @ Z(8) @ Z(10) @ X(11)) + 7.061174268006357e-05 * (Y(7) @ Z(8) @ Z(10) @ Z(11) @ Z(12) @ Y(13)) + 7.061174268006357e-05 * (X(7) @ Z(8) @ Z(10) @ Z(11) @ Z(12) @ X(13)) + 0.13735126663612907 * (Z(7) @ Z(10)) + 0.011308061483323941 * (Y(7) @ Z(8) @ Z(9) @ Y(11)) + 0.011308061483323941 * (X(7) @ Z(8) @ Z(9) @ X(11)) + 0.0001275908349419338 * (Y(7) @ Z(8) @ Z(9) @ Z(11) @ Z(12) @ Y(13)) + 0.0001275908349419338 * (X(7) @ Z(8) @ Z(9) @ Z(11) @ Z(12) @ X(13)) + 5.6517814404867344e-05 * (Z(7) @ Y(10) @ Z(11) @ Y(12)) + 5.6517814404867344e-05 * (Z(7) @ X(10) @ Z(11) @ X(12)) + -6.479002510625793e-05 * (Y(7) @ Z(8) @ Z(9) @ X(10) @ X(11) @ Y(12)) + 6.479002510625793e-05 * (Y(7) @ Z(8) @ Z(9) @ Y(10) @ X(11) @ X(12)) + 6.479002510625793e-05 * (X(7) @ Z(8) @ Z(9) @ X(10) @ Y(11) @ Y(12)) + -6.479002510625793e-05 * (X(7) @ Z(8) @ Z(9) @ Y(10) @ Y(11) @ X(12)) + -0.014564757847846287 * (Y(7) @ Z(8) @ Z(9) @ X(10) @ X(12) @ Y(13)) + -0.014564757847846287 * (Y(7) @ Z(8) @ Z(9) @ Y(10) @ Y(12) @ Y(13)) + -0.014564757847846287 * (X(7) @ Z(8) @ Z(9) @ X(10) @ X(12) @ X(13)) + -0.014564757847846287 * (X(7) @ Z(8) @ Z(9) @ Y(10) @ Y(12) @ X(13)) + 6.28008098356742e-05 * (Y(7) @ Z(8) @ Z(9) @ Z(10) @ Z(12) @ Y(13)) + 6.28008098356742e-05 * (X(7) @ Z(8) @ Z(9) @ Z(10) @ Z(12) @ X(13)) + 0.025106395160105592 * (Y(7) @ Z(8) @ Z(9) @ Z(10) @ Y(11) @ Z(13)) + 0.025106395160105592 * (X(7) @ Z(8) @ Z(9) @ Z(10) @ X(11) @ Z(13)) + 0.15137754035331907 * (Z(7) @ Z(12)) + 0.010541637312259301 * (Y(7) @ Z(8) @ Z(9) @ Z(10) @ Y(11) @ Z(12)) + 0.010541637312259301 * (X(7) @ Z(8) @ Z(9) @ Z(10) @ X(11) @ Z(12)) + -8.18190452969234e-05 * (Y(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ Y(13)) + -8.18190452969234e-05 * (X(7) @ Z(8) @ Z(9) @ Z(10) @ Z(11) @ X(13)) + 0.13767060913150608 * (Z(8) @ Z(10)) + 3.1221831942918505e-05 * (Z(8) @ Y(10) @ Z(11) @ Y(12)) + 3.1221831942918505e-05 * (Z(8) @ X(10) @ Z(11) @ X(12)) + 0.1497311355679549 * (Z(8) @ Z(12)) + 0.2200397733437613 * (Z(8) @ Z(9)) + 0.009560047991369048 * (Y(8) @ X(9) @ X(10) @ Y(11)) + -0.009560047991369048 * (Y(8) @ Y(9) @ X(10) @ X(11)) + -0.009560047991369048 * (X(8) @ X(9) @ Y(10) @ Y(11)) + 0.009560047991369048 * (X(8) @ Y(9) @ Y(10) @ X(11)) + -1.2959889312167333e-05 * (Y(8) @ X(9) @ X(11) @ Y(12)) + -1.2959889312167333e-05 * (Y(8) @ Y(9) @ Y(11) @ Y(12)) + -1.2959889312167333e-05 * (X(8) @ X(9) @ X(11) @ X(12)) + -1.2959889312167333e-05 * (X(8) @ Y(9) @ Y(11) @ X(12)) + 1.2959889312167333e-05 * (Y(8) @ X(9) @ X(10) @ Z(11) @ Z(12) @ Y(13)) + -1.2959889312167333e-05 * (Y(8) @ Y(9) @ X(10) @ Z(11) @ Z(12) @ X(13)) + -1.2959889312167333e-05 * (X(8) @ X(9) @ Y(10) @ Z(11) @ Z(12) @ Y(13)) + 1.2959889312167333e-05 * (X(8) @ Y(9) @ Y(10) @ Z(11) @ Z(12) @ X(13)) + 0.006087763625219276 * (Y(8) @ X(9) @ X(12) @ Y(13)) + -0.006087763625219276 * (Y(8) @ Y(9) @ X(12) @ X(13)) + -0.006087763625219276 * (X(8) @ X(9) @ Y(12) @ Y(13)) + 0.006087763625219276 * (X(8) @ Y(9) @ Y(12) @ X(13)) + 0.14723065712287514 * (Z(8) @ Z(11)) + 1.826194263075155e-05 * (Z(8) @ Y(11) @ Z(12) @ Y(13)) + 1.826194263075155e-05 * (Z(8) @ X(11) @ Z(12) @ X(13)) + 0.15581889919317415 * (Z(8) @ Z(13)) + 0.13767060913150608 * (Z(9) @ Z(11)) + 3.1221831942918505e-05 * (Z(9) @ Y(11) @ Z(12) @ Y(13)) + 3.1221831942918505e-05 * (Z(9) @ X(11) @ Z(12) @ X(13)) + 0.1497311355679549 * (Z(9) @ Z(13)) + 0.14723065712287514 * (Z(9) @ Z(10)) + 1.826194263075155e-05 * (Z(9) @ Y(10) @ Z(11) @ Y(12)) + 1.826194263075155e-05 * (Z(9) @ X(10) @ Z(11) @ X(12)) + 0.15581889919317415 * (Z(9) @ Z(12)) + 0.11270485037291339 * (Z(10) @ Z(12)) + 0.14926575904805267 * (Z(10) @ Z(11)) + 0.00016021927829049307 * (Y(10) @ Y(12)) + 0.00016021927829049307 * (X(10) @ X(12)) + 0.00016021927829049307 * (Z(10) @ Y(11) @ Z(12) @ Y(13)) + 0.00016021927829049307 * (Z(10) @ X(11) @ Z(12) @ X(13)) + 0.028682469959295236 * (Y(10) @ X(11) @ X(12) @ Y(13)) + -0.028682469959295236 * (Y(10) @ Y(11) @ X(12) @ X(13)) + -0.028682469959295236 * (X(10) @ X(11) @ Y(12) @ Y(13)) + 0.028682469959295236 * (X(10) @ Y(11) @ Y(12) @ X(13)) + 0.14138732033220863 * (Z(10) @ Z(13)) + -0.0001555449403911595 * (Y(10) @ Z(11) @ Y(12) @ Z(13)) + -0.0001555449403911595 * (X(10) @ Z(11) @ X(12) @ Z(13)) + 0.11270485037291339 * (Z(11) @ Z(13)) + 0.14138732033220863 * (Z(11) @ Z(12)) + -0.0001555449403911595 * (Y(11) @ Y(13)) + -0.0001555449403911595 * (X(11) @ X(13)) + 0.15435443925447362 * (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 = -74.9645256918536

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:

  1. We can construct molecular Hamiltonians in PennyLane by using a user-installed version of PySCF by passing the argument method=pyscf to the molecular_hamiltonian() function.

  2. 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.

  3. We can easily convert between OpenFermion operators and PennyLane operators using the from_openfermion() and to_openfermion() functions.

  4. Finally, we can convert PySCF wave functions to PennyLane state vectors using the import_state() function.

About the author

Soran Jahangiri
Soran Jahangiri

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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Building molecular Hamiltonians

A brief overview of VQE

Givens rotations for quantum chemistry

Adaptive circuits for quantum chemistry

Measurement optimization

VQE in different spin sectors

Qubit tapering

VQE with parallel QPUs with Rigetti

Estimating observables with classical shadows in the Pauli basis

Fermionic operators

PennyLane

PennyLane is a cross-platform Python library for quantum computing, quantum machine learning, and quantum chemistry. Built by researchers, for research. Created with ❤️ by Xanadu.

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