Molecular Hamiltonian#

Central to quantum chemical calculations for molecules is the fermionic spin-free nonrelativistic two-body Hamiltonian in second quantized form in the absence of external fields. Under the Born-Oppenheimer approximation, the nuclear repulsion energy can be taken to be constant, which allows us to write the molecular Hamiltonian as:

\[\hat{H} = \sum_{pq} h_{pq} a^\dagger_p a_q + \frac12 \sum_{pqrs} g_{pqrs} a^\dagger_p a^\dagger_r a_s a_q \]

The integrals \(h_{pq}\) and \(g_{pqrs}\) are the one- and two-electron integrals in atomic units. For spin-orbitals \(\phi_j\) that make up the basis, the integrals are [1]:

\[h_{pq} = \int \phi^*_p(\mathbf{x}_1)\left( -\frac12 \nabla^2 - \sum_A \frac{Z_A}{r_{A}} \right) \phi_q(\mathbf{x}_1) d\mathbf{x}_1\]

\[g_{pqrs} = \int \int \phi^*_p(\mathbf{x}_1)\phi^*_r(\mathbf{x}_2) \frac{1}{r_{12}} \phi_s(\mathbf{x}_2)\phi_q(\mathbf{x}_1) dx_1 dx_2\]

Fermionic Hamiltonian#

The process for obtaining the molecular integrals using PySCF has been given in an earlier example. Qibochem then uses these integrals and the OpenFermion package to construct the second quantized fermionic Hamiltonian for the molecular system in terms of creation and annihilation operators.

from qibochem.driver.molecule import Molecule

mol = Molecule([('H', (0.0, 0.0, 0.0)), ('H', (0.0, 0.0, 0.74804))])
mol.run_pyscf()
mol_ferm_ham = mol.hamiltonian("ferm") # or mol.hamiltonian("f")
print(mol_ferm_ham.terms)

Output:

{(): 0.7074183344740923, ((0, 1), (0, 0)): -1.248461959132892, ((1, 1), (1, 0)): -1.248461959132892, ((2, 1), (2, 0)): -0.48007161818330846, ((3, 1), (3, 0)): -0.48007161818330846, ((0, 1), (0, 1), (0, 0), (0, 0)): 0.3366109237586995, ((0, 1), (0, 1), (2, 0), (2, 0)): 0.09083064962340165, ((0, 1), (1, 1), (1, 0), (0, 0)): 0.3366109237586995, ((0, 1), (1, 1), (3, 0), (2, 0)): 0.09083064962340165, ((0, 1), (2, 1), (0, 0), (2, 0)): 0.09083064962340165, ((0, 1), (2, 1), (2, 0), (0, 0)): 0.33115823068165495, ((0, 1), (3, 1), (1, 0), (2, 0)): 0.09083064962340165, ((0, 1), (3, 1), (3, 0), (0, 0)): 0.33115823068165495, ((1, 1), (0, 1), (0, 0), (1, 0)): 0.3366109237586995, ((1, 1), (0, 1), (2, 0), (3, 0)): 0.09083064962340165, ((1, 1), (1, 1), (1, 0), (1, 0)): 0.3366109237586995, ((1, 1), (1, 1), (3, 0), (3, 0)): 0.09083064962340165, ((1, 1), (2, 1), (0, 0), (3, 0)): 0.09083064962340165, ((1, 1), (2, 1), (2, 0), (1, 0)): 0.33115823068165495, ((1, 1), (3, 1), (1, 0), (3, 0)): 0.09083064962340165, ((1, 1), (3, 1), (3, 0), (1, 0)): 0.33115823068165495, ((2, 1), (0, 1), (0, 0), (2, 0)): 0.3311582306816552, ((2, 1), (0, 1), (2, 0), (0, 0)): 0.09083064962340165, ((2, 1), (1, 1), (1, 0), (2, 0)): 0.3311582306816552, ((2, 1), (1, 1), (3, 0), (0, 0)): 0.09083064962340165, ((2, 1), (2, 1), (0, 0), (0, 0)): 0.09083064962340165, ((2, 1), (2, 1), (2, 0), (2, 0)): 0.348087115228365, ((2, 1), (3, 1), (1, 0), (0, 0)): 0.09083064962340165, ((2, 1), (3, 1), (3, 0), (2, 0)): 0.348087115228365, ((3, 1), (0, 1), (0, 0), (3, 0)): 0.3311582306816552, ((3, 1), (0, 1), (2, 0), (1, 0)): 0.09083064962340165, ((3, 1), (1, 1), (1, 0), (3, 0)): 0.3311582306816552, ((3, 1), (1, 1), (3, 0), (1, 0)): 0.09083064962340165, ((3, 1), (2, 1), (0, 0), (1, 0)): 0.09083064962340165, ((3, 1), (2, 1), (2, 0), (3, 0)): 0.348087115228365, ((3, 1), (3, 1), (1, 0), (1, 0)): 0.09083064962340165, ((3, 1), (3, 1), (3, 0), (3, 0)): 0.348087115228365}

Fermion to Qubit mapping#

Next, in order to run chemistry simulations, the fermionic Hamiltonian must first be mapped onto a qubit Hamiltonian. Qibochem supports the Jordan-Wigner and Bravyi-Kitaev mapping schemes (as implemented in OpenFermion), and this can be specified using the keyword argument ferm_qubit_map as follows:

mol_qubit_ham = mol.hamiltonian("qubit", ferm_qubit_map="jw") # or just mol.hamiltonian("q")
print(mol_qubit_ham.terms)

Output:

{(): -0.10728041160866736, ((0, 'Z'),): 0.17018261181714206, ((1, 'Z'),): 0.17018261181714206, ((2, 'Z'),): -0.21975065439248248, ((3, 'Z'),): -0.21975065439248248, ((0, 'Z'), (1, 'Z')): 0.16830546187934975, ((0, 'Z'), (2, 'Z')): 0.1201637905291267, ((0, 'Z'), (3, 'Z')): 0.16557911534082753, ((1, 'Z'), (2, 'Z')): 0.16557911534082753, ((1, 'Z'), (3, 'Z')): 0.1201637905291267, ((2, 'Z'), (3, 'Z')): 0.1740435576141825, ((0, 'X'), (1, 'X'), (2, 'Y'), (3, 'Y')): -0.045415324811700825, ((0, 'X'), (1, 'Y'), (2, 'Y'), (3, 'X')): 0.045415324811700825, ((0, 'Y'), (1, 'X'), (2, 'X'), (3, 'Y')): 0.045415324811700825, ((0, 'Y'), (1, 'Y'), (2, 'X'), (3, 'X')): -0.045415324811700825}

Qibo SymbolicHamiltonian#

Lastly, to carry out quantum simulations of the molecular electronic structure using Qibo, the qubit Hamiltonian can be returned as a Qibo SymbolicHamiltonian:

mol_sym_ham = mol.hamiltonian("sym")     # or mol.hamiltonian("s")
print(mol_sym_ham.form)

-0.107280411608667 - 0.0454153248117008*X0*X1*Y2*Y3 + 0.0454153248117008*X0*Y1*Y2*X3 + 0.0454153248117008*Y0*X1*X2*Y3 - 0.0454153248117008*Y0*Y1*X2*X3 + 0.170182611817142*Z0 + 0.16830546187935*Z0*Z1 + 0.120163790529127*Z0*Z2 + 0.165579115340828*Z0*Z3 + 0.170182611817142*Z1 + 0.165579115340828*Z1*Z2 + 0.120163790529127*Z1*Z3 - 0.219750654392482*Z2 + 0.174043557614182*Z2*Z3 - 0.219750654392482*Z3

By default, the molecular Hamiltonian is returned as a SymbolicHamiltonian, i.e. if no arguments are given in mol.hamiltonian(). In addition, if HF embedding has been applied, the embedded values of the one-/two- electron integrals will be used to construct the molecular Hamiltonian as well.

Otherwise, using the "f"/"ferm" and "q"/"qubit" arguments will return the molecular Hamiltonian as an OpenFermion FermionOperator and QubitOperator respectively. Additional information about the data structure of these two classes can be found here.

References