Source code for pennylane.qchem.openfermion_obs
# Copyright 2018-2020 Xanadu Quantum Technologies Inc.
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
# http://www.apache.org/licenses/LICENSE-2.0
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""This module contains functions to construct many-body observables with ``OpenFermion-PySCF``.
"""
# pylint: disable=too-many-arguments, too-few-public-methods, too-many-branches, unused-variable
# pylint: disable=consider-using-generator, protected-access
import os
import numpy as np
import pennylane as qml
from pennylane.operation import active_new_opmath
# Bohr-Angstrom correlation coefficient (https://physics.nist.gov/cgi-bin/cuu/Value?bohrrada0)
bohr_angs = 0.529177210903
def _import_of():
"""Import openfermion and openfermionpyscf."""
try:
# pylint: disable=import-outside-toplevel, unused-import, multiple-imports
import openfermion, openfermionpyscf
except ImportError as Error:
raise ImportError(
"This feature requires openfermionpyscf. "
"It can be installed with: pip install openfermionpyscf."
) from Error
return openfermion, openfermionpyscf
[docs]def observable(fermion_ops, init_term=0, mapping="jordan_wigner", wires=None):
r"""Builds the fermionic many-body observable whose expectation value can be
measured in PennyLane.
The second-quantized operator of the fermionic many-body system can combine one-particle
and two-particle operators as in the case of electronic Hamiltonians :math:`\hat{H}`:
.. math::
\hat{H} = \sum_{\alpha, \beta} \langle \alpha \vert \hat{t}^{(1)} +
\cdots + \hat{t}^{(n)} \vert \beta \rangle ~ \hat{c}_\alpha^\dagger \hat{c}_\beta
+ \frac{1}{2} \sum_{\alpha, \beta, \gamma, \delta}
\langle \alpha, \beta \vert \hat{v}^{(1)} + \cdots + \hat{v}^{(n)}
\vert \gamma, \delta \rangle ~ \hat{c}_\alpha^\dagger \hat{c}_\beta^\dagger
\hat{c}_\gamma \hat{c}_\delta
In the latter equations the indices :math:`\alpha, \beta, \gamma, \delta` run over the
basis of single-particle states. The operators :math:`\hat{c}^\dagger` and :math:`\hat{c}`
are the particle creation and annihilation operators, respectively.
:math:`\langle \alpha \vert \hat{t} \vert \beta \rangle` denotes the matrix element of
the single-particle operator :math:`\hat{t}` entering the observable. For example,
in electronic structure calculations, this is the case for the kinetic energy operator,
the nuclei Coulomb potential, or any other external fields included in the Hamiltonian.
On the other hand, :math:`\langle \alpha, \beta \vert \hat{v} \vert \gamma, \delta \rangle`
denotes the matrix element of the two-particle operator :math:`\hat{v}`, for example, the
Coulomb interaction between the electrons.
- The observable is built by adding the operators
:math:`\sum_{\alpha, \beta} t_{\alpha\beta}^{(i)}
\hat{c}_\alpha^\dagger \hat{c}_\beta` and
:math:`\frac{1}{2} \sum_{\alpha, \beta, \gamma, \delta}
v_{\alpha\beta\gamma\delta}^{(i)}
\hat{c}_\alpha^\dagger \hat{c}_\beta^\dagger \hat{c}_\gamma \hat{c}_\delta`.
- Second-quantized operators contributing to the
many-body observable must be represented using the `FermionOperator
<https://github.com/quantumlib/OpenFermion/blob/master/docs/
tutorials/intro_to_openfermion.ipynb>`_ data structure as implemented in OpenFermion.
See the functions :func:`~.one_particle` and :func:`~.two_particle` to build the
FermionOperator representations of one-particle and two-particle operators.
- The function uses tools of `OpenFermion <https://github.com/quantumlib/OpenFermion>`_
to map the resulting fermionic Hamiltonian to the basis of Pauli matrices via the
Jordan-Wigner or Bravyi-Kitaev transformation. Finally, the qubit operator is converted
to a PennyLane observable by the function :func:`~.convert_observable`.
Args:
fermion_ops (list[FermionOperator]): list containing the FermionOperator data structures
representing the one-particle and/or two-particle operators entering the many-body
observable
init_term (float): Any quantity required to initialize the many-body observable. For
example, this can be used to pass the nuclear-nuclear repulsion energy :math:`V_{nn}`
which is typically included in the electronic Hamiltonian of molecules.
mapping (str): Specifies the fermion-to-qubit mapping. Input values can
be ``'jordan_wigner'`` or ``'bravyi_kitaev'``.
wires (Wires, list, tuple, dict): Custom wire mapping used to convert the qubit operator
to an observable measurable in a PennyLane ansatz.
For types Wires/list/tuple, each item in the iterable represents a wire label
corresponding to the qubit number equal to its index.
For type dict, only int-keyed dict (for qubit-to-wire conversion) is accepted.
If None, will use identity map (e.g. 0->0, 1->1, ...).
Returns:
pennylane.Hamiltonian: the fermionic-to-qubit transformed observable
**Example**
>>> t = FermionOperator("0^ 0", 0.5) + FermionOperator("1^ 1", 0.25)
>>> v = FermionOperator("1^ 0^ 0 1", -0.15) + FermionOperator("2^ 0^ 2 0", 0.3)
>>> print(observable([t, v], mapping="jordan_wigner"))
(0.2625) [I0]
+ (-0.1375) [Z0]
+ (-0.0875) [Z1]
+ (-0.0375) [Z0 Z1]
+ (0.075) [Z2]
+ (-0.075) [Z0 Z2]
"""
openfermion, _ = _import_of()
if mapping.strip().lower() not in ("jordan_wigner", "bravyi_kitaev"):
raise TypeError(
f"The '{mapping}' transformation is not available. \n "
f"Please set 'mapping' to 'jordan_wigner' or 'bravyi_kitaev'."
)
# Initialize the FermionOperator
mb_obs = openfermion.ops.FermionOperator("") * init_term
for ops in fermion_ops:
if not isinstance(ops, openfermion.ops.FermionOperator):
raise TypeError(
f"Elements in the lists are expected to be of type 'FermionOperator'; got {type(ops)}"
)
mb_obs += ops
# Map the fermionic operator to a qubit operator
if mapping.strip().lower() == "bravyi_kitaev":
return qml.qchem.convert.import_operator(
openfermion.transforms.bravyi_kitaev(mb_obs), wires=wires
)
return qml.qchem.convert.import_operator(
openfermion.transforms.jordan_wigner(mb_obs), wires=wires
)
[docs]def one_particle(matrix_elements, core=None, active=None, cutoff=1.0e-12):
r"""Generates the `FermionOperator <https://github.com/quantumlib/OpenFermion/blob/master/docs/
tutorials/intro_to_openfermion.ipynb>`_ representing a given one-particle operator
required to build many-body qubit observables.
Second quantized one-particle operators are expanded in the basis of single-particle
states as
.. math::
\hat{T} = \sum_{\alpha, \beta} \langle \alpha \vert \hat{t} \vert \beta \rangle
[\hat{c}_{\alpha\uparrow}^\dagger \hat{c}_{\beta\uparrow} +
\hat{c}_{\alpha\downarrow}^\dagger \hat{c}_{\beta\downarrow}].
In the equation above the indices :math:`\alpha, \beta` run over the basis of spatial
orbitals :math:`\phi_\alpha(r)`. Since the operator :math:`\hat{t}` acts only on the
spatial coordinates, the spin quantum numbers are indicated explicitly with the up/down arrows.
The operators :math:`\hat{c}^\dagger` and :math:`\hat{c}` are the particle creation
and annihilation operators, respectively, and
:math:`\langle \alpha \vert \hat{t} \vert \beta \rangle` denotes the matrix elements of
the operator :math:`\hat{t}`
.. math::
\langle \alpha \vert \hat{t} \vert \beta \rangle = \int dr ~ \phi_\alpha^*(r)
\hat{t}(r) \phi_\beta(r).
If an active space is defined (see :func:`~.active_space`), the summation indices
run over the active orbitals and the contribution due to core orbitals is computed as
:math:`t_\mathrm{core} = 2 \sum_{\alpha\in \mathrm{core}}
\langle \alpha \vert \hat{t} \vert \alpha \rangle`.
Args:
matrix_elements (array[float]): 2D NumPy array with the matrix elements
:math:`\langle \alpha \vert \hat{t} \vert \beta \rangle`
core (list): indices of core orbitals, i.e., the orbitals that are
not correlated in the many-body wave function
active (list): indices of active orbitals, i.e., the orbitals used to
build the correlated many-body wave function
cutoff (float): Cutoff value for including matrix elements. The
matrix elements with absolute value less than ``cutoff`` are neglected.
Returns:
FermionOperator: an instance of OpenFermion's ``FermionOperator`` representing the
one-particle operator :math:`\hat{T}`.
**Example**
>>> import numpy as np
>>> matrix_elements = np.array([[-1.27785301e+00, 0.00000000e+00],
... [ 1.52655666e-16, -4.48299696e-01]])
>>> t_op = one_particle(matrix_elements)
>>> print(t_op)
-1.277853006156875 [0^ 0] +
-1.277853006156875 [1^ 1] +
-0.44829969610163756 [2^ 2] +
-0.44829969610163756 [3^ 3]
"""
openfermion, _ = _import_of()
orbitals = matrix_elements.shape[0]
if matrix_elements.ndim != 2:
raise ValueError(
f"'matrix_elements' must be a 2D array; got matrix_elements.ndim = {matrix_elements.ndim}"
)
if not core:
t_core = 0
else:
if any(i > orbitals - 1 or i < 0 for i in core):
raise ValueError(
f"Indices of core orbitals must be between 0 and {orbitals - 1}; got core = {core}"
)
# Compute contribution due to core orbitals
t_core = 2 * np.sum(matrix_elements[np.array(core), np.array(core)])
if active is None:
if not core:
active = list(range(orbitals))
else:
active = [i for i in range(orbitals) if i not in core]
if any(i > orbitals - 1 or i < 0 for i in active):
raise ValueError(
f"Indices of active orbitals must be between 0 and {orbitals - 1}; got active = {active}"
)
# Indices of the matrix elements with absolute values >= cutoff
indices = np.nonzero(np.abs(matrix_elements) >= cutoff)
# Single out the indices of active orbitals
num_indices = len(indices[0])
pairs = [
[indices[0][i], indices[1][i]]
for i in range(num_indices)
if all(indices[j][i] in active for j in range(len(indices)))
]
# Build the FermionOperator representing T
t_op = openfermion.ops.FermionOperator("") * t_core
for pair in pairs:
alpha, beta = pair
element = matrix_elements[alpha, beta]
# spin-up term
a = 2 * active.index(alpha)
b = 2 * active.index(beta)
t_op += openfermion.ops.FermionOperator(((a, 1), (b, 0)), element)
# spin-down term
t_op += openfermion.ops.FermionOperator(((a + 1, 1), (b + 1, 0)), element)
return t_op
[docs]def two_particle(matrix_elements, core=None, active=None, cutoff=1.0e-12):
r"""Generates the `FermionOperator <https://github.com/quantumlib/OpenFermion/blob/master/docs/
tutorials/intro_to_openfermion.ipynb>`_ representing a given two-particle operator
required to build many-body qubit observables.
Second quantized two-particle operators are expanded in the basis of single-particle
states as
.. math::
\hat{V} = \frac{1}{2} \sum_{\alpha, \beta, \gamma, \delta}
\langle \alpha, \beta \vert \hat{v} \vert \gamma, \delta \rangle
~ &[& \hat{c}_{\alpha\uparrow}^\dagger \hat{c}_{\beta\uparrow}^\dagger
\hat{c}_{\gamma\uparrow} \hat{c}_{\delta\uparrow} + \hat{c}_{\alpha\uparrow}^\dagger
\hat{c}_{\beta\downarrow}^\dagger \hat{c}_{\gamma\downarrow} \hat{c}_{\delta\uparrow} \\
&+& \hat{c}_{\alpha\downarrow}^\dagger \hat{c}_{\beta\uparrow}^\dagger
\hat{c}_{\gamma\uparrow} \hat{c}_{\delta\downarrow} + \hat{c}_{\alpha\downarrow}^\dagger
\hat{c}_{\beta\downarrow}^\dagger \hat{c}_{\gamma\downarrow} \hat{c}_{\delta\downarrow}~].
In the equation above the indices :math:`\alpha, \beta, \gamma, \delta` run over the basis
of spatial orbitals :math:`\phi_\alpha(r)`. Since the operator :math:`v` acts only on the
spatial coordinates the spin quantum numbers are indicated explicitly with the up/down arrows.
The operators :math:`\hat{c}^\dagger` and :math:`\hat{c}` are the particle creation and
annihilation operators, respectively, and
:math:`\langle \alpha, \beta \vert \hat{v} \vert \gamma, \delta \rangle` denotes the
matrix elements of the operator :math:`\hat{v}`
.. math::
\langle \alpha, \beta \vert \hat{v} \vert \gamma, \delta \rangle =
\int dr_1 \int dr_2 ~ \phi_\alpha^*(r_1) \phi_\beta^*(r_2) ~\hat{v}(r_1, r_2)~
\phi_\gamma(r_2) \phi_\delta(r_1).
If an active space is defined (see :func:`~.active_space`), the summation indices
run over the active orbitals and the contribution due to core orbitals is computed as
.. math::
&& \hat{V}_\mathrm{core} = v_\mathrm{core} +
\sum_{\alpha, \beta \in \mathrm{active}} \sum_{i \in \mathrm{core}}
(2 \langle i, \alpha \vert \hat{v} \vert \beta, i \rangle -
\langle i, \alpha \vert \hat{v} \vert i, \beta \rangle)~
[\hat{c}_{\alpha\uparrow}^\dagger \hat{c}_{\beta\uparrow} +
\hat{c}_{\alpha\downarrow}^\dagger \hat{c}_{\beta\downarrow}] \\
&& v_\mathrm{core} = \sum_{\alpha,\beta \in \mathrm{core}}
[2 \langle \alpha, \beta \vert \hat{v} \vert \beta, \alpha \rangle -
\langle \alpha, \beta \vert \hat{v} \vert \alpha, \beta \rangle].
Args:
matrix_elements (array[float]): 4D NumPy array with the matrix elements
:math:`\langle \alpha, \beta \vert \hat{v} \vert \gamma, \delta \rangle`
core (list): indices of core orbitals, i.e., the orbitals that are
not correlated in the many-body wave function
active (list): indices of active orbitals, i.e., the orbitals used to
build the correlated many-body wave function
cutoff (float): Cutoff value for including matrix elements. The
matrix elements with absolute value less than ``cutoff`` are neglected.
Returns:
FermionOperator: an instance of OpenFermion's ``FermionOperator`` representing the
two-particle operator :math:`\hat{V}`.
**Example**
>>> import numpy as np
>>> matrix_elements = np.array([[[[ 6.82389533e-01, -1.45716772e-16],
... [-2.77555756e-17, 1.79000576e-01]],
... [[-2.77555756e-17, 1.79000576e-16],
... [ 6.70732778e-01, 0.00000000e+00]]],
... [[[-1.45716772e-16, 6.70732778e-16],
... [ 1.79000576e-01, -8.32667268e-17]],
... [[ 1.79000576e-16, -8.32667268e-17],
... [ 0.00000000e+00, 7.05105632e-01]]]])
>>> v_op = two_particle(matrix_elements)
>>> print(v_op)
0.3411947665 [0^ 0^ 0 0] +
0.089500288 [0^ 0^ 2 2] +
0.3411947665 [0^ 1^ 1 0] +
0.089500288 [0^ 1^ 3 2] +
0.335366389 [0^ 2^ 2 0] +
0.335366389 [0^ 3^ 3 0] +
0.3411947665 [1^ 0^ 0 1] +
0.089500288 [1^ 0^ 2 3] +
0.3411947665 [1^ 1^ 1 1] +
0.089500288 [1^ 1^ 3 3] +
0.335366389 [1^ 2^ 2 1] +
0.335366389 [1^ 3^ 3 1] +
0.089500288 [2^ 0^ 2 0] +
0.089500288 [2^ 1^ 3 0] +
0.352552816 [2^ 2^ 2 2] +
0.352552816 [2^ 3^ 3 2] +
0.089500288 [3^ 0^ 2 1] +
0.089500288 [3^ 1^ 3 1] +
0.352552816 [3^ 2^ 2 3] +
0.352552816 [3^ 3^ 3 3]
"""
openfermion, _ = _import_of()
orbitals = matrix_elements.shape[0]
if matrix_elements.ndim != 4:
raise ValueError(
f"'matrix_elements' must be a 4D array; got 'matrix_elements.ndim = ' {matrix_elements.ndim}"
)
if not core:
v_core = 0
else:
if any(i > orbitals - 1 or i < 0 for i in core):
raise ValueError(
f"Indices of core orbitals must be between 0 and {orbitals - 1}; got core = {core}"
)
# Compute the contribution of core orbitals
v_core = sum(
[
2 * matrix_elements[alpha, beta, beta, alpha]
- matrix_elements[alpha, beta, alpha, beta]
for alpha in core
for beta in core
]
)
if active is None:
if not core:
active = list(range(orbitals))
else:
active = [i for i in range(orbitals) if i not in core]
if any(i > orbitals - 1 or i < 0 for i in active):
raise ValueError(
f"Indices of active orbitals must be between 0 and {orbitals - 1}; got active = {active}"
)
# Indices of the matrix elements with absolute values >= cutoff
indices = np.nonzero(np.abs(matrix_elements) >= cutoff)
# Single out the indices of active orbitals
num_indices = len(indices[0])
quads = [
[indices[0][i], indices[1][i], indices[2][i], indices[3][i]]
for i in range(num_indices)
if all(indices[j][i] in active for j in range(len(indices)))
]
# Build the FermionOperator representing V
v_op = openfermion.ops.FermionOperator("") * v_core
# add renormalized (due to core orbitals) "one-particle" operators
if core:
for alpha in active:
for beta in active:
element = 2 * np.sum(
matrix_elements[np.array(core), alpha, beta, np.array(core)]
) - np.sum(matrix_elements[np.array(core), alpha, np.array(core), beta])
# up-up term
a = 2 * active.index(alpha)
b = 2 * active.index(beta)
v_op += openfermion.ops.FermionOperator(((a, 1), (b, 0)), element)
# down-down term
v_op += openfermion.ops.FermionOperator(((a + 1, 1), (b + 1, 0)), element)
# add two-particle operators
for quad in quads:
alpha, beta, gamma, delta = quad
element = matrix_elements[alpha, beta, gamma, delta]
# up-up-up-up term
a = 2 * active.index(alpha)
b = 2 * active.index(beta)
g = 2 * active.index(gamma)
d = 2 * active.index(delta)
v_op += openfermion.ops.FermionOperator(((a, 1), (b, 1), (g, 0), (d, 0)), 0.5 * element)
# up-down-down-up term
b = 2 * active.index(beta) + 1
g = 2 * active.index(gamma) + 1
v_op += openfermion.ops.FermionOperator(((a, 1), (b, 1), (g, 0), (d, 0)), 0.5 * element)
# down-up-up-down term
a = 2 * active.index(alpha) + 1
b = 2 * active.index(beta)
g = 2 * active.index(gamma)
d = 2 * active.index(delta) + 1
v_op += openfermion.ops.FermionOperator(((a, 1), (b, 1), (g, 0), (d, 0)), 0.5 * element)
# down-down-down-down term
b = 2 * active.index(beta) + 1
g = 2 * active.index(gamma) + 1
v_op += openfermion.ops.FermionOperator(((a, 1), (b, 1), (g, 0), (d, 0)), 0.5 * element)
return v_op
[docs]def dipole_of(
symbols,
coordinates,
name="molecule",
charge=0,
mult=1,
basis="sto-3g",
package="pyscf",
core=None,
active=None,
mapping="jordan_wigner",
cutoff=1.0e-12,
outpath=".",
wires=None,
):
r"""Computes the electric dipole moment operator in the Pauli basis.
The second quantized dipole moment operator :math:`\hat{D}` of a molecule is given by
.. math::
\hat{D} = -\sum_{\alpha, \beta} \langle \alpha \vert \hat{{\bf r}} \vert \beta \rangle
[\hat{c}_{\alpha\uparrow}^\dagger \hat{c}_{\beta\uparrow} +
\hat{c}_{\alpha\downarrow}^\dagger \hat{c}_{\beta\downarrow}] + \hat{D}_\mathrm{n}.
In the equation above, the indices :math:`\alpha, \beta` run over the basis of Hartree-Fock
molecular orbitals and the operators :math:`\hat{c}^\dagger` and :math:`\hat{c}` are the
electron creation and annihilation operators, respectively. The matrix elements of the
position operator :math:`\hat{{\bf r}}` are computed as
.. math::
\langle \alpha \vert \hat{{\bf r}} \vert \beta \rangle = \sum_{i, j}
C_{\alpha i}^*C_{\beta j} \langle i \vert \hat{{\bf r}} \vert j \rangle,
where :math:`\vert i \rangle` is the wave function of the atomic orbital,
:math:`C_{\alpha i}` are the coefficients defining the molecular orbitals,
and :math:`\langle i \vert \hat{{\bf r}} \vert j \rangle`
is the representation of operator :math:`\hat{{\bf r}}` in the atomic basis.
The contribution of the nuclei to the dipole operator is given by
.. math::
\hat{D}_\mathrm{n} = \sum_{i=1}^{N_\mathrm{atoms}} Z_i {\bf R}_i \hat{I},
where :math:`Z_i` and :math:`{\bf R}_i` denote, respectively, the atomic number and the
nuclear coordinates of the :math:`i`-th atom of the molecule.
Args:
symbols (list[str]): symbols of the atomic species in the molecule
coordinates (array[float]): 1D array with the atomic positions in Cartesian
coordinates. The coordinates must be given in atomic units and the size of the array
should be ``3*N`` where ``N`` is the number of atoms.
name (str): name of the molecule
charge (int): charge of the molecule
mult (int): spin multiplicity :math:`\mathrm{mult}=N_\mathrm{unpaired} + 1` of the
Hartree-Fock (HF) state based on the number of unpaired electrons occupying the
HF orbitals
basis (str): Atomic basis set used to represent the molecular orbitals. Basis set
availability per element can be found
`here <www.psicode.org/psi4manual/master/basissets_byelement.html#apdx-basiselement>`_
package (str): quantum chemistry package (pyscf) used to solve the
mean field electronic structure problem
core (list): indices of core orbitals
active (list): indices of active orbitals
mapping (str): transformation (``'jordan_wigner'`` or ``'bravyi_kitaev'``) used to
map the fermionic operator to the Pauli basis
cutoff (float): Cutoff value for including the matrix elements
:math:`\langle \alpha \vert \hat{{\bf r}} \vert \beta \rangle`. The matrix elements
with absolute value less than ``cutoff`` are neglected.
outpath (str): path to the directory containing output files
wires (Wires, list, tuple, dict): Custom wire mapping used to convert the qubit operator
to an observable measurable in a PennyLane ansatz.
For types Wires/list/tuple, each item in the iterable represents a wire label
corresponding to the qubit number equal to its index.
For type dict, only int-keyed dict (for qubit-to-wire conversion) is accepted.
If None, will use identity map (e.g. 0->0, 1->1, ...).
Returns:
list[pennylane.Hamiltonian]: the qubit observables corresponding to the components
:math:`\hat{D}_x`, :math:`\hat{D}_y` and :math:`\hat{D}_z` of the dipole operator in
atomic units.
**Example**
>>> symbols = ["H", "H", "H"]
>>> coordinates = np.array([0.028, 0.054, 0.0, 0.986, 1.610, 0.0, 1.855, 0.002, 0.0])
>>> dipole_obs = dipole_of(symbols, coordinates, charge=1)
>>> print(dipole_obs)
[<Hamiltonian: terms=18, wires=[0, 1, 2, 3, 4, 5]>,
<Hamiltonian: terms=18, wires=[0, 1, 2, 3, 4, 5]>,
<Hamiltonian: terms=1, wires=[0]>]
>>> print(dipole_obs[0]) # x-component of D
(0.24190977644628117) [Z4]
+ (0.24190977644628117) [Z5]
+ (0.4781123173263878) [Z0]
+ (0.4781123173263878) [Z1]
+ (0.714477906181248) [Z2]
+ (0.714477906181248) [Z3]
+ (-0.3913638489487808) [Y0 Z1 Y2]
+ (-0.3913638489487808) [X0 Z1 X2]
+ (-0.3913638489487808) [Y1 Z2 Y3]
+ (-0.3913638489487808) [X1 Z2 X3]
+ (-0.1173495878099553) [Y2 Z3 Y4]
+ (-0.1173495878099553) [X2 Z3 X4]
+ (-0.1173495878099553) [Y3 Z4 Y5]
+ (-0.1173495878099553) [X3 Z4 X5]
+ (0.26611147045300276) [Y0 Z1 Z2 Z3 Y4]
+ (0.26611147045300276) [X0 Z1 Z2 Z3 X4]
+ (0.26611147045300276) [Y1 Z2 Z3 Z4 Y5]
+ (0.26611147045300276) [X1 Z2 Z3 Z4 X5]
"""
openfermion, _ = _import_of()
atomic_numbers = {
"H": 1,
"He": 2,
"Li": 3,
"Be": 4,
"B": 5,
"C": 6,
"N": 7,
"O": 8,
"F": 9,
"Ne": 10,
}
if mult != 1:
raise ValueError(
f"Currently, this functionality is constrained to Hartree-Fock states with spin multiplicity = 1;"
f" got multiplicity 2S+1 = {mult}"
)
for i in symbols:
if i not in atomic_numbers:
raise ValueError(
f"Currently, only first- or second-row elements of the periodic table are supported;"
f" got element {i}"
)
hf_file = qml.qchem.meanfield(symbols, coordinates, name, charge, mult, basis, package, outpath)
hf = openfermion.MolecularData(filename=hf_file.strip())
# Load dipole matrix elements in the atomic basis
# pylint: disable=import-outside-toplevel
from pyscf import gto
mol = gto.M(
atom=hf.geometry, basis=hf.basis, charge=hf.charge, spin=0.5 * (hf.multiplicity - 1)
)
dip_ao = mol.intor_symmetric("int1e_r", comp=3).real
# Transform dipole matrix elements to the MO basis
n_orbs = hf.n_orbitals
c_hf = hf.canonical_orbitals
dip_mo = np.zeros((3, n_orbs, n_orbs))
for comp in range(3):
for alpha in range(n_orbs):
for beta in range(alpha + 1):
dip_mo[comp, alpha, beta] = c_hf[:, alpha] @ dip_ao[comp] @ c_hf[:, beta]
dip_mo[comp] += dip_mo[comp].T - np.diag(np.diag(dip_mo[comp]))
# Compute the nuclear contribution
dip_n = np.zeros(3)
for comp in range(3):
for i, symb in enumerate(symbols):
dip_n[comp] += atomic_numbers[symb] * coordinates[3 * i + comp]
# Build the observable
dip = []
for i in range(3):
fermion_obs = one_particle(dip_mo[i], core=core, active=active, cutoff=cutoff)
dip.append(observable([-fermion_obs], init_term=dip_n[i], mapping=mapping, wires=wires))
return dip
[docs]def meanfield(
symbols,
coordinates,
name="molecule",
charge=0,
mult=1,
basis="sto-3g",
package="pyscf",
outpath=".",
): # pylint: disable=too-many-arguments
r"""Generates a file from which the mean field electronic structure
of the molecule can be retrieved.
This function uses OpenFermion-PySCF plugins to
perform the Hartree-Fock (HF) calculation for the polyatomic system using the quantum
chemistry packages ``PySCF``. The mean field electronic
structure is saved in an hdf5-formatted file.
The charge of the molecule can be given to simulate cationic/anionic systems.
Also, the spin multiplicity can be input to determine the number of unpaired electrons
occupying the HF orbitals as illustrated in the figure below.
|
.. figure:: ../../_static/qchem/hf_references.png
:align: center
:width: 50%
|
Args:
symbols (list[str]): symbols of the atomic species in the molecule
coordinates (array[float]): 1D array with the atomic positions in Cartesian
coordinates. The coordinates must be given in atomic units and the size of the array
should be ``3*N`` where ``N`` is the number of atoms.
name (str): molecule label
charge (int): net charge of the system
mult (int): Spin multiplicity :math:`\mathrm{mult}=N_\mathrm{unpaired} + 1` for
:math:`N_\mathrm{unpaired}` unpaired electrons occupying the HF orbitals.
Possible values for ``mult`` are :math:`1, 2, 3, \ldots`. If not specified,
a closed-shell HF state is assumed.
basis (str): Atomic basis set used to represent the HF orbitals. Basis set
availability per element can be found
`here <www.psicode.org/psi4manual/master/basissets_byelement.html#apdx-basiselement>`_
package (str): Quantum chemistry package used to solve the Hartree-Fock equations.
outpath (str): path to output directory
Returns:
str: absolute path to the file containing the mean field electronic structure
**Example**
>>> symbols, coordinates = (['H', 'H'], np.array([0., 0., -0.66140414, 0., 0., 0.66140414]))
>>> meanfield(symbols, coordinates, name="h2")
./h2_pyscf_sto-3g
"""
openfermion, openfermionpyscf = _import_of()
if coordinates.size != 3 * len(symbols):
raise ValueError(
f"The size of the array 'coordinates' has to be 3*len(symbols) = {3 * len(symbols)};"
f" got 'coordinates.size' = {coordinates.size}"
)
package = package.strip().lower()
if package not in "pyscf":
error_message = (
f"Integration with quantum chemistry package '{package}' is not available. \n Please set"
f" 'package' to 'pyscf'."
)
raise TypeError(error_message)
filename = name + "_" + package.lower() + "_" + basis.strip()
path_to_file = os.path.join(outpath.strip(), filename)
geometry = [
[symbol, tuple(np.array(coordinates)[3 * i : 3 * i + 3] * bohr_angs)]
for i, symbol in enumerate(symbols)
]
molecule = openfermion.MolecularData(geometry, basis, mult, charge, filename=path_to_file)
if package == "pyscf":
# pylint: disable=import-outside-toplevel
from openfermionpyscf import run_pyscf
run_pyscf(molecule, run_scf=1, verbose=0)
return path_to_file
[docs]def decompose(hf_file, mapping="jordan_wigner", core=None, active=None):
r"""Decomposes the molecular Hamiltonian into a linear combination of Pauli operators using
OpenFermion tools.
This function uses OpenFermion functions to build the second-quantized electronic Hamiltonian
of the molecule and map it to the Pauli basis using the Jordan-Wigner or Bravyi-Kitaev
transformation.
Args:
hf_file (str): absolute path to the hdf5-formatted file with the
Hartree-Fock electronic structure
mapping (str): Specifies the transformation to map the fermionic Hamiltonian to the
Pauli basis. Input values can be ``'jordan_wigner'`` or ``'bravyi_kitaev'``.
core (list): indices of core orbitals, i.e., the orbitals that are
not correlated in the many-body wave function
active (list): indices of active orbitals, i.e., the orbitals used to
build the correlated many-body wave function
Returns:
QubitOperator: an instance of OpenFermion's ``QubitOperator``
**Example**
>>> decompose('./pyscf/sto-3g/h2', mapping='bravyi_kitaev')
(-0.04207897696293986+0j) [] + (0.04475014401986122+0j) [X0 Z1 X2] +
(0.04475014401986122+0j) [X0 Z1 X2 Z3] +(0.04475014401986122+0j) [Y0 Z1 Y2] +
(0.04475014401986122+0j) [Y0 Z1 Y2 Z3] +(0.17771287459806262+0j) [Z0] +
(0.17771287459806265+0j) [Z0 Z1] +(0.1676831945625423+0j) [Z0 Z1 Z2] +
(0.1676831945625423+0j) [Z0 Z1 Z2 Z3] +(0.12293305054268105+0j) [Z0 Z2] +
(0.12293305054268105+0j) [Z0 Z2 Z3] +(0.1705973832722409+0j) [Z1] +
(-0.2427428049645989+0j) [Z1 Z2 Z3] +(0.1762764080276107+0j) [Z1 Z3] +
(-0.2427428049645989+0j) [Z2]
"""
openfermion, _ = _import_of()
# loading HF data from the hdf5 file
molecule = openfermion.MolecularData(filename=hf_file.strip())
# getting the terms entering the second-quantized Hamiltonian
terms_molecular_hamiltonian = molecule.get_molecular_hamiltonian(
occupied_indices=core, active_indices=active
)
# generating the fermionic Hamiltonian
fermionic_hamiltonian = openfermion.transforms.get_fermion_operator(terms_molecular_hamiltonian)
mapping = mapping.strip().lower()
if mapping not in ("jordan_wigner", "bravyi_kitaev"):
raise TypeError(
f"The '{mapping}' transformation is not available. \n "
f"Please set 'mapping' to 'jordan_wigner' or 'bravyi_kitaev'."
)
# fermionic-to-qubit transformation of the Hamiltonian
if mapping == "bravyi_kitaev":
return openfermion.transforms.bravyi_kitaev(fermionic_hamiltonian)
return openfermion.transforms.jordan_wigner(fermionic_hamiltonian)
[docs]def molecular_hamiltonian(
symbols,
coordinates,
name="molecule",
charge=0,
mult=1,
basis="sto-3g",
method="dhf",
active_electrons=None,
active_orbitals=None,
mapping="jordan_wigner",
outpath=".",
wires=None,
alpha=None,
coeff=None,
args=None,
load_data=False,
convert_tol=1e012,
): # pylint:disable=too-many-arguments
r"""Generate the qubit Hamiltonian of a molecule.
This function drives the construction of the second-quantized electronic Hamiltonian
of a molecule and its transformation to the basis of Pauli matrices.
The net charge of the molecule can be given to simulate cationic/anionic systems. Also, the
spin multiplicity can be input to determine the number of unpaired electrons occupying the HF
orbitals as illustrated in the left panel of the figure below.
The basis of Gaussian-type *atomic* orbitals used to represent the *molecular* orbitals can be
specified to go beyond the minimum basis approximation.
An active space can be defined for a given number of *active electrons* occupying a reduced set
of *active orbitals* as sketched in the right panel of the figure below.
|
.. figure:: ../../_static/qchem/fig_mult_active_space.png
:align: center
:width: 90%
|
Args:
symbols (list[str]): symbols of the atomic species in the molecule
coordinates (array[float]): atomic positions in Cartesian coordinates.
The atomic coordinates must be in atomic units and can be given as either a 1D array of
size ``3*N``, or a 2D array of shape ``(N, 3)`` where ``N`` is the number of atoms.
name (str): name of the molecule
charge (int): Net charge of the molecule. If not specified a neutral system is assumed.
mult (int): Spin multiplicity :math:`\mathrm{mult}=N_\mathrm{unpaired} + 1`
for :math:`N_\mathrm{unpaired}` unpaired electrons occupying the HF orbitals.
Possible values of ``mult`` are :math:`1, 2, 3, \ldots`. If not specified,
a closed-shell HF state is assumed.
basis (str): atomic basis set used to represent the molecular orbitals
method (str): Quantum chemistry method used to solve the
mean field electronic structure problem. Available options are ``method="dhf"``
to specify the built-in differentiable Hartree-Fock solver, or ``method="pyscf"``
to use the OpenFermion-PySCF plugin (this requires ``openfermionpyscf`` to be installed).
active_electrons (int): Number of active electrons. If not specified, all electrons
are considered to be active.
active_orbitals (int): Number of active orbitals. If not specified, all orbitals
are considered to be active.
mapping (str): transformation used to map the fermionic Hamiltonian to the qubit Hamiltonian
outpath (str): path to the directory containing output files
wires (Wires, list, tuple, dict): Custom wire mapping for connecting to Pennylane ansatz.
For types ``Wires``/``list``/``tuple``, each item in the iterable represents a wire label
corresponding to the qubit number equal to its index.
For type dict, only int-keyed dict (for qubit-to-wire conversion) is accepted for
partial mapping. If None, will use identity map.
alpha (array[float]): exponents of the primitive Gaussian functions
coeff (array[float]): coefficients of the contracted Gaussian functions
args (array[array[float]]): initial values of the differentiable parameters
load_data (bool): flag to load data from the basis-set-exchange library
convert_tol (float): Tolerance in `machine epsilon <https://numpy.org/doc/stable/reference/generated/numpy.real_if_close.html>`_
for the imaginary part of the Hamiltonian coefficients created by openfermion.
Coefficients with imaginary part less than 2.22e-16*tol are considered to be real.
Returns:
tuple[pennylane.Hamiltonian, int]: the fermionic-to-qubit transformed Hamiltonian
and the number of qubits
**Example**
>>> symbols, coordinates = (['H', 'H'], np.array([0., 0., -0.66140414, 0., 0., 0.66140414]))
>>> H, qubits = molecular_hamiltonian(symbols, coordinates)
>>> print(qubits)
4
>>> print(H)
(-0.04207897647782188) [I0]
+ (0.17771287465139934) [Z0]
+ (0.1777128746513993) [Z1]
+ (-0.24274280513140484) [Z2]
+ (-0.24274280513140484) [Z3]
+ (0.17059738328801055) [Z0 Z1]
+ (0.04475014401535161) [Y0 X1 X2 Y3]
+ (-0.04475014401535161) [Y0 Y1 X2 X3]
+ (-0.04475014401535161) [X0 X1 Y2 Y3]
+ (0.04475014401535161) [X0 Y1 Y2 X3]
+ (0.12293305056183801) [Z0 Z2]
+ (0.1676831945771896) [Z0 Z3]
+ (0.1676831945771896) [Z1 Z2]
+ (0.12293305056183801) [Z1 Z3]
+ (0.176276408043196) [Z2 Z3]
"""
if method not in ["dhf", "pyscf"]:
raise ValueError("Only 'dhf' and 'pyscf' backends are supported.")
if len(coordinates) == len(symbols) * 3:
geometry_dhf = qml.numpy.array(coordinates.reshape(len(symbols), 3))
geometry_hf = coordinates
elif len(coordinates) == len(symbols):
geometry_dhf = qml.numpy.array(coordinates)
geometry_hf = coordinates.flatten()
if method == "dhf":
if wires:
wires_new = qml.qchem.convert._process_wires(wires)
wires_map = dict(zip(range(len(wires_new)), list(wires_new.labels)))
if mapping != "jordan_wigner":
raise ValueError(
"Only 'jordan_wigner' mapping is supported for the differentiable workflow."
)
if mult != 1:
raise ValueError(
"Openshell systems are not supported for the differentiable workflow. Use "
"`method = 'pyscf'` or change the charge or spin multiplicity of the molecule."
)
if args is None and isinstance(geometry_dhf, qml.numpy.tensor):
geometry_dhf.requires_grad = False
mol = qml.qchem.Molecule(
symbols,
geometry_dhf,
charge=charge,
mult=mult,
basis_name=basis,
load_data=load_data,
alpha=alpha,
coeff=coeff,
)
core, active = qml.qchem.active_space(
mol.n_electrons, mol.n_orbitals, mult, active_electrons, active_orbitals
)
requires_grad = args is not None
h = (
qml.qchem.diff_hamiltonian(mol, core=core, active=active)(*args)
if requires_grad
else qml.qchem.diff_hamiltonian(mol, core=core, active=active)()
)
if active_new_opmath():
h_as_ps = qml.pauli.pauli_sentence(h)
coeffs = qml.numpy.real(list(h_as_ps.values()), requires_grad=requires_grad)
h_as_ps = qml.pauli.PauliSentence(dict(zip(h_as_ps.keys(), coeffs)))
h = (
qml.s_prod(0, qml.Identity(h.wires[0]))
if len(h_as_ps) == 0
else h_as_ps.operation()
)
else:
coeffs = qml.numpy.real(h.coeffs, requires_grad=requires_grad)
h = qml.Hamiltonian(coeffs, h.ops)
if wires:
h = qml.map_wires(h, wires_map)
return h, 2 * len(active)
openfermion, _ = _import_of()
hf_file = meanfield(symbols, geometry_hf, name, charge, mult, basis, method, outpath)
molecule = openfermion.MolecularData(filename=hf_file)
core, active = qml.qchem.active_space(
molecule.n_electrons, molecule.n_orbitals, mult, active_electrons, active_orbitals
)
h_of, qubits = (decompose(hf_file, mapping, core, active), 2 * len(active))
h_pl = qml.qchem.convert.import_operator(h_of, wires=wires, tol=convert_tol)
return h_pl, qubits
_modules/pennylane/qchem/openfermion_obs
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