# Copyright 20182020 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/LICENSE2.0
# 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 the core functions for electronic structure calculations,
and converting the resulting data structures to forms understood by PennyLane."""
import os
import subprocess
from shutil import copyfile
import numpy as np
import pennylane as qml
from pennylane import Hamiltonian
from pennylane.wires import Wires
from . import openfermion
# BohrAngstrom correlation coefficient (https://physics.nist.gov/cgibin/cuu/Value?bohrrada0)
bohr_angs = 0.529177210903
def _process_wires(wires, n_wires=None):
r"""
Checks and consolidates custom user wire mapping into a consistent, directionfree, `Wires`
format. Used for converting between OpenFermion qubit numbering and Pennylane wire labels.
Since OpenFermion's qubit numbering is always consecutive int, simple iterable types such as
list, tuple, or Wires can be used to specify the 2way `qubit index` <> `wire label` mapping
with indices representing qubits. Dict can also be used as a mapping, but does not provide any
advantage over lists other than the ability to do partial mapping/permutation in the
`qubit index` > `wire label` direction.
It is recommended to pass Wires/list/tuple `wires` since it's directionfree, i.e. the same
`wires` argument can be used to convert both ways between OpenFermion and Pennylane. Only use
dict for partial or unordered mapping.
Args:
wires (Wires, list, tuple, dict): User wire labels or mapping for Pennylane ansatz.
For types Wires, list, or tuple, each item in the iterable represents a wire label
corresponding to the qubit number equal to its index.
For type dict, only intkeyed dict (for qubittowire conversion) or
consecutiveintvalued dict (for wiretoqubit conversion) is accepted.
If None, will be set to consecutive int based on ``n_wires``.
n_wires (int): Number of wires used if known. If None, will be inferred from ``wires``; if
``wires`` is not available, will be set to 1.
Returns:
Wires: Cleaned wire mapping with indices corresponding to qubits and values
corresponding to wire labels.
**Example**
>>> # consec int wires if no wires mapping provided, ie. identity map: 0<>0, 1<>1, 2<>2
>>> _process_wires(None, 3)
<Wires = [0, 1, 2]>
>>> # List as mapping, qubit indices with wire label values: 0<>w0, 1<>w1, 2<>w2
>>> _process_wires(['w0','w1','w2'])
<Wires = ['w0', 'w1', 'w2']>
>>> # Wires as mapping, qubit indices with wire label values: 0<>w0, 1<>w1, 2<>w2
>>> _process_wires(Wires(['w0', 'w1', 'w2']))
<Wires = ['w0', 'w1', 'w2']>
>>> # Dict as partial mapping, int qubits keys to wire label values: 0>w0, 1 unchanged, 2>w2
>>> _process_wires({0:'w0',2:'w2'})
<Wires = ['w0', 1, 'w2']>
>>> # Dict as mapping, wires label keys to consec int qubit values: w2>2, w0>0, w1>1
>>> _process_wires({'w2':2, 'w0':0, 'w1':1})
<Wires = ['w0', 'w1', 'w2']>
"""
# infer from wires, or assume 1 if wires is not of accepted types.
if n_wires is None:
n_wires = len(wires) if isinstance(wires, (Wires, list, tuple, dict)) else 1
# defaults to no mapping.
if wires is None:
return Wires(range(n_wires))
if isinstance(wires, (Wires, list, tuple)):
# does not care about the tail if more wires are provided than n_wires.
wires = Wires(wires[:n_wires])
elif isinstance(wires, dict):
if all(isinstance(w, int) for w in wires.keys()):
# Assuming keys are taken from consecutive int wires. Allows for partial mapping.
n_wires = max(wires) + 1
labels = list(range(n_wires)) # used for completing potential partial mapping.
for k, v in wires.items():
if k < n_wires:
labels[k] = v
wires = Wires(labels)
elif set(range(n_wires)).issubset(set(wires.values())):
# Assuming values are consecutive int wires (up to n_wires, ignores the rest).
# Does NOT allow for partial mapping.
wires = {v: k for k, v in wires.items()} # flip for easy indexing
wires = Wires([wires[i] for i in range(n_wires)])
else:
raise ValueError("Expected only intkeyed or consecutive intvalued dict for `wires`")
else:
raise ValueError(
"Expected type Wires, list, tuple, or dict for `wires`, got {}".format(type(wires))
)
if len(wires) != n_wires:
# check length consistency when all checking and cleaning are done.
raise ValueError(
"Length of `wires` ({}) does not match `n_wires` ({})".format(len(wires), n_wires)
)
return wires
def _exec_exists(prog):
r"""Checks whether the executable program ``prog`` exists in any of the directories
set in the ``PATH`` environment variable.
Args:
prog (str): name of the executable program
Returns:
boolean: ``True`` if the executable ``prog`` is found; ``False`` otherwise
"""
for dir_in_path in os.environ["PATH"].split(os.pathsep):
path_to_prog = os.path.join(dir_in_path, prog)
if os.path.exists(path_to_prog):
try:
subprocess.call([path_to_prog], stdout=subprocess.PIPE, stderr=subprocess.STDOUT)
except OSError:
return False
return True
return False
[docs]def read_structure(filepath, outpath="."):
r"""Reads the structure of the polyatomic system from a file and returns
a list with the symbols of the atoms in the molecule and a 1D array
with their positions :math:`[x_1, y_1, z_1, x_2, y_2, z_2, \dots]` in
atomic units (Bohr radius = 1).
The atomic coordinates in the file must be in Angstroms.
The `xyz <https://en.wikipedia.org/wiki/XYZ_file_format>`_ format is supported out of the box.
If `Open Babel <https://openbabel.org/>`_ is installed,
`any format recognized by Open Babel <https://openbabel.org/wiki/Category:Formats>`_
is also supported. Additionally, the new file ``structure.xyz``,
containing the input geometry, is created in a directory with path given by ``outpath``.
Open Babel can be installed using ``apt`` if on Ubuntu:
.. codeblock:: bash
sudo apt install openbabel
or using Anaconda:
.. codeblock:: bash
conda install c condaforge openbabel
See the Open Babel documentation for more details on installation.
Args:
filepath (str): name of the molecular structure file in the working directory
or the absolute path to the file if it is located in a different folder
outpath (str): path to the output directory
Returns:
tuple[list, array]: symbols of the atoms in the molecule and a 1D array with their
positions in atomic units.
**Example**
>>> symbols, coordinates = read_structure('h2.xyz')
>>> print(symbols, coordinates)
['H', 'H'] [0. 0. 0.66140414 0. 0. 0.66140414]
"""
obabel_error_message = (
"Open Babel converter not found:\n"
"If using Ubuntu or Debian, try: 'sudo apt install openbabel' \n"
"If using openSUSE, try: 'sudo zypper install openbabel' \n"
"If using CentOS or Fedora, try: 'sudo snap install openbabel' "
"Open Babel can also be downloaded from http://openbabel.org/wiki/Main_Page, \n"
"make sure you add it to the PATH environment variable. \n"
"If Anaconda is installed, try: 'conda install c condaforge openbabel'"
)
extension = filepath.split(".")[1].strip().lower()
file_in = filepath.strip()
file_out = os.path.join(outpath, "structure.xyz")
if extension != "xyz":
if not _exec_exists("obabel"):
raise TypeError(obabel_error_message)
try:
subprocess.run(
["obabel", "i" + extension, file_in, "oxyz", "O", file_out], check=True
)
except subprocess.CalledProcessError as e:
raise RuntimeError(
"Open Babel error. See the following Open Babel "
"output for details:\n\n {}\n{}".format(e.stdout, e.stderr)
) from e
else:
copyfile(file_in, file_out)
symbols = []
coordinates = []
with open(file_out) as f:
for line in f.readlines()[2:]:
symbol, x, y, z = line.split()
symbols.append(symbol)
coordinates.append(float(x))
coordinates.append(float(y))
coordinates.append(float(z))
return symbols, np.array(coordinates) / bohr_angs
[docs]def meanfield(
symbols,
coordinates,
name="molecule",
charge=0,
mult=1,
basis="sto3g",
package="pyscf",
outpath=".",
): # pylint: disable=toomanyarguments
r"""Generates a file from which the mean field electronic structure
of the molecule can be retrieved.
This function uses OpenFermionPySCF and OpenFermionPsi4 plugins to
perform the HartreeFock (HF) calculation for the polyatomic system using the quantum
chemistry packages ``PySCF`` and ``Psi4``, respectively. The mean field electronic
structure is saved in an hdf5formatted file in the directory
``os.path.join(outpath, package, basis)``.
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 closedshell 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#apdxbasiselement>`_
package (str): Quantum chemistry package used to solve the HartreeFock equations.
Either ``'pyscf'`` or ``'psi4'`` can be used.
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")
./pyscf/sto3g/h2
"""
if coordinates.size != 3 * len(symbols):
raise ValueError(
"The size of the array 'coordinates' has to be 3*len(symbols) = {};"
" got 'coordinates.size' = {}".format(3 * len(symbols), coordinates.size)
)
package = package.strip().lower()
if package not in ("psi4", "pyscf"):
error_message = (
"Integration with quantum chemistry package '{}' is not available. \n Please set"
" 'package' to 'pyscf' or 'psi4'.".format(package)
)
raise TypeError(error_message)
package_dir = os.path.join(outpath.strip(), package)
basis_dir = os.path.join(package_dir, basis.strip())
if not os.path.isdir(package_dir):
os.mkdir(package_dir)
os.mkdir(basis_dir)
elif not os.path.isdir(basis_dir):
os.mkdir(basis_dir)
path_to_file = os.path.join(basis_dir, name.strip())
geometry = [
[symbol, tuple(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 == "psi4":
# pylint: disable=importoutsidetoplevel
from openfermionpsi4 import run_psi4
run_psi4(molecule, run_scf=1, verbose=0, tolerate_error=1)
if package == "pyscf":
# pylint: disable=importoutsidetoplevel
from openfermionpyscf import run_pyscf
run_pyscf(molecule, run_scf=1, verbose=0)
return path_to_file
[docs]def active_space(electrons, orbitals, mult=1, active_electrons=None, active_orbitals=None):
r"""Builds the active space for a given number of active electrons and active orbitals.
PostHartreeFock (HF) electron correlation methods expand the manybody wave function
as a linear combination of Slater determinants, commonly referred to as configurations.
This configurations are generated by exciting electrons from the occupied to the
unoccupied HF orbitals as sketched in the figure below. Since the number of configurations
increases combinatorially with the number of electrons and orbitals this expansion can be
truncated by defining an active space.
The active space is created by classifying the HF orbitals as core, active and
external orbitals:
 Core orbitals are always occupied by two electrons
 Active orbitals can be occupied by zero, one, or two electrons
 The external orbitals are never occupied

.. figure:: ../../_static/qchem/sketch_active_space.png
:align: center
:width: 50%

.. note::
The number of active *spin*orbitals ``2*active_orbitals`` determines the number of
qubits required to perform the quantum simulations of the electronic structure
of the manyelectron system.
Args:
electrons (int): total number of electrons
orbitals (int): total number of orbitals
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 closedshell HF state is assumed.
active_electrons (int): Number of active electrons. If not specified, all electrons
are treated as active.
active_orbitals (int): Number of active orbitals. If not specified, all orbitals
are treated as active.
Returns:
tuple: lists of indices for core and active orbitals
**Example**
>>> electrons = 4
>>> orbitals = 4
>>> core, active = active_space(electrons, orbitals, active_electrons=2, active_orbitals=2)
>>> print(core) # core orbitals
[0]
>>> print(active) # active orbitals
[1, 2]
"""
# pylint: disable=toomanybranches
if active_electrons is None:
ncore_orbs = 0
core = []
else:
if active_electrons <= 0:
raise ValueError(
"The number of active electrons ({}) "
"has to be greater than 0.".format(active_electrons)
)
if active_electrons > electrons:
raise ValueError(
"The number of active electrons ({}) "
"can not be greater than the total "
"number of electrons ({}).".format(active_electrons, electrons)
)
if active_electrons < mult  1:
raise ValueError(
"For a reference state with multiplicity {}, "
"the number of active electrons ({}) should be "
"greater than or equal to {}.".format(mult, active_electrons, mult  1)
)
if mult % 2 == 1:
if active_electrons % 2 != 0:
raise ValueError(
"For a reference state with multiplicity {}, "
"the number of active electrons ({}) should be even.".format(
mult, active_electrons
)
)
else:
if active_electrons % 2 != 1:
raise ValueError(
"For a reference state with multiplicity {}, "
"the number of active electrons ({}) should be odd.".format(
mult, active_electrons
)
)
ncore_orbs = (electrons  active_electrons) // 2
core = list(range(ncore_orbs))
if active_orbitals is None:
active = list(range(ncore_orbs, orbitals))
else:
if active_orbitals <= 0:
raise ValueError(
"The number of active orbitals ({}) "
"has to be greater than 0.".format(active_orbitals)
)
if ncore_orbs + active_orbitals > orbitals:
raise ValueError(
"The number of core ({}) + active orbitals ({}) can not be "
"greater than the total number of orbitals ({})".format(
ncore_orbs, active_orbitals, orbitals
)
)
homo = (electrons + mult  1) / 2
if ncore_orbs + active_orbitals <= homo:
raise ValueError(
"For n_active_orbitals={}, there are no virtual orbitals "
"in the active space.".format(active_orbitals)
)
active = list(range(ncore_orbs, ncore_orbs + active_orbitals))
return core, active
[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 secondquantized electronic Hamiltonian
of the molecule and map it to the Pauli basis using the JordanWigner or BravyiKitaev
transformation.
Args:
hf_file (str): absolute path to the hdf5formatted file with the
HartreeFock 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 manybody wave function
active (list): indices of active orbitals, i.e., the orbitals used to
build the correlated manybody wave function
Returns:
QubitOperator: an instance of OpenFermion's ``QubitOperator``
**Example**
>>> decompose('./pyscf/sto3g/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]
"""
# loading HF data from the hdf5 file
molecule = openfermion.MolecularData(filename=hf_file.strip())
# getting the terms entering the secondquantized 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(
"The '{}' transformation is not available. \n "
"Please set 'mapping' to 'jordan_wigner' or 'bravyi_kitaev'.".format(mapping)
)
# fermionictoqubit transformation of the Hamiltonian
if mapping == "bravyi_kitaev":
return openfermion.transforms.bravyi_kitaev(fermionic_hamiltonian)
return openfermion.transforms.jordan_wigner(fermionic_hamiltonian)
[docs]def _qubit_operator_to_terms(qubit_operator, wires=None):
r"""Converts OpenFermion ``QubitOperator`` to a 2tuple of coefficients and
PennyLane Pauli observables.
Args:
qubit_operator (QubitOperator): Fermionictoqubit transformed operator in terms of
Pauli matrices
wires (Wires, list, tuple, dict): Custom wire mapping used to convert the qubit operator
to an observable terms 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 intkeyed dict (for qubittowire conversion) is accepted.
If None, will use identity map (e.g. 0>0, 1>1, ...).
Returns:
tuple[array[float], Iterable[pennylane.operation.Observable]]: coefficients and their
corresponding PennyLane observables in the Pauli basis
**Example**
>>> q_op = 0.1*QubitOperator('X0') + 0.2*QubitOperator('Y0 Z2')
>>> q_op
0.1 [X0] +
0.2 [Y0 Z2]
>>> _qubit_operator_to_terms(q_op, wires=['w0','w1','w2','extra_wire'])
(array([0.1, 0.2]), [Tensor(PauliX(wires=['w0'])), Tensor(PauliY(wires=['w0']), PauliZ(wires=['w2']))])
"""
n_wires = (
1 + max([max([i for i, _ in t]) if t else 1 for t in qubit_operator.terms])
if qubit_operator.terms
else 1
)
wires = _process_wires(wires, n_wires=n_wires)
if not qubit_operator.terms: # added since can't unpack empty zip to (coeffs, ops) below
return np.array([0.0]), [qml.operation.Tensor(qml.Identity(wires[0]))]
xyz2pauli = {"X": qml.PauliX, "Y": qml.PauliY, "Z": qml.PauliZ}
coeffs, ops = zip(
*[
(
coef,
qml.operation.Tensor(*[xyz2pauli[q[1]](wires=wires[q[0]]) for q in term])
if term
else qml.operation.Tensor(qml.Identity(wires[0]))
# example term: ((0,'X'), (2,'Z'), (3,'Y'))
)
for term, coef in qubit_operator.terms.items()
]
)
return np.real(np.array(coeffs)), list(ops)
[docs]def _terms_to_qubit_operator(coeffs, ops, wires=None):
r"""Converts a 2tuple of complex coefficients and PennyLane operations to
OpenFermion ``QubitOperator``.
This function is the inverse of ``_qubit_operator_to_terms``.
Args:
coeffs (array[complex]):
coefficients for each observable, same length as ops
ops (Iterable[pennylane.operation.Observable]): List of PennyLane observables as
Tensor products of Pauli observables
wires (Wires, list, tuple, dict): Custom wire mapping used to convert to qubit operator
from an observable terms 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 consecutiveintvalued dict (for wiretoqubit conversion) is
accepted. If None, will map sorted wires from all `ops` to consecutive int.
Returns:
QubitOperator: an instance of OpenFermion's ``QubitOperator``.
**Example**
>>> coeffs = np.array([0.1, 0.2])
>>> ops = [
... qml.operation.Tensor(qml.PauliX(wires=['w0'])),
... qml.operation.Tensor(qml.PauliY(wires=['w0']), qml.PauliZ(wires=['w2']))
... ]
>>> _terms_to_qubit_operator(coeffs, ops, wires=Wires(['w0', 'w1', 'w2']))
0.1 [X0] +
0.2 [Y0 Z2]
"""
all_wires = Wires.all_wires([op.wires for op in ops], sort=True)
if wires is not None:
qubit_indexed_wires = _process_wires(
wires,
)
if not set(all_wires).issubset(set(qubit_indexed_wires)):
raise ValueError("Supplied `wires` does not cover all wires defined in `ops`.")
else:
qubit_indexed_wires = all_wires
q_op = openfermion.QubitOperator()
for coeff, op in zip(coeffs, ops):
extra_obsvbs = set(op.name)  {"PauliX", "PauliY", "PauliZ", "Identity"}
if extra_obsvbs != set():
raise ValueError(
"Expected only PennyLane observables PauliX/Y/Z or Identity, "
+ "but also got {}.".format(extra_obsvbs)
)
# Pauli axis names, note s[1] expects only 'Pauli{X,Y,Z}'
pauli_names = [s[1] if s != "Identity" else s for s in op.name]
all_identity = all(obs.name == "Identity" for obs in op.obs)
if (op.name == ["Identity"] and len(op.wires) == 1) or all_identity:
term_str = ""
else:
term_str = " ".join(
[
"{}{}".format(pauli, qubit_indexed_wires.index(wire))
for pauli, wire in zip(pauli_names, op.wires)
if pauli != "Identity"
]
)
# This is how one makes QubitOperator in OpenFermion
q_op += coeff * openfermion.QubitOperator(term_str)
return q_op
[docs]def _qubit_operators_equivalent(openfermion_qubit_operator, pennylane_qubit_operator, wires=None):
r"""Checks equivalence between OpenFermion :class:`~.QubitOperator` and Pennylane VQE
``Hamiltonian`` (Tensor product of Pauli matrices).
Equality is based on OpenFermion :class:`~.QubitOperator`'s equality.
Args:
openfermion_qubit_operator (QubitOperator): OpenFermion qubit operator represented as
a Pauli summation
pennylane_qubit_operator (pennylane.Hamiltonian): PennyLane
Hamiltonian object
wires (Wires, list, tuple, dict): Custom wire mapping used to convert to qubit operator
from an observable terms 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 intkeyed dict (for qubittowire conversion) is accepted.
If None, will map sorted wires from all Pennylane terms to consecutive int.
Returns:
(bool): True if equivalent
"""
coeffs, ops = pennylane_qubit_operator.terms
return openfermion_qubit_operator == _terms_to_qubit_operator(coeffs, ops, wires=wires)
[docs]def convert_observable(qubit_observable, wires=None, tol=1e08):
r"""Converts an OpenFermion :class:`~.QubitOperator` operator to a Pennylane VQE observable
Args:
qubit_observable (QubitOperator): Observable represented as an OpenFermion ``QubitOperator``
wires (Wires, list, tuple, dict): Custom wire mapping used to convert the qubit operator
to an observable terms 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 intkeyed dict (for qubittowire conversion) is accepted.
If None, will use identity map (e.g. 0>0, 1>1, ...).
tol (float): Tolerance in machine epsilons for the imaginary part of the
coefficients in ``qubit_observable``. Coefficients with imaginary part
less than 2.22e16*tol are considered to be real.
Returns:
(pennylane.Hamiltonian): Pennylane VQE observable. PennyLane :class:`~.Hamiltonian`
represents any operator expressed as linear combinations of observables, e.g.,
:math:`\sum_{k=0}^{N1} c_k O_k`.
**Example**
>>> h_of = decompose('./pyscf/sto3g/h2')
>>> h_pl = convert_observable(h_of)
>>> print(h_pl.coeffs)
[0.04207898 0.17771287 0.17771287 0.24274281 0.24274281 0.17059738
0.04475014 0.04475014 0.04475014 0.04475014 0.12293305 0.16768319
0.16768319 0.12293305 0.17627641]
"""
if any(
np.iscomplex(np.real_if_close(coef, tol=tol)) for coef in qubit_observable.terms.values()
):
raise TypeError(
"The coefficients entering the QubitOperator must be real;"
" got complex coefficients in the operator {}".format(qubit_observable)
)
return Hamiltonian(*_qubit_operator_to_terms(qubit_observable, wires=wires))
[docs]def molecular_hamiltonian(
symbols,
coordinates,
name="molecule",
charge=0,
mult=1,
basis="sto3g",
package="pyscf",
active_electrons=None,
active_orbitals=None,
mapping="jordan_wigner",
outpath=".",
wires=None,
): # pylint:disable=toomanyarguments
r"""Generates the qubit Hamiltonian of a molecule.
This function drives the construction of the secondquantized electronic Hamiltonian
of a molecule and its transformation to the basis of Pauli matrices.
#. OpenFermionPySCF or OpenFermionPsi4 plugins are used to launch
the HartreeFock (HF) calculation for the polyatomic system using the quantum
chemistry package ``PySCF`` or ``Psi4``, respectively.
 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 Gaussiantype *atomic* orbitals used to represent the *molecular* orbitals
can be specified to go beyond the minimum basis approximation. Basis set availability
per element can be found
`here <www.psicode.org/psi4manual/master/basissets_byelement.html#apdxbasiselement>`_
#. An active space can be defined for a given number of *active electrons*
occupying a reduced set of *active orbitals* in the vicinity of the frontier
orbitals as sketched in the right panel of the figure below.
#. Finally, the secondquantized Hamiltonian is mapped to the Pauli basis and
converted to a PennyLane observable.

.. 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]): 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): Net charge of the molecule. If not specified a 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 closedshell HF state is assumed.
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#apdxbasiselement>`_
package (str): quantum chemistry package (pyscf or psi4) used to solve the
mean field electronic structure problem
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 (``'jordan_wigner'`` or ``'bravyi_kitaev'``) 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 intkeyed dict (for qubittowire conversion) is accepted for
partial mapping. If None, will use identity map.
Returns:
tuple[pennylane.Hamiltonian, int]: the fermionictoqubit 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]
"""
hf_file = meanfield(symbols, coordinates, name, charge, mult, basis, package, outpath)
molecule = openfermion.MolecularData(filename=hf_file)
core, active = 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))
return convert_observable(h_of, wires=wires), qubits
[docs]def excitations(electrons, orbitals, delta_sz=0):
r"""Generates single and double excitations from a HartreeFock reference state.
Single and double excitations can be generated by acting with the operators
:math:`\hat T_1` and :math:`\hat T_2` on the HartreeFock reference state:
.. math::
&& \hat{T}_1 = \sum_{r \in \mathrm{occ} \\ p \in \mathrm{unocc}}
\hat{c}_p^\dagger \hat{c}_r \\
&& \hat{T}_2 = \sum_{r>s \in \mathrm{occ} \\ p>q \in
\mathrm{unocc}} \hat{c}_p^\dagger \hat{c}_q^\dagger \hat{c}_r \hat{c}_s.
In the equations above the indices :math:`r, s` and :math:`p, q` run over the
occupied (occ) and unoccupied (unocc) *spin* orbitals and :math:`\hat c` and
:math:`\hat c^\dagger` are the electron annihilation and creation operators,
respectively.

.. figure:: ../../_static/qchem/sd_excitations.png
:align: center
:width: 80%

Args:
electrons (int): Number of electrons. If an active space is defined, this
is the number of active electrons.
orbitals (int): Number of *spin* orbitals. If an active space is defined,
this is the number of active spinorbitals.
delta_sz (int): Specifies the selection rules ``sz[p]  sz[r] = delta_sz`` and
``sz[p] + sz[p]  sz[r]  sz[s] = delta_sz`` for the spinprojection ``sz`` of
the orbitals involved in the single and double excitations, respectively.
``delta_sz`` can take the values :math:`0`, :math:`\pm 1` and :math:`\pm 2`.
Returns:
tuple(list, list): lists with the indices of the spin orbitals involved in the
single and double excitations
**Example**
>>> electrons = 2
>>> orbitals = 4
>>> singles, doubles = excitations(electrons, orbitals)
>>> print(singles)
[[0, 2], [1, 3]]
>>> print(doubles)
[[0, 1, 2, 3]]
"""
if not electrons > 0:
raise ValueError(
"The number of active electrons has to be greater than 0 \n"
"Got n_electrons = {}".format(electrons)
)
if orbitals <= electrons:
raise ValueError(
"The number of active spinorbitals ({}) "
"has to be greater than the number of active electrons ({}).".format(
orbitals, electrons
)
)
if delta_sz not in (0, 1, 1, 2, 2):
raise ValueError(
"Expected values for 'delta_sz' are 0, +/ 1 and +/ 2 but got ({}).".format(delta_sz)
)
# define the spin projection 'sz' of the singleparticle states
sz = np.array([0.5 if (i % 2 == 0) else 0.5 for i in range(orbitals)])
singles = [
[r, p]
for r in range(electrons)
for p in range(electrons, orbitals)
if sz[p]  sz[r] == delta_sz
]
doubles = [
[s, r, q, p]
for s in range(electrons  1)
for r in range(s + 1, electrons)
for q in range(electrons, orbitals  1)
for p in range(q + 1, orbitals)
if (sz[p] + sz[q]  sz[r]  sz[s]) == delta_sz
]
return singles, doubles
[docs]def hf_state(electrons, orbitals):
r"""Generates the occupationnumber vector representing the HartreeFock state.
The manyparticle wave function in the HartreeFock (HF) approximation is a `Slater determinant
<https://en.wikipedia.org/wiki/Slater_determinant>`_. In Fock space, a Slater determinant
for :math:`N` electrons is represented by the occupationnumber vector:
.. math::
\vert {\bf n} \rangle = \vert n_1, n_2, \dots, n_\mathrm{orbs} \rangle,
n_i = \left\lbrace \begin{array}{ll} 1 & i \leq N \\ 0 & i > N \end{array} \right.,
where :math:`n_i` indicates the occupation of the :math:`i`th orbital.
Args:
electrons (int): Number of electrons. If an active space is defined, this
is the number of active electrons.
orbitals (int): Number of *spin* orbitals. If an active space is defined,
this is the number of active spinorbitals.
Returns:
array: NumPy array containing the vector :math:`\vert {\bf n} \rangle`
**Example**
>>> state = hf_state(2, 6)
>>> print(state)
[1 1 0 0 0 0]
"""
if electrons <= 0:
raise ValueError(
"The number of active electrons has to be larger than zero; got 'electrons' = {}".format(
electrons
)
)
if electrons > orbitals:
raise ValueError(
"The number of active orbitals cannot be smaller than the number of active electrons;"
" got 'orbitals'={} < 'electrons'={}".format(orbitals, electrons)
)
state = np.where(np.arange(orbitals) < electrons, 1, 0)
return np.array(state)
[docs]def excitations_to_wires(singles, doubles, wires=None):
r"""Map the indices representing the single and double excitations
generated with the function :func:`~.excitations` to the wires that
the Unitary CoupledCluster (UCCSD) template will act on.
Args:
singles (list[list[int]]): list with the indices ``r``, ``p`` of the two qubits
representing the single excitation
:math:`\vert r, p \rangle = \hat{c}_p^\dagger \hat{c}_r \vert \mathrm{HF}\rangle`
doubles (list[list[int]]): list with the indices ``s``, ``r``, ``q``, ``p`` of the four
qubits representing the double excitation
:math:`\vert s, r, q, p \rangle = \hat{c}_p^\dagger \hat{c}_q^\dagger
\hat{c}_r \hat{c}_s \vert \mathrm{HF}\rangle`
wires (Iterable[Any]): Wires of the quantum device. If None, will use consecutive wires.
The indices :math:`r, s` and :math:`p, q` in these lists correspond, respectively, to the
occupied and virtual orbitals involved in the generated single and double excitations.
Returns:
tuple[list[list[Any]], list[list[list[Any]]]]: lists with the sequence of wires,
resulting from the single and double excitations, that the Unitary CoupledCluster
(UCCSD) template will act on.
**Example**
>>> singles = [[0, 2], [1, 3]]
>>> doubles = [[0, 1, 2, 3]]
>>> singles_wires, doubles_wires = excitations_to_wires(singles, doubles)
>>> print(single_wires)
[[0, 1, 2], [1, 2, 3]]
>>> print(doubles_wires)
[[[0, 1], [2, 3]]]
>>> wires=['a0', 'b1', 'c2', 'd3']
>>> singles_wires, doubles_wires = excitations_to_wires(singles, doubles, wires=wires)
>>> print(singles_wires)
[['a0', 'b1', 'c2'], ['b1', 'c2', 'd3']]
>>> print(doubles_wires)
[[['a0', 'b1'], ['c2', 'd3']]]
"""
if (not singles) and (not doubles):
raise ValueError(
"'singles' and 'doubles' lists can not be both empty;\
got singles = {}, doubles = {}".format(
singles, doubles
)
)
expected_shape = (2,)
for single_ in singles:
if np.array(single_).shape != expected_shape:
raise ValueError(
"Expected entries of 'singles' to be of shape (2,); got {}".format(
np.array(single_).shape
)
)
expected_shape = (4,)
for double_ in doubles:
if np.array(double_).shape != expected_shape:
raise ValueError(
"Expected entries of 'doubles' to be of shape (4,); got {}".format(
np.array(double_).shape
)
)
max_idx = 0
if singles:
max_idx = np.max(singles)
if doubles:
max_idx = max(np.max(doubles), max_idx)
if wires is None:
wires = range(max_idx + 1)
elif len(wires) != max_idx + 1:
raise ValueError("Expected number of wires is {}; got {}".format(max_idx + 1, len(wires)))
singles_wires = []
for r, p in singles:
s_wires = [wires[i] for i in range(r, p + 1)]
singles_wires.append(s_wires)
doubles_wires = []
for s, r, q, p in doubles:
d1_wires = [wires[i] for i in range(s, r + 1)]
d2_wires = [wires[i] for i in range(q, p + 1)]
doubles_wires.append([d1_wires, d2_wires])
return singles_wires, doubles_wires
__all__ = [
"read_structure",
"meanfield",
"active_space",
"decompose",
"convert_observable",
"molecular_hamiltonian",
"hf_state",
"excitations",
"excitations_to_wires",
"_qubit_operator_to_terms",
"_terms_to_qubit_operator",
"_qubit_operators_equivalent",
]
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