Source code for pennylane.templates.subroutines.fermionic_double_excitation
# Copyright 2018-2021 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.
r"""
Contains the FermionicDoubleExcitation template.
"""
# pylint: disable-msg=too-many-branches,too-many-arguments,protected-access
import numpy as np
import pennylane as qml
from pennylane.operation import Operation, AnyWires
from pennylane.ops import RZ, RX, CNOT, Hadamard
def _layer1(weight, s, r, q, p, set_cnot_wires):
r"""Implement the first layer of the circuit to exponentiate the double-excitation
operator entering the UCCSD ansatz.
.. math::
\hat{U}_{pqrs}^{(1)}(\theta) = \mathrm{exp} \Big\{ \frac{i\theta}{8}
\bigotimes_{b=s+1}^{r-1} \hat{Z}_b \bigotimes_{a=q+1}^{p-1} \hat{Z}_a
(\hat{X}_s \hat{X}_r \hat{Y}_q \hat{X}_p) \Big\}
Args:
weight (float): angle :math:`\theta` entering the Z rotation acting on wire ``p``
s (int): qubit index ``s``
r (int): qubit index ``r``
q (int): qubit index ``q``
p (int): qubit index ``p``
set_cnot_wires (list[Wires]): list of CNOT wires
Returns:
list[.Operator]: sequence of operators defined by this function
"""
# U_1, U_2, U_3, U_4 acting on wires 's', 'r', 'q' and 'p'
op_list = [Hadamard(wires=s), Hadamard(wires=r), RX(-np.pi / 2, wires=q), Hadamard(wires=p)]
# Applying CNOTs
for cnot_wires in set_cnot_wires:
op_list.append(CNOT(wires=cnot_wires))
# Z rotation acting on wire 'p'
op_list.append(RZ(weight / 8, wires=p))
# Applying CNOTs in reverse order
for cnot_wires in reversed(set_cnot_wires):
op_list.append(CNOT(wires=cnot_wires))
# U_1^+, U_2^+, U_3^+, U_4^+ acting on wires 's', 'r', 'q' and 'p'
op_list.append(Hadamard(wires=s))
op_list.append(Hadamard(wires=r))
op_list.append(RX(np.pi / 2, wires=q))
op_list.append(Hadamard(wires=p))
return op_list
def _layer2(weight, s, r, q, p, set_cnot_wires):
r"""Implement the second layer of the circuit to exponentiate the double-excitation
operator entering the UCCSD ansatz.
.. math::
\hat{U}_{pqrs}^{(2)}(\theta) = \mathrm{exp} \Big\{ \frac{i\theta}{8}
\bigotimes_{b=s+1}^{r-1} \hat{Z}_b \bigotimes_{a=q+1}^{p-1} \hat{Z}_a
(\hat{Y}_s \hat{X}_r \hat{Y}_q \hat{Y}_p) \Big\}
Args:
weight (float): angle :math:`\theta` entering the Z rotation acting on wire ``p``
s (int): qubit index ``s``
r (int): qubit index ``r``
q (int): qubit index ``q``
p (int): qubit index ``p``
set_cnot_wires (list[Wires]): list of CNOT wires
Returns:
list[.Operator]: sequence of operators defined by this function
"""
# U_1, U_2, U_3, U_4 acting on wires 's', 'r', 'q' and 'p'
op_list = [
RX(-np.pi / 2, wires=s),
Hadamard(wires=r),
RX(-np.pi / 2, wires=q),
RX(-np.pi / 2, wires=p),
]
# Applying CNOTs
for cnot_wires in set_cnot_wires:
op_list.append(CNOT(wires=cnot_wires))
# Z rotation acting on wire 'p'
op_list.append(RZ(weight / 8, wires=p))
# Applying CNOTs in reverse order
for cnot_wires in reversed(set_cnot_wires):
op_list.append(CNOT(wires=cnot_wires))
# U_1^+, U_2^+, U_3^+, U_4^+ acting on wires 's', 'r', 'q' and 'p'
op_list.append(RX(np.pi / 2, wires=s))
op_list.append(Hadamard(wires=r))
op_list.append(RX(np.pi / 2, wires=q))
op_list.append(RX(np.pi / 2, wires=p))
return op_list
def _layer3(weight, s, r, q, p, set_cnot_wires):
r"""Implement the third layer of the circuit to exponentiate the double-excitation
operator entering the UCCSD ansatz.
.. math::
\hat{U}_{pqrs}^{(3)}(\theta) = \mathrm{exp} \Big\{ \frac{i\theta}{8}
\bigotimes_{b=s+1}^{r-1} \hat{Z}_b \bigotimes_{a=q+1}^{p-1} \hat{Z}_a
(\hat{X}_s \hat{Y}_r \hat{Y}_q \hat{Y}_p) \Big\}
Args:
weight (float): angle :math:`\theta` entering the Z rotation acting on wire ``p``
s (int): qubit index ``s``
r (int): qubit index ``r``
q (int): qubit index ``q``
p (int): qubit index ``p``
set_cnot_wires (list[Wires]): list of CNOT wires
Returns:
list[.Operator]: sequence of operators defined by this function
"""
# U_1, U_2, U_3, U_4 acting on wires 's', 'r', 'q' and 'p'
op_list = [
Hadamard(wires=s),
RX(-np.pi / 2, wires=r),
RX(-np.pi / 2, wires=q),
RX(-np.pi / 2, wires=p),
]
# Applying CNOTs
for cnot_wires in set_cnot_wires:
op_list.append(CNOT(wires=cnot_wires))
# Z rotation acting on wire 'p'
op_list.append(RZ(weight / 8, wires=p))
# Applying CNOTs in reverse order
for cnot_wires in reversed(set_cnot_wires):
op_list.append(CNOT(wires=cnot_wires))
# U_1^+, U_2^+, U_3^+, U_4^+ acting on wires 's', 'r', 'q' and 'p'
op_list.append(Hadamard(wires=s))
op_list.append(RX(np.pi / 2, wires=r))
op_list.append(RX(np.pi / 2, wires=q))
op_list.append(RX(np.pi / 2, wires=p))
return op_list
def _layer4(weight, s, r, q, p, set_cnot_wires):
r"""Implement the fourth layer of the circuit to exponentiate the double-excitation
operator entering the UCCSD ansatz.
.. math::
\hat{U}_{pqrs}^{(4)}(\theta) = \mathrm{exp} \Big\{ \frac{i\theta}{8}
\bigotimes_{b=s+1}^{r-1} \hat{Z}_b \bigotimes_{a=q+1}^{p-1} \hat{Z}_a
(\hat{X}_s \hat{X}_r \hat{X}_q \hat{Y}_p) \Big\}
Args:
weight (float): angle :math:`\theta` entering the Z rotation acting on wire ``p``
s (int): qubit index ``s``
r (int): qubit index ``r``
q (int): qubit index ``q``
p (int): qubit index ``p``
set_cnot_wires (list[Wires]): list of CNOT wires
Returns:
list[.Operator]: sequence of operators defined by this function
"""
# U_1, U_2, U_3, U_4 acting on wires 's', 'r', 'q' and 'p'
op_list = [Hadamard(wires=s), Hadamard(wires=r), Hadamard(wires=q), RX(-np.pi / 2, wires=p)]
# Applying CNOTs
for cnot_wires in set_cnot_wires:
op_list.append(CNOT(wires=cnot_wires))
# Z rotation acting on wire 'p'
op_list.append(RZ(weight / 8, wires=p))
# Applying CNOTs in reverse order
for cnot_wires in reversed(set_cnot_wires):
op_list.append(CNOT(wires=cnot_wires))
# U_1^+, U_2^+, U_3^+, U_4^+ acting on wires 's', 'r', 'q' and 'p'
op_list.append(Hadamard(wires=s))
op_list.append(Hadamard(wires=r))
op_list.append(Hadamard(wires=q))
op_list.append(RX(np.pi / 2, wires=p))
return op_list
def _layer5(weight, s, r, q, p, set_cnot_wires):
r"""Implement the fifth layer of the circuit to exponentiate the double-excitation
operator entering the UCCSD ansatz.
.. math::
\hat{U}_{pqrs}^{(5)}(\theta) = \mathrm{exp} \Big\{ -\frac{i\theta}{8}
\bigotimes_{b=s+1}^{r-1} \hat{Z}_b \bigotimes_{a=q+1}^{p-1} \hat{Z}_a
(\hat{Y}_s \hat{X}_r \hat{X}_q \hat{X}_p) \Big\}
Args:
weight (float): angle :math:`\theta` entering the Z rotation acting on wire ``p``
s (int): qubit index ``s``
r (int): qubit index ``r``
q (int): qubit index ``q``
p (int): qubit index ``p``
set_cnot_wires (list[Wires]): list of CNOT wires
Returns:
list[.Operator]: sequence of operators defined by this function
"""
# U_1, U_2, U_3, U_4 acting on wires 's', 'r', 'q' and 'p'
op_list = [RX(-np.pi / 2, wires=s), Hadamard(wires=r), Hadamard(wires=q), Hadamard(wires=p)]
# Applying CNOTs
for cnot_wires in set_cnot_wires:
op_list.append(CNOT(wires=cnot_wires))
# Z rotation acting on wire 'p'
op_list.append(RZ(-weight / 8, wires=p))
# Applying CNOTs in reverse order
for cnot_wires in reversed(set_cnot_wires):
op_list.append(CNOT(wires=cnot_wires))
# U_1^+, U_2^+, U_3^+, U_4^+ acting on wires 's', 'r', 'q' and 'p'
op_list.append(RX(np.pi / 2, wires=s))
op_list.append(Hadamard(wires=r))
op_list.append(Hadamard(wires=q))
op_list.append(Hadamard(wires=p))
return op_list
def _layer6(weight, s, r, q, p, set_cnot_wires):
r"""Implement the sixth layer of the circuit to exponentiate the double-excitation
operator entering the UCCSD ansatz.
.. math::
\hat{U}_{pqrs}^{(6)}(\theta) = \mathrm{exp} \Big\{ -\frac{i\theta}{8}
\bigotimes_{b=s+1}^{r-1} \hat{Z}_b \bigotimes_{a=q+1}^{p-1} \hat{Z}_a
(\hat{X}_s \hat{Y}_r \hat{X}_q \hat{X}_p) \Big\}
Args:
weight (float): angle :math:`\theta` entering the Z rotation acting on wire ``p``
s (int): qubit index ``s``
r (int): qubit index ``r``
q (int): qubit index ``q``
p (int): qubit index ``p``
set_cnot_wires (list[Wires]): list of CNOT wires
Returns:
list[.Operator]: sequence of operators defined by this function
"""
# U_1, U_2, U_3, U_4 acting on wires 's', 'r', 'q' and 'p'
op_list = [Hadamard(wires=s), RX(-np.pi / 2, wires=r), Hadamard(wires=q), Hadamard(wires=p)]
# Applying CNOTs
for cnot_wires in set_cnot_wires:
op_list.append(CNOT(wires=cnot_wires))
# Z rotation acting on wire 'p'
op_list.append(RZ(-weight / 8, wires=p))
# Applying CNOTs in reverse order
for cnot_wires in reversed(set_cnot_wires):
op_list.append(CNOT(wires=cnot_wires))
# U_1^+, U_2^+, U_3^+, U_4^+ acting on wires 's', 'r', 'q' and 'p'
op_list.append(Hadamard(wires=s))
op_list.append(RX(np.pi / 2, wires=r))
op_list.append(Hadamard(wires=q))
op_list.append(Hadamard(wires=p))
return op_list
def _layer7(weight, s, r, q, p, set_cnot_wires):
r"""Implement the seventh layer of the circuit to exponentiate the double-excitation
operator entering the UCCSD ansatz.
.. math::
\hat{U}_{pqrs}^{(7)}(\theta) = \mathrm{exp} \Big\{ -\frac{i\theta}{8}
\bigotimes_{b=s+1}^{r-1} \hat{Z}_b \bigotimes_{a=q+1}^{p-1} \hat{Z}_a
(\hat{Y}_s \hat{Y}_r \hat{Y}_q \hat{X}_p) \Big\}
Args:
weight (float): angle :math:`\theta` entering the Z rotation acting on wire ``p``
s (int): qubit index ``s``
r (int): qubit index ``r``
q (int): qubit index ``q``
p (int): qubit index ``p``
set_cnot_wires (list[Wires]): list of CNOT wires
Returns:
list[.Operator]: sequence of operators defined by this function
"""
# U_1, U_2, U_3, U_4 acting on wires 's', 'r', 'q' and 'p'
op_list = [
RX(-np.pi / 2, wires=s),
RX(-np.pi / 2, wires=r),
RX(-np.pi / 2, wires=q),
Hadamard(wires=p),
]
# Applying CNOTs
for cnot_wires in set_cnot_wires:
op_list.append(CNOT(wires=cnot_wires))
# Z rotation acting on wire 'p'
op_list.append(RZ(-weight / 8, wires=p))
# Applying CNOTs in reverse order
for cnot_wires in reversed(set_cnot_wires):
op_list.append(CNOT(wires=cnot_wires))
# U_1^+, U_2^+, U_3^+, U_4^+ acting on wires 's', 'r', 'q' and 'p'
op_list.append(RX(np.pi / 2, wires=s))
op_list.append(RX(np.pi / 2, wires=r))
op_list.append(RX(np.pi / 2, wires=q))
op_list.append(Hadamard(wires=p))
return op_list
def _layer8(weight, s, r, q, p, set_cnot_wires):
r"""Implement the eighth layer of the circuit to exponentiate the double-excitation
operator entering the UCCSD ansatz.
.. math::
\hat{U}_{pqrs}^{(8)}(\theta) = \mathrm{exp} \Big\{ -\frac{i\theta}{8}
\bigotimes_{b=s+1}^{r-1} \hat{Z}_b \bigotimes_{a=q+1}^{p-1} \hat{Z}_a
(\hat{Y}_s \hat{Y}_r \hat{X}_q \hat{Y}_p) \Big\}
Args:
weight (float): angle :math:`\theta` entering the Z rotation acting on wire ``p``
s (int): qubit index ``s``
r (int): qubit index ``r``
q (int): qubit index ``q``
p (int): qubit index ``p``
set_cnot_wires (list[Wires]): list of CNOT wires
Returns:
list[.Operator]: sequence of operators defined by this function
"""
# U_1, U_2, U_3, U_4 acting on wires 's', 'r', 'q' and 'p'
op_list = [
RX(-np.pi / 2, wires=s),
RX(-np.pi / 2, wires=r),
Hadamard(wires=q),
RX(-np.pi / 2, wires=p),
]
# Applying CNOTs
for cnot_wires in set_cnot_wires:
op_list.append(CNOT(wires=cnot_wires))
# Z rotation acting on wire 'p'
op_list.append(RZ(-weight / 8, wires=p))
# Applying CNOTs in reverse order
for cnot_wires in reversed(set_cnot_wires):
op_list.append(CNOT(wires=cnot_wires))
# U_1^+, U_2^+, U_3^+, U_4^+ acting on wires 's', 'r', 'q' and 'p'
op_list.append(RX(np.pi / 2, wires=s))
op_list.append(RX(np.pi / 2, wires=r))
op_list.append(Hadamard(wires=q))
op_list.append(RX(np.pi / 2, wires=p))
return op_list
[docs]class FermionicDoubleExcitation(Operation):
r"""Circuit to exponentiate the tensor product of Pauli matrices representing the
double-excitation operator entering the Unitary Coupled-Cluster Singles
and Doubles (UCCSD) ansatz. UCCSD is a VQE ansatz commonly used to run quantum
chemistry simulations.
The CC double-excitation operator is given by
.. math::
\hat{U}_{pqrs}(\theta) = \mathrm{exp} \{ \theta (\hat{c}_p^\dagger \hat{c}_q^\dagger
\hat{c}_r \hat{c}_s - \mathrm{H.c.}) \},
where :math:`\hat{c}` and :math:`\hat{c}^\dagger` are the fermionic annihilation and
creation operators and the indices :math:`r, s` and :math:`p, q` run over the occupied and
unoccupied molecular orbitals, respectively. Using the `Jordan-Wigner transformation
<https://arxiv.org/abs/1208.5986>`_ the fermionic operator defined above can be written
in terms of Pauli matrices (for more details see
`arXiv:1805.04340 <https://arxiv.org/abs/1805.04340>`_):
.. math::
\hat{U}_{pqrs}(\theta) = \mathrm{exp} \Big\{
\frac{i\theta}{8} \bigotimes_{b=s+1}^{r-1} \hat{Z}_b \bigotimes_{a=q+1}^{p-1}
\hat{Z}_a (\hat{X}_s \hat{X}_r \hat{Y}_q \hat{X}_p +
\hat{Y}_s \hat{X}_r \hat{Y}_q \hat{Y}_p + \hat{X}_s \hat{Y}_r \hat{Y}_q \hat{Y}_p +
\hat{X}_s \hat{X}_r \hat{X}_q \hat{Y}_p - \mathrm{H.c.} ) \Big\}
The quantum circuit to exponentiate the tensor product of Pauli matrices entering
the latter equation is shown below (see `arXiv:1805.04340 <https://arxiv.org/abs/1805.04340>`_):
|
.. figure:: ../../_static/templates/subroutines/double_excitation_unitary.png
:align: center
:width: 60%
:target: javascript:void(0);
|
As explained in `Seely et al. (2012) <https://arxiv.org/abs/1208.5986>`_,
the exponential of a tensor product of Pauli-Z operators can be decomposed in terms of
:math:`2(n-1)` CNOT gates and a single-qubit Z-rotation referred to as :math:`U_\theta` in
the figure above. If there are :math:`X` or:math:`Y` Pauli matrices in the product, the
Hadamard (:math:`H`) or :math:`R_x` gate has to be applied to change to the :math:`X`
or :math:`Y` basis, respectively. The latter operations are denoted as
:math:`U_1`, :math:`U_2`, :math:`U_3` and :math:`U_4` in the figure above. See the
Usage Details section for more details.
Args:
weight (float or tensor_like): angle :math:`\theta` entering the Z rotation acting on wire ``p``
wires1 (Iterable): Wires of the qubits representing the subset of occupied orbitals
in the interval ``[s, r]``. The first wire is interpreted as ``s``
and the last wire as ``r``.
Wires in between are acted on with CNOT gates to compute the parity of the set of qubits.
wires2 (Iterable): Wires of the qubits representing the subset of unoccupied
orbitals in the interval ``[q, p]``. The first wire is interpreted as ``q`` and
the last wire is interpreted as ``p``. Wires in between are acted on with CNOT gates
to compute the parity of the set of qubits.
.. details::
:title: Usage Details
Notice that:
#. :math:`\hat{U}_{pqrs}(\theta)` involves eight exponentiations where
:math:`\hat{U}_1`, :math:`\hat{U}_2`, :math:`\hat{U}_3`, :math:`\hat{U}_4` and
:math:`\hat{U}_\theta` are defined as follows,
.. math::
[U_1, && U_2, U_3, U_4, U_{\theta}] = \\
&& \Bigg\{\bigg[H, H, R_x(-\frac{\pi}{2}), H, R_z(\theta/8)\bigg],
\bigg[R_x(-\frac{\pi}{2}), H, R_x(-\frac{\pi}{2}), R_x(-\frac{\pi}{2}),
R_z(\frac{\theta}{8}) \bigg], \\
&& \bigg[H, R_x(-\frac{\pi}{2}), R_x(-\frac{\pi}{2}), R_x(-\frac{\pi}{2}),
R_z(\frac{\theta}{8}) \bigg], \bigg[H, H, H, R_x(-\frac{\pi}{2}),
R_z(\frac{\theta}{8}) \bigg], \\
&& \bigg[R_x(-\frac{\pi}{2}), H, H, H, R_z(-\frac{\theta}{8}) \bigg],
\bigg[H, R_x(-\frac{\pi}{2}), H, H, R_z(-\frac{\theta}{8}) \bigg], \\
&& \bigg[R_x(-\frac{\pi}{2}), R_x(-\frac{\pi}{2}), R_x(-\frac{\pi}{2}),
H, R_z(-\frac{\theta}{8}) \bigg], \bigg[R_x(-\frac{\pi}{2}), R_x(-\frac{\pi}{2}),
H, R_x(-\frac{\pi}{2}), R_z(-\frac{\theta}{8}) \bigg] \Bigg\}
#. For a given quadruple ``[s, r, q, p]`` with :math:`p>q>r>s`, seventy-two single-qubit
and ``16*(len(wires1)-1 + len(wires2)-1 + 1)`` CNOT operations are applied.
Consecutive CNOT gates act on qubits with indices between ``s`` and ``r`` and
``q`` and ``p`` while a single CNOT acts on wires ``r`` and ``q``. The operations
performed across these qubits are shown in dashed lines in the figure above.
An example of how to use this template is shown below:
.. code-block:: python
import pennylane as qml
dev = qml.device('default.qubit', wires=5)
@qml.qnode(dev)
def circuit(weight, wires1=None, wires2=None):
qml.FermionicDoubleExcitation(weight, wires1=wires1, wires2=wires2)
return qml.expval(qml.Z(0))
weight = 1.34817
print(circuit(weight, wires1=[0, 1], wires2=[2, 3, 4]))
"""
num_wires = AnyWires
grad_method = "A"
parameter_frequencies = [(0.5, 1.0)]
def _flatten(self):
return self.data, (self.hyperparameters["wires1"], self.hyperparameters["wires2"])
@classmethod
def _unflatten(cls, data, metadata) -> "FermionicDoubleExcitation":
return cls(data[0], wires1=metadata[0], wires2=metadata[1])
def __init__(self, weight, wires1=None, wires2=None, id=None):
if len(wires1) < 2:
raise ValueError(
f"expected at least two wires representing the occupied orbitals; "
f"got {len(wires1)}"
)
if len(wires2) < 2:
raise ValueError(
f"expected at least two wires representing the unoccupied orbitals; "
f"got {len(wires2)}"
)
shape = qml.math.shape(weight)
if shape != ():
raise ValueError(f"Weight must be a scalar; got shape {shape}.")
wires1 = qml.wires.Wires(wires1)
wires2 = qml.wires.Wires(wires2)
self._hyperparameters = {
"wires1": wires1,
"wires2": wires2,
}
wires = wires1 + wires2
super().__init__(weight, wires=wires, id=id)
@property
def num_params(self):
return 1
[docs] @staticmethod
def compute_decomposition(
weight, wires, wires1, wires2
): # pylint: disable=arguments-differ,unused-argument
r"""Representation of the operator as a product of other operators.
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.FermionicDoubleExcitation.decomposition`.
Args:
weight (float or tensor_like): angle :math:`\theta` entering the Z rotation
wires (Any or Iterable[Any]): full set of wires that the operator acts on
wires1 (Iterable): Wires of the qubits representing the subset of occupied orbitals
in the interval ``[s, r]``.
wires2 (Iterable): Wires of the qubits representing the subset of unoccupied
orbitals in the interval ``[q, p]``.
Returns:
list[.Operator]: decomposition of the operator
"""
s = wires1[0]
r = wires1[-1]
q = wires2[0]
p = wires2[-1]
# Sequence of the wires entering the CNOTs
cnots_occ = [wires1[l : l + 2] for l in range(len(wires1) - 1)]
cnots_unocc = [wires2[l : l + 2] for l in range(len(wires2) - 1)]
set_cnot_wires = cnots_occ + [[r, q]] + cnots_unocc
op_list = []
op_list.extend(_layer1(weight, s, r, q, p, set_cnot_wires))
op_list.extend(_layer2(weight, s, r, q, p, set_cnot_wires))
op_list.extend(_layer3(weight, s, r, q, p, set_cnot_wires))
op_list.extend(_layer4(weight, s, r, q, p, set_cnot_wires))
op_list.extend(_layer5(weight, s, r, q, p, set_cnot_wires))
op_list.extend(_layer6(weight, s, r, q, p, set_cnot_wires))
op_list.extend(_layer7(weight, s, r, q, p, set_cnot_wires))
op_list.extend(_layer8(weight, s, r, q, p, set_cnot_wires))
return op_list
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