def decompose_ua(phi, wires=None):
r"""Implements the circuit decomposing the controlled application of the unitary
:math:`U_A(\phi)`
.. math::
U_A(\phi) = \left(\begin{array}{cc} 0 & e^{-i\phi} \\ e^{-i\phi} & 0 \\ \end{array}\right)
in terms of the quantum operations supported by PennyLane.
:math:`U_A(\phi)` is used in `arXiv:1805.04340 <https://arxiv.org/abs/1805.04340>`_,
to define two-qubit exchange gates required to build particle-conserving
VQE ansatze for quantum chemistry simulations. See :func:`~.ParticleConservingU1`.
:math:`U_A(\phi)` is expressed in terms of ``PhaseShift``, ``Rot`` and ``PauliZ`` operations
:math:`U_A(\phi) = R_\phi(-2\phi) R(-\phi, \pi, \phi) \sigma_z`.
Args:
phi (float): angle :math:`\phi` defining the unitary :math:`U_A(\phi)`
wires (list[Wires]): the wires ``n`` and ``m`` the circuit acts on
"""
n, m = wires
CZ(wires=wires)
CRot(-phi, np.pi, phi, wires=wires)
# decomposition of C-PhaseShift(2*phi) gate
PhaseShift(-phi, wires=m)
CNOT(wires=wires)
PhaseShift(phi, wires=m)
CNOT(wires=wires)
PhaseShift(-phi, wires=n)
def expand(self):
weight = self.parameters[0]
# Interpret first and last wire as r and p
r = self.wires[0]
p = self.wires[-1]
# Sequence of the wires entering the CNOTs between wires 'r' and 'p'
set_cnot_wires = [
self.wires[l:l + 2] for l in range(len(self.wires) - 1)
]
with qml.tape.QuantumTape() as tape:
# ------------------------------------------------------------------
# Apply the first layer
# U_1, U_2 acting on wires 'r' and 'p'
RX(-np.pi / 2, wires=r)
Hadamard(wires=p)
# Applying CNOTs between wires 'r' and 'p'
for cnot_wires in set_cnot_wires:
CNOT(wires=cnot_wires)
# Z rotation acting on wire 'p'
RZ(weight / 2, wires=p)
# Applying CNOTs in reverse order
for cnot_wires in reversed(set_cnot_wires):
CNOT(wires=cnot_wires)
# U_1^+, U_2^+ acting on wires 'r' and 'p'
RX(np.pi / 2, wires=r)
Hadamard(wires=p)
# ------------------------------------------------------------------
# Apply the second layer
# U_1, U_2 acting on wires 'r' and 'p'
Hadamard(wires=r)
RX(-np.pi / 2, wires=p)
# Applying CNOTs between wires 'r' and 'p'
for cnot_wires in set_cnot_wires:
CNOT(wires=cnot_wires)
# Z rotation acting on wire 'p'
RZ(-weight / 2, wires=p)
# Applying CNOTs in reverse order
for cnot_wires in reversed(set_cnot_wires):
CNOT(wires=cnot_wires)
# U_1^+, U_2^+ acting on wires 'r' and 'p'
Hadamard(wires=r)
RX(np.pi / 2, wires=p)
return tape
def zz(weight, wires):
"""Template for decomposition of ZZ coupling.
Args:
wires (list[int]): qubit indices that the template acts on
"""
CNOT(wires=wires)
RZ(2 * weight, wires=wires[0])
CNOT(wires=wires)
def test_bell_state(self, tol):
"""Tests that we correctly prepare a Bell state by applying a Hadamard then a CNOT"""
dev = qml.device("default.mixed", wires=2)
ops = [Hadamard(0), CNOT(wires=[0, 1])]
dev.apply(ops)
bell = np.zeros((4, 4))
bell[0, 0] = bell[0, 3] = bell[3, 0] = bell[3, 3] = 1 / 2
assert np.allclose(bell, dev.state, atol=tol, rtol=0)
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 (sequence[int]): two-element sequence with the indices of the qubits
the CNOT gates act on
"""
# U_1, U_2, U_3, U_4 acting on wires 's', 'r', 'q' and 'p'
RX(-np.pi / 2, wires=s)
Hadamard(wires=r)
RX(-np.pi / 2, wires=q)
RX(-np.pi / 2, wires=p)
# Applying CNOTs between wires 's' and 'p'
for cnot_wires in set_cnot_wires:
CNOT(wires=cnot_wires)
# Z rotation acting on wire 'p'
RZ(weight / 8, wires=p)
# Applying CNOTs in reverse order
for cnot_wires in reversed(set_cnot_wires):
CNOT(wires=cnot_wires)
# U_1^+, U_2^+, U_3^+, U_4^+ acting on wires 's', 'r', 'q' and 'p'
RX(np.pi / 2, wires=s)
Hadamard(wires=r)
RX(np.pi / 2, wires=q)
RX(np.pi / 2, wires=p)
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
"""
# U_1, U_2, U_3, U_4 acting on wires 's', 'r', 'q' and 'p'
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:
CNOT(wires=cnot_wires)
# Z rotation acting on wire 'p'
RZ(-weight / 8, wires=p)
# Applying CNOTs in reverse order
for cnot_wires in reversed(set_cnot_wires):
CNOT(wires=cnot_wires)
# U_1^+, U_2^+, U_3^+, U_4^+ acting on wires 's', 'r', 'q' and 'p'
RX(np.pi / 2, wires=s)
RX(np.pi / 2, wires=r)
Hadamard(wires=q)
RX(np.pi / 2, wires=p)
def u2_ex_gate(phi, wires=None):
r"""Implements the two-qubit exchange gate :math:`U_{2,\mathrm{ex}}` proposed in
`arXiv:1805.04340 <https://arxiv.org/abs/1805.04340>`_ to build particle-conserving VQE ansatze
for Quantum Chemistry simulations.
The unitary matrix :math:`U_{2, \mathrm{ex}}` acts on the Hilbert space of two qubits
.. math::
U_{2, \mathrm{ex}}(\phi) = \left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \mathrm{cos}(\phi) & -i\;\mathrm{sin}(\phi) & 0 \\
0 & -i\;\mathrm{sin}(\phi) & \mathrm{cos}(\phi) & 0 \\
0 & 0 & 0 & 1 \\
\end{array}\right).
Args:
phi (float): angle entering the controlled-RX operator :math:`CRX(2\phi)`
wires (list[Wires]): the two wires ``n`` and ``m`` the circuit acts on
"""
CNOT(wires=wires)
CRX(2 * phi, wires=wires[::-1])
CNOT(wires=wires)
def qaoa_ising_hamiltonian(weights, wires, local_fields, l):
"""Implements the Ising-like Hamiltonian of the QAOA embedding.
Args:
weights (array): array of weights
wires (Sequence[int] or int): `n` qubit indices that the template acts on
local_fields (str): gate implementing the local field
l (int): layer index
"""
# trainable "Ising" ansatz
if len(wires) == 1:
local_fields(weights[l][0], wires=wires[0])
elif len(wires) == 2:
# ZZ coupling
CNOT(wires=[wires[0], wires[1]])
RZ(2 * weights[l][0], wires=wires[0])
CNOT(wires=[wires[0], wires[1]])
# local fields
for i, _ in enumerate(wires):
local_fields(weights[l][i + 1], wires=wires[i])
else:
for i, _ in enumerate(wires):
if i < len(wires) - 1:
# ZZ coupling
CNOT(wires=[wires[i], wires[i + 1]])
RZ(2 * weights[l][i], wires=wires[i])
CNOT(wires=[wires[i], wires[i + 1]])
else:
# ZZ coupling to enforce periodic boundary condition
CNOT(wires=[wires[i], wires[0]])
RZ(2 * weights[l][i], wires=wires[i])
CNOT(wires=[wires[i], wires[0]])
# local fields
for i, _ in enumerate(wires):
local_fields(weights[l][len(wires) + i], wires=wires[i])
def SingleExcitationUnitary(weight, wires=None):
r"""Circuit to exponentiate the tensor product of Pauli matrices representing the
single-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 single-excitation operator is given by
.. math::
\hat{U}_{pr}(\theta) = \mathrm{exp} \{ \theta_{pr} (\hat{c}_p^\dagger \hat{c}_r
-\mathrm{H.c.}) \},
where :math:`\hat{c}` and :math:`\hat{c}^\dagger` are the fermionic annihilation and
creation operators and the indices :math:`r` and :math:`p` 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}_{pr}(\theta) = \mathrm{exp} \Big\{ \frac{i\theta}{2}
\bigotimes_{a=r+1}^{p-1}\hat{Z}_a (\hat{Y}_r \hat{X}_p) \Big\}
\mathrm{exp} \Big\{ -\frac{i\theta}{2}
\bigotimes_{a=r+1}^{p-1} \hat{Z}_a (\hat{X}_r \hat{Y}_p) \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/single_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` and :math:`U_2` in the figure above. See the Usage Details section for more
information.
Args:
weight (float): angle :math:`\theta` entering the Z rotation acting on wire ``p``
wires (Iterable or Wires): Wires that the template acts on.
The wires represent the subset of orbitals in the interval ``[r, p]``. Must be of
minimum length 2. The first wire is interpreted as ``r`` and the last wire as ``p``.
Wires in between are acted on with CNOT gates to compute the parity of the set
of qubits.
Raises:
ValueError: if inputs do not have the correct format
.. UsageDetails::
Notice that:
#. :math:`\hat{U}_{pr}(\theta)` involves two exponentiations where :math:`\hat{U}_1`,
:math:`\hat{U}_2`, and :math:`\hat{U}_\theta` are defined as follows,
.. math::
[U_1, U_2, U_{\theta}] = \Bigg\{\bigg[R_x(-\pi/2), H, R_z(\theta/2)\bigg],
\bigg[H, R_x(-\frac{\pi}{2}), R_z(-\theta/2) \bigg] \Bigg\}
#. For a given pair ``[r, p]``, ten single-qubit and ``4*(len(wires)-1)`` CNOT
operations are applied. Notice also that CNOT gates act only on qubits
``wires[1]`` to ``wires[-2]``. 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
from pennylane.templates import SingleExcitationUnitary
dev = qml.device('default.qubit', wires=3)
@qml.qnode(dev)
def circuit(weight, wires=None):
SingleExcitationUnitary(weight, wires=wires)
return qml.expval(qml.PauliZ(0))
weight = 0.56
print(circuit(weight, wires=[0, 1, 2]))
"""
##############
# Input checks
wires = Wires(wires)
if len(wires) < 2:
raise ValueError("expected at least two wires; got {}".format(
len(wires)))
expected_shape = ()
check_shape(
weight,
expected_shape,
msg="'weight' must be of shape {}; got {}".format(
expected_shape, get_shape(weight)),
)
###############
# Interpret first and last wire as r and p
r = wires[0]
p = wires[-1]
# Sequence of the wires entering the CNOTs between wires 'r' and 'p'
set_cnot_wires = [wires.subset([l, l + 1]) for l in range(len(wires) - 1)]
# ------------------------------------------------------------------
# Apply the first layer
# U_1, U_2 acting on wires 'r' and 'p'
RX(-np.pi / 2, wires=r)
Hadamard(wires=p)
# Applying CNOTs between wires 'r' and 'p'
for cnot_wires in set_cnot_wires:
CNOT(wires=cnot_wires)
# Z rotation acting on wire 'p'
RZ(weight / 2, wires=p)
# Applying CNOTs in reverse order
for cnot_wires in reversed(set_cnot_wires):
CNOT(wires=cnot_wires)
# U_1^+, U_2^+ acting on wires 'r' and 'p'
RX(np.pi / 2, wires=r)
Hadamard(wires=p)
# ------------------------------------------------------------------
# Apply the second layer
# U_1, U_2 acting on wires 'r' and 'p'
Hadamard(wires=r)
RX(-np.pi / 2, wires=p)
# Applying CNOTs between wires 'r' and 'p'
for cnot_wires in set_cnot_wires:
CNOT(wires=cnot_wires)
# Z rotation acting on wire 'p'
RZ(-weight / 2, wires=p)
# Applying CNOTs in reverse order
for cnot_wires in reversed(set_cnot_wires):
CNOT(wires=cnot_wires)
# U_1^+, U_2^+ acting on wires 'r' and 'p'
Hadamard(wires=r)
RX(np.pi / 2, wires=p)
def ConstantTemplateDouble(wires):
T(wires=wires[0])
CNOT(wires=wires)