qml.Reflection Operation

doc/introduction/templates.rst

@@ -211,6 +211,10 @@ Other useful templates which do not belong to the previous categories can be fou
     :description: :doc:`Grover Diffusion Operator <../code/api/pennylane.GroverOperator>`
     :figure: _static/templates/subroutines/grover.svg
 
+.. gallery-item::
+    :description: :doc:`Reflection Operator <../code/api/pennylane.Reflection>`
+    :figure: _static/templates/subroutines/reflection.png
+
 .. gallery-item::
     :description: :doc:`Interferometer <../code/api/pennylane.Interferometer>`
     :figure: _static/templates/subroutines/interferometer.png

doc/releases/changelog-dev.md

@@ -223,6 +223,10 @@
 
 * The module `pennylane/math/quantum.py` has now support for the min-entropy.
   [(#3959)](https://github.com/PennyLaneAI/pennylane/pull/3959/)
+
+* Create the `qml.Reflection` operator, useful for amplitude amplification and its variants.
+  [(##5159)](https://github.com/PennyLaneAI/pennylane/pull/5159)
+
 
 * A function called `apply_operation` has been added to the new `qutrit_mixed` module found in `qml.devices` that applies operations to device-compatible states.
   [(#5032)](https://github.com/PennyLaneAI/pennylane/pull/5032)
@@ -638,6 +642,7 @@
 This release contains contributions from (in alphabetical order):
 
 Abhishek Abhishek,
+Guillermo Alonso,
 Mikhail Andrenkov,
 Utkarsh Azad,
 Trenten Babcock,

pennylane/templates/subroutines/__init__.py

@@ -39,3 +39,4 @@
 from .controlled_sequence import ControlledSequence
 from .trotter import TrotterProduct
 from .aqft import AQFT
+from .reflection import Reflection

pennylane/templates/subroutines/reflection.py

@@ -0,0 +1,166 @@
+# Copyright 2018-2024 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 submodule contains the template for the Reflection operation.
+"""
+
+import numpy as np
+import pennylane as qml
+from pennylane.operation import Operation
+from pennylane.queuing import QueuingManager
+
+
+class Reflection(Operation):
+    r"""Apply a reflection about a state :math:`|\Psi\rangle`.
+
+    Given an operator :math:`U` such that :math:`|\Psi\rangle = U|0\rangle`  and a reflection angle :math:`\alpha`,
+    this template creates the operation:
+
+    .. math::
+
+       R(U, \alpha) = -I + (1 - e^{i\alpha}) |\Psi\rangle \langle \Psi|
+
+    This operator is an important component of quantum algorithms such as amplitude amplification [`arXiv:quant-ph/0005055 <https://arxiv.org/abs/quant-ph/0005055>`__]
+    and oblivious amplitude amplification [`arXiv:1312.1414 <https://arxiv.org/abs/1312.1414>`__].
+
+    Args:
+        U (Operator): The operator that generates the state :math:`|\Psi\rangle`.
+        alpha (float): the angle of the operator. Default is :math:`\pi`.
+        reflection_wires (Any or Iterable[Any]): Subsystem of wires on which to reflect. The default is None and the reflection will be applied on the U wires.
+
+    **Example**
+
+    This example shows how to apply the reflection :math:`-I + 2|+\rangle \langle +|` to the state :math:`|1\rangle`.
+
+    .. code-block::
+
+        @qml.prod
+        def generator(wires):
+            qml.Hadamard(wires=wires)
+
+        U = generator(wires=0)
+
+        dev = qml.device('default.qubit')
+        @qml.qnode(dev)
+        def circuit():
+
+            # Initialize to the state |1>
+            qml.PauliX(wires=0)
+
+            # Apply the reflection
+            qml.Reflection(U)
+
+            return qml.state()
+
+        >>> circuit()
+        tensor([1.+6.123234e-17j, 0.-6.123234e-17j], requires_grad=True)
+
+
+
+    .. details::
+        :title: Theory
+
+        The operator is built as follows:
+
+        .. math::
+
+            \text{Reflection}(U, \alpha) = -I + (1 - e^{i\alpha}) |\Psi\rangle \langle \Psi| = U(-I + (1 - e^{i\alpha}) |0\rangle \langle 0|)U^{\dagger}.
+
+        The central block is obtained through a :class:`~.PhaseShift` controlled operator.
+
+        In the case of specifying the reflection wires, the operator would have the following expression.
+
+        .. math::
+
+            U(-I + (1 - e^{i\alpha}) |0\rangle^{\otimes m} \langle 0|^{\otimes m}\otimes I^{n-m})U^{\dagger},
+
+        where :math:`m` is the number of reflection wires and :math:`n` is the total number of wires.
+
+    """
+
+    def _flatten(self):
+        data = (self.hyperparameters["base"], self.parameters[0])
+        metadata = tuple(value for key, value in self.hyperparameters.items() if key != "base")
+        return data, metadata
+
+    @classmethod
+    def _unflatten(cls, data, metadata):
+        U, alpha = (data[0], data[1])
+        return cls(U, alpha=alpha, reflection_wires=metadata[0])
+
+    def __init__(self, U, alpha=np.pi, reflection_wires=None, id=None):
+        self._name = "Reflection"
+        wires = U.wires
+
+        if reflection_wires is None:
+            reflection_wires = U.wires
+
+        if not set(reflection_wires).issubset(set(U.wires)):
+            raise ValueError("The reflection wires must be a subset of the operation wires.")
+
+        self._hyperparameters = {
+            "base": U,
+            "reflection_wires": reflection_wires,
+        }
+
+        super().__init__(alpha, wires=wires, id=id)
+
+    @property
+    def alpha(self):
+        """The alpha angle for the operation."""
+        return self.parameters[0]
+
+    @property
+    def reflection_wires(self):
+        """The reflection wires for the operation."""
+        return self.hyperparameters["reflection_wires"]
+
+    def queue(self, context=QueuingManager):
+        context.remove(self.hyperparameters["base"])
+        context.append(self)
+        return self
+
+    @staticmethod
+    def compute_decomposition(*parameters, wires=None, **hyperparameters):
+        alpha = parameters[0]
+        U = hyperparameters["base"]
+        reflection_wires = hyperparameters["reflection_wires"]
+
+        wires = qml.wires.Wires(reflection_wires) if reflection_wires is not None else wires
+
+        ops = []
+
+        ops.append(qml.GlobalPhase(np.pi))
+        ops.append(qml.adjoint(U))
+
+        if len(wires) > 1:
+            ops.append(qml.PauliX(wires=wires[-1]))
+            ops.append(
+                qml.ctrl(
+                    qml.PhaseShift(alpha, wires=wires[-1]),
+                    control=wires[:-1],
+                    control_values=[0] * (len(wires) - 1),
+                )
+            )
+            ops.append(qml.PauliX(wires=wires[-1]))
+
+        else:
+            ops.append(qml.PauliX(wires=wires))
+            ops.append(qml.PhaseShift(alpha, wires=wires))
+            ops.append(qml.PauliX(wires=wires))
+
+        ops.append(U)
+
+        return ops