Source code for pennylane.ops.op_math.controlled_ops
# 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 controlled operators based on the ControlledOp class.
"""
import warnings
from typing import Iterable
from functools import lru_cache
import numpy as np
from scipy.linalg import block_diag
import pennylane as qml
from pennylane.operation import (
AnyWires,
Wires,
)
from pennylane.ops.qubit.matrix_ops import QubitUnitary
from pennylane.ops.qubit.parametric_ops_single_qubit import stack_last
from .controlled import ControlledOp
from .controlled_decompositions import decompose_mcx
INV_SQRT2 = 1 / qml.math.sqrt(2)
# pylint: disable=too-few-public-methods
[docs]class ControlledQubitUnitary(ControlledOp):
r"""ControlledQubitUnitary(U, control_wires, wires, control_values)
Apply an arbitrary fixed unitary to ``wires`` with control from the ``control_wires``.
In addition to default ``Operation`` instance attributes, the following are
available for ``ControlledQubitUnitary``:
* ``control_wires``: wires that act as control for the operation
* ``control_values``: the state on which to apply the controlled operation (see below)
* ``target_wires``: the wires the unitary matrix will be applied to
* ``active_wires``: Wires modified by the operator. This is the control wires followed
by the target wires.
**Details:**
* Number of wires: Any (the operation can act on any number of wires)
* Number of parameters: 1
* Number of dimensions per parameter: (2,)
* Gradient recipe: None
Args:
base (Union[array[complex], QubitUnitary]): square unitary matrix or a QubitUnitary
operation. If passing a matrix, this will be used to construct a QubitUnitary
operator that will be used as the base operator. If providing a ``qml.QubitUnitary``,
this will be used as the base directly.
control_wires (Union[Wires, Sequence[int], or int]): the control wire(s)
wires (Union[Wires, Sequence[int], or int]): the wire(s) the unitary acts on
(optional if U is provided as a QubitUnitary)
control_values (List[int, bool]): a list providing the state of the control qubits to
control on (default is the all 1s state)
unitary_check (bool): whether to check whether an array U is unitary when creating the
operator (default False)
**Example**
The following shows how a single-qubit unitary can be applied to wire ``2`` with control on
both wires ``0`` and ``1``:
>>> U = np.array([[ 0.94877869, 0.31594146], [-0.31594146, 0.94877869]])
>>> qml.ControlledQubitUnitary(U, control_wires=[0, 1], wires=2)
Controlled(QubitUnitary(array([[ 0.94877869, 0.31594146],
[-0.31594146, 0.94877869]]), wires=[2]), control_wires=[0, 1])
Alternatively, the same operator can be constructed with a QubitUnitary:
>>> base = qml.QubitUnitary(U, wires=2)
>>> qml.ControlledQubitUnitary(base, control_wires=[0, 1])
Controlled(QubitUnitary(array([[ 0.94877869, 0.31594146],
[-0.31594146, 0.94877869]]), wires=[2]), control_wires=[0, 1])
Typically, controlled operations apply a desired gate if the control qubits
are all in the state :math:`\vert 1\rangle`. However, there are some situations where
it is necessary to apply a gate conditioned on all qubits being in the
:math:`\vert 0\rangle` state, or a mix of the two.
The state on which to control can be changed by passing a string of bits to
`control_values`. For example, if we want to apply a single-qubit unitary to
wire ``3`` conditioned on three wires where the first is in state ``0``, the
second is in state ``1``, and the third in state ``1``, we can write:
>>> qml.ControlledQubitUnitary(U, control_wires=[0, 1, 2], wires=3, control_values=[0, 1, 1])
or
>>> qml.ControlledQubitUnitary(U, control_wires=[0, 1, 2], wires=3, control_values=[False, True, True])
"""
num_wires = AnyWires
"""int: Number of wires that the operator acts on."""
num_params = 1
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = (2,)
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
grad_method = None
"""Gradient computation method."""
@classmethod
def _unflatten(cls, data, metadata):
return cls(
data[0], control_wires=metadata[0], control_values=metadata[1], work_wires=metadata[2]
)
# pylint: disable= too-many-arguments
def __init__(
self,
base,
control_wires,
wires=None,
control_values=None,
unitary_check=False,
work_wires=None,
):
if getattr(base, "wires", False) and wires is not None:
warnings.warn(
"base operator already has wires; values specified through wires kwarg will be ignored."
)
if isinstance(base, Iterable):
base = QubitUnitary(base, wires=wires, unitary_check=unitary_check)
super().__init__(
base,
control_wires,
control_values=control_values,
work_wires=work_wires,
)
self._name = "ControlledQubitUnitary"
def _controlled(self, wire):
ctrl_wires = wire + self.control_wires
values = None if self.control_values is None else [True] + self.control_values
return ControlledQubitUnitary(
self.base,
control_wires=ctrl_wires,
control_values=values,
work_wires=self.work_wires,
)
@property
def has_decomposition(self):
if not super().has_decomposition:
return False
with qml.QueuingManager.stop_recording():
# we know this is using try-except as logical control, but are favouring
# certainty in it being correct over explicitness in an edge case.
try:
self.decomposition()
except qml.operation.DecompositionUndefinedError:
return False
return True
[docs]class CH(ControlledOp):
r"""CH(wires)
The controlled-Hadamard operator
.. math:: CH = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}
\end{bmatrix}.
.. note:: The first wire provided corresponds to the **control qubit**.
**Details:**
* Number of wires: 2
* Number of parameters: 0
Args:
wires (Sequence[int]): the wires the operation acts on
"""
num_wires = 2
"""int: Number of wires that the operation acts on."""
num_params = 0
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = ()
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
name = "CH"
def _flatten(self):
return tuple(), (self.wires,)
@classmethod
def _unflatten(cls, data, metadata):
return cls(metadata[0])
def __init__(self, wires, id=None):
control_wires = wires[:1]
target_wires = wires[1:]
super().__init__(qml.Hadamard(wires=target_wires), control_wires, id=id)
def __repr__(self):
return f"CH(wires={self.wires.tolist()})"
[docs] @staticmethod
@lru_cache()
def compute_matrix(): # pylint: disable=arguments-differ
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.CH.matrix`
Returns:
ndarray: matrix
**Example**
>>> print(qml.CH.compute_matrix())
[[ 1. 0. 0. 0. ]
[ 0. 1. 0. 0. ]
[ 0. 0. 0.70710678 0.70710678]
[ 0. 0. 0.70710678 -0.70710678]]
"""
return np.array(
[
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, INV_SQRT2, INV_SQRT2],
[0, 0, INV_SQRT2, -INV_SQRT2],
]
)
[docs] @staticmethod
def compute_decomposition(wires): # pylint: disable=arguments-differ
r"""Representation of the operator as a product of other operators (static method).
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.CH.decomposition`.
Args:
wires (Iterable, Wires): wires that the operator acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> print(qml.CH.compute_decomposition([0, 1]))
[RY(-0.7853981633974483, wires=[1]), CZ(wires=[0, 1]), RY(0.7853981633974483, wires=[1])]
"""
return [
qml.RY(-np.pi / 4, wires=wires[1]),
qml.CZ(wires=wires),
qml.RY(+np.pi / 4, wires=wires[1]),
]
[docs]class CY(ControlledOp):
r"""CY(wires)
The controlled-Y operator
.. math:: CY = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0\\
0 & 0 & 0 & -i\\
0 & 0 & i & 0
\end{bmatrix}.
.. note:: The first wire provided corresponds to the **control qubit**.
**Details:**
* Number of wires: 2
* Number of parameters: 0
Args:
wires (Sequence[int]): the wires the operation acts on
id (str): custom label given to an operator instance,
can be useful for some applications where the instance has to be identified.
"""
num_wires = 2
"""int: Number of wires that the operator acts on."""
num_params = 0
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = ()
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
name = "CY"
def _flatten(self):
return tuple(), (self.wires,)
@classmethod
def _unflatten(cls, data, metadata):
return cls(metadata[0])
def __init__(self, wires, id=None):
control_wire, wire = wires
super().__init__(qml.Y(wire), control_wire, id=id)
def __repr__(self):
return f"CY(wires={self.wires.tolist()})"
[docs] @staticmethod
@lru_cache()
def compute_matrix(): # pylint: disable=arguments-differ
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.CY.matrix`
Returns:
ndarray: matrix
**Example**
>>> print(qml.CY.compute_matrix())
[[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 1.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j -0.-1.j]
[ 0.+0.j 0.+0.j 0.+1.j 0.+0.j]]
"""
return np.array(
[
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 0, -1j],
[0, 0, 1j, 0],
]
)
[docs] @staticmethod
def compute_decomposition(wires): # pylint: disable=arguments-differ
r"""Representation of the operator as a product of other operators (static method).
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.CY.decomposition`.
Args:
wires (Iterable, Wires): wires that the operator acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> print(qml.CY.compute_decomposition([0, 1]))
[CRY(3.141592653589793, wires=[0, 1])), S(wires=[0])]
"""
return [qml.CRY(np.pi, wires=wires), qml.S(wires=wires[0])]
[docs]class CZ(ControlledOp):
r"""CZ(wires)
The controlled-Z operator
.. math:: CZ = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & -1
\end{bmatrix}.
.. note:: The first wire provided corresponds to the **control qubit**.
**Details:**
* Number of wires: 2
* Number of parameters: 0
Args:
wires (Sequence[int]): the wires the operation acts on
"""
num_wires = 2
"""int: Number of wires that the operator acts on."""
num_params = 0
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = ()
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
name = "CZ"
def _flatten(self):
return tuple(), (self.wires,)
@classmethod
def _unflatten(cls, data, metadata):
return cls(metadata[0])
def __init__(self, wires, id=None):
control_wire, wire = wires
super().__init__(qml.Z(wires=wire), control_wire, id=id)
def __repr__(self):
return f"CZ(wires={self.wires.tolist()})"
[docs] @staticmethod
@lru_cache()
def compute_matrix(): # pylint: disable=arguments-differ
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.CZ.matrix`
Returns:
ndarray: matrix
**Example**
>>> print(qml.CZ.compute_matrix())
[[ 1 0 0 0]
[ 0 1 0 0]
[ 0 0 1 0]
[ 0 0 0 -1]]
"""
return np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, -1]])
def _controlled(self, wire):
return qml.CCZ(wires=wire + self.wires)
[docs] @staticmethod
def compute_decomposition(wires): # pylint: disable=arguments-differ
return [qml.ControlledPhaseShift(np.pi, wires=wires)]
[docs]class CSWAP(ControlledOp):
r"""CSWAP(wires)
The controlled-swap operator
.. math:: CSWAP = \begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{bmatrix}.
.. note:: The first wire provided corresponds to the **control qubit**.
**Details:**
* Number of wires: 3
* Number of parameters: 0
Args:
wires (Sequence[int]): the wires the operation acts on
"""
num_wires = 3
"""int : Number of wires that the operation acts on."""
num_params = 0
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = ()
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
name = "CSWAP"
def _flatten(self):
return tuple(), (self.wires,)
@classmethod
def _unflatten(cls, data, metadata):
return cls(metadata[0])
def __init__(self, wires, id=None):
control_wires = wires[:1]
target_wires = wires[1:]
super().__init__(qml.SWAP(wires=target_wires), control_wires, id=id)
def __repr__(self):
return f"CSWAP(wires={self.wires.tolist()})"
[docs] @staticmethod
@lru_cache()
def compute_matrix(): # pylint: disable=arguments-differ
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.CSWAP.matrix`
Returns:
ndarray: matrix
**Example**
>>> print(qml.CSWAP.compute_matrix())
[[1 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0]
[0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 1 0]
[0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 1]]
"""
return np.array(
[
[1, 0, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1],
]
)
[docs] @staticmethod
def compute_decomposition(wires): # pylint: disable=arguments-differ
r"""Representation of the operator as a product of other operators (static method).
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.CSWAP.decomposition`.
Args:
wires (Iterable, Wires): wires that the operator acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> print(qml.CSWAP.compute_decomposition((0,1,2)))
[Toffoli(wires=[0, 2, 1]), Toffoli(wires=[0, 1, 2]), Toffoli(wires=[0, 2, 1])]
"""
decomp_ops = [
qml.Toffoli(wires=[wires[0], wires[2], wires[1]]),
qml.Toffoli(wires=[wires[0], wires[1], wires[2]]),
qml.Toffoli(wires=[wires[0], wires[2], wires[1]]),
]
return decomp_ops
[docs]class CCZ(ControlledOp):
r"""CCZ(wires)
CCZ (controlled-controlled-Z) gate.
.. math::
CCZ =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1
\end{pmatrix}
**Details:**
* Number of wires: 3
* Number of parameters: 0
Args:
wires (Sequence[int]): the subsystem the gate acts on
"""
def _flatten(self):
return tuple(), (self.wires,)
@classmethod
def _unflatten(cls, data, metadata):
return cls(metadata[0])
num_wires = 3
"""int: Number of wires that the operator acts on."""
num_params = 0
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = ()
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
name = "CCZ"
def __init__(self, wires, id=None):
control_wires = wires[:2]
target_wires = wires[2:]
super().__init__(qml.PauliZ(wires=target_wires), control_wires, id=id)
def __repr__(self):
return f"CCZ(wires={self.wires.tolist()})"
[docs] @staticmethod
@lru_cache()
def compute_matrix(): # pylint: disable=arguments-differ
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.CCZ.matrix`
Returns:
ndarray: matrix
**Example**
>>> print(qml.CCZ.compute_matrix())
[[1 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0]
[0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 -1]]
"""
return np.array(
[
[1, 0, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 0, -1],
]
)
[docs] @staticmethod
def compute_decomposition(wires): # pylint: disable=arguments-differ
r"""Representation of the operator as a product of other operators (static method).
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.Toffoli.decomposition`.
Args:
wires (Iterable, Wires): wires that the operator acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> qml.CCZ.compute_decomposition((0,1,2))
[CNOT(wires=[1, 2]),
Adjoint(T(wires=[2])),
CNOT(wires=[0, 2]),
T(wires=[2]),
CNOT(wires=[1, 2]),
Adjoint(T(wires=[2])),
CNOT(wires=[0, 2]),
T(wires=[2]),
T(wires=[1]),
CNOT(wires=[0, 1]),
Hadamard(wires=[2]),
T(wires=[0]),
Adjoint(T(wires=[1])),
CNOT(wires=[0, 1]),
Hadamard(wires=[2])]
"""
return [
qml.CNOT(wires=[wires[1], wires[2]]),
qml.adjoint(qml.T(wires=wires[2])),
qml.CNOT(wires=[wires[0], wires[2]]),
qml.T(wires=wires[2]),
qml.CNOT(wires=[wires[1], wires[2]]),
qml.adjoint(qml.T(wires=wires[2])),
qml.CNOT(wires=[wires[0], wires[2]]),
qml.T(wires=wires[2]),
qml.T(wires=wires[1]),
qml.CNOT(wires=[wires[0], wires[1]]),
qml.Hadamard(wires=wires[2]),
qml.T(wires=wires[0]),
qml.adjoint(qml.T(wires=wires[1])),
qml.CNOT(wires=[wires[0], wires[1]]),
qml.Hadamard(wires=wires[2]),
]
[docs]class CNOT(ControlledOp):
r"""CNOT(wires)
The controlled-NOT operator
.. math:: CNOT = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0
\end{bmatrix}.
.. note:: The first wire provided corresponds to the **control qubit**.
**Details:**
* Number of wires: 2
* Number of parameters: 0
Args:
wires (Sequence[int]): the wires the operation acts on
"""
num_wires = 2
"""int: Number of wires that the operator acts on."""
num_params = 0
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = ()
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
arithmetic_depth = 0
name = "CNOT"
def _flatten(self):
return tuple(), (self.wires,)
@classmethod
def _unflatten(cls, data, metadata):
return cls(metadata[0])
def __init__(self, wires, id=None):
control_wire, wire = wires
super().__init__(qml.PauliX(wires=wire), control_wire, id=id)
def __repr__(self):
return f"CNOT(wires={self.wires.tolist()})"
[docs] @staticmethod
@lru_cache()
def compute_matrix(): # pylint: disable=arguments-differ
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.CNOT.matrix`
Returns:
ndarray: matrix
**Example**
>>> print(qml.CNOT.compute_matrix())
[[1 0 0 0]
[0 1 0 0]
[0 0 0 1]
[0 0 1 0]]
"""
return np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]])
def _controlled(self, wire):
return qml.Toffoli(wires=wire + self.wires)
[docs]class Toffoli(ControlledOp):
r"""Toffoli(wires)
Toffoli (controlled-controlled-X) gate.
.. math::
Toffoli =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0
\end{pmatrix}
**Details:**
* Number of wires: 3
* Number of parameters: 0
Args:
wires (Sequence[int]): the subsystem the gate acts on
"""
num_wires = 3
"""int: Number of wires that the operator acts on."""
num_params = 0
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = ()
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
arithmetic_depth = 0
name = "Toffoli"
def _flatten(self):
return tuple(), (self.wires,)
@classmethod
def _unflatten(cls, _, metadata):
return cls(metadata[0])
def __init__(self, wires, id=None):
control_wires = wires[:2]
target_wires = wires[2:]
super().__init__(qml.PauliX(wires=target_wires), control_wires, id=id)
def __repr__(self):
return f"Toffoli(wires={self.wires.tolist()})"
[docs] @staticmethod
@lru_cache()
def compute_matrix(): # pylint: disable=arguments-differ
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.Toffoli.matrix`
Returns:
ndarray: matrix
**Example**
>>> print(qml.Toffoli.compute_matrix())
[[1 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0]
[0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 1]
[0 0 0 0 0 0 1 0]]
"""
return np.array(
[
[1, 0, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 0, 1, 0],
]
)
[docs] @staticmethod
def compute_decomposition(wires): # pylint: disable=arguments-differ
r"""Representation of the operator as a product of other operators (static method).
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.Toffoli.decomposition`.
Args:
wires (Iterable, Wires): wires that the operator acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> qml.Toffoli.compute_decomposition((0,1,2))
[Hadamard(wires=[2]),
CNOT(wires=[1, 2]),
Adjoint(T(wires=[2])),
CNOT(wires=[0, 2]),
T(wires=[2]),
CNOT(wires=[1, 2]),
Adjoint(T(wires=[2])),
CNOT(wires=[0, 2]),
T(wires=[2]),
T(wires=[1]),
CNOT(wires=[0, 1]),
Hadamard(wires=[2]),
T(wires=[0]),
Adjoint(T(wires=[1])),
CNOT(wires=[0, 1])]
"""
return [
qml.Hadamard(wires=wires[2]),
CNOT(wires=[wires[1], wires[2]]),
qml.adjoint(qml.T(wires=wires[2])),
CNOT(wires=[wires[0], wires[2]]),
qml.T(wires=wires[2]),
CNOT(wires=[wires[1], wires[2]]),
qml.adjoint(qml.T(wires=wires[2])),
CNOT(wires=[wires[0], wires[2]]),
qml.T(wires=wires[2]),
qml.T(wires=wires[1]),
CNOT(wires=[wires[0], wires[1]]),
qml.Hadamard(wires=wires[2]),
qml.T(wires=wires[0]),
qml.adjoint(qml.T(wires=wires[1])),
CNOT(wires=[wires[0], wires[1]]),
]
def _check_and_convert_control_values(control_values, control_wires):
if isinstance(control_values, str):
# Make sure all values are either 0 or 1
if not set(control_values).issubset({"1", "0"}):
raise ValueError("String of control values can contain only '0' or '1'.")
control_values = [int(x) for x in control_values]
if control_values is None:
return [1] * len(control_wires)
if len(control_values) != len(control_wires):
raise ValueError("Length of control values must equal number of control wires.")
return control_values
[docs]class MultiControlledX(ControlledOp):
r"""MultiControlledX(control_wires, wires, control_values)
Apply a Pauli X gate controlled on an arbitrary computational basis state.
**Details:**
* Number of wires: Any (the operation can act on any number of wires)
* Number of parameters: 0
* Gradient recipe: None
Args:
control_wires (Union[Wires, Sequence[int], or int]): Deprecated way to indicate the control wires.
Now users should use "wires" to indicate both the control wires and the target wire.
wires (Union[Wires, Sequence[int], or int]): control wire(s) followed by a single target wire where
the operation acts on
control_values (Union[bool, list[bool], int, list[int]]): The value(s) the control wire(s)
should take. Integers other than 0 or 1 will be treated as ``int(bool(x))``.
work_wires (Union[Wires, Sequence[int], or int]): optional work wires used to decompose
the operation into a series of Toffoli gates
.. note::
If ``MultiControlledX`` is not supported on the targeted device, PennyLane will decompose
the operation into :class:`~.Toffoli` and/or :class:`~.CNOT` gates. When controlling on
three or more wires, the Toffoli-based decompositions described in Lemmas 7.2 and 7.3 of
`Barenco et al. <https://arxiv.org/abs/quant-ph/9503016>`__ will be used. These methods
require at least one work wire.
The number of work wires provided determines the decomposition method used and the resulting
number of Toffoli gates required. When ``MultiControlledX`` is controlling on :math:`n`
wires:
#. If at least :math:`n - 2` work wires are provided, the decomposition in Lemma 7.2 will be
applied using the first :math:`n - 2` work wires.
#. If fewer than :math:`n - 2` work wires are provided, a combination of Lemmas 7.3 and 7.2
will be applied using only the first work wire.
These methods present a tradeoff between qubit number and depth. The method in point 1
requires fewer Toffoli gates but a greater number of qubits.
Note that the state of the work wires before and after the decomposition takes place is
unchanged.
"""
is_self_inverse = True
"""bool: Whether or not the operator is self-inverse."""
num_wires = AnyWires
"""int: Number of wires the operation acts on."""
num_params = 0
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = ()
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
name = "MultiControlledX"
def _flatten(self):
return (), (self.active_wires, tuple(self.control_values), self.work_wires)
@classmethod
def _unflatten(cls, _, metadata):
return cls(wires=metadata[0], control_values=metadata[1], work_wires=metadata[2])
# pylint: disable=too-many-arguments
def __init__(self, control_wires=None, wires=None, control_values=None, work_wires=None):
if wires is None:
raise ValueError("Must specify the wires where the operation acts on")
wires = wires if isinstance(wires, Wires) else Wires(wires)
if control_wires is not None:
warnings.warn(
"The control_wires keyword will be removed soon. Use wires = (control_wires, "
"target_wire) instead. See the documentation for more information.",
UserWarning,
)
if len(wires) != 1:
raise ValueError("MultiControlledX accepts a single target wire.")
else:
if len(wires) < 2:
raise ValueError(
f"MultiControlledX: wrong number of wires. {len(wires)} wire(s) given. Need at least 2."
)
control_wires = wires[:-1]
wires = wires[-1:]
control_values = _check_and_convert_control_values(control_values, control_wires)
super().__init__(
qml.X(wires),
control_wires=control_wires,
control_values=control_values,
work_wires=work_wires,
)
def __repr__(self):
return f"MultiControlledX(wires={self.active_wires.tolist()}, control_values={self.control_values})"
@property
def wires(self):
return self.active_wires
# pylint: disable=unused-argument, arguments-differ
[docs] @staticmethod
def compute_matrix(control_wires, control_values=None, **kwargs):
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.MultiControlledX.matrix`
Args:
control_wires (Any or Iterable[Any]): wires to place controls on
control_values (Union[bool, list[bool], int, list[int]]): The value(s) the control wire(s)
should take. Integers other than 0 or 1 will be treated as ``int(bool(x))``.
Returns:
tensor_like: matrix representation
**Example**
>>> print(qml.MultiControlledX.compute_matrix([0], [1]))
[[1. 0. 0. 0.]
[0. 1. 0. 0.]
[0. 0. 0. 1.]
[0. 0. 1. 0.]]
>>> print(qml.MultiControlledX.compute_matrix([1], [0]))
[[0. 1. 0. 0.]
[1. 0. 0. 0.]
[0. 0. 1. 0.]
[0. 0. 0. 1.]]
"""
control_values = _check_and_convert_control_values(control_values, control_wires)
padding_left = sum(2**i * int(val) for i, val in enumerate(reversed(control_values))) * 2
padding_right = 2 ** (len(control_wires) + 1) - 2 - padding_left
return block_diag(np.eye(padding_left), qml.X.compute_matrix(), np.eye(padding_right))
[docs] def matrix(self, wire_order=None):
canonical_matrix = self.compute_matrix(self.control_wires, self.control_values)
wire_order = wire_order or self.wires
return qml.math.expand_matrix(
canonical_matrix, wires=self.active_wires, wire_order=wire_order
)
# pylint: disable=unused-argument, arguments-differ
[docs] @staticmethod
def compute_decomposition(wires=None, work_wires=None, control_values=None, **kwargs):
r"""Representation of the operator as a product of other operators (static method).
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.MultiControlledX.decomposition`.
Args:
wires (Iterable[Any] or Wires): wires that the operation acts on
work_wires (Wires): optional work wires used to decompose
the operation into a series of Toffoli gates.
control_values (Union[bool, list[bool], int, list[int]]): The value(s) the control wire(s)
should take. Integers other than 0 or 1 will be treated as ``int(bool(x))``.
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> print(qml.MultiControlledX.compute_decomposition(
... wires=[0,1,2,3], control_values=[1,1,1], work_wires=qml.wires.Wires("aux")))
[Toffoli(wires=[2, 'aux', 3]),
Toffoli(wires=[0, 1, 'aux']),
Toffoli(wires=[2, 'aux', 3]),
Toffoli(wires=[0, 1, 'aux'])]
"""
if len(wires) < 2:
raise ValueError(f"Wrong number of wires. {len(wires)} given. Need at least 2.")
target_wire = wires[-1]
control_wires = wires[:-1]
if control_values is None:
control_values = [True] * len(control_wires)
work_wires = work_wires or []
if len(control_wires) > 2 and len(work_wires) == 0:
raise ValueError(
"At least one work wire is required to decompose operation: MultiControlledX"
)
flips1 = [qml.X(w) for w, val in zip(control_wires, control_values) if not val]
if len(control_wires) == 1:
decomp = [qml.CNOT(wires=wires)]
elif len(control_wires) == 2:
decomp = qml.Toffoli.compute_decomposition(wires=wires)
else:
decomp = decompose_mcx(control_wires, target_wire, work_wires)
flips2 = [qml.X(w) for w, val in zip(control_wires, control_values) if not val]
return flips1 + decomp + flips2
[docs] def decomposition(self):
return self.compute_decomposition(self.active_wires, self.work_wires, self.control_values)
[docs]class CRX(ControlledOp):
r"""The controlled-RX operator
.. math::
\begin{align}
CR_x(\phi) &=
\begin{bmatrix}
& 1 & 0 & 0 & 0 \\
& 0 & 1 & 0 & 0\\
& 0 & 0 & \cos(\phi/2) & -i\sin(\phi/2)\\
& 0 & 0 & -i\sin(\phi/2) & \cos(\phi/2)
\end{bmatrix}.
\end{align}
**Details:**
* Number of wires: 2
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
* Gradient recipe: The controlled-RX operator satisfies a four-term parameter-shift rule
(see Appendix F, https://doi.org/10.1088/1367-2630/ac2cb3):
.. math::
\frac{d}{d\phi}f(CR_x(\phi)) = c_+ \left[f(CR_x(\phi+a)) - f(CR_x(\phi-a))\right] - c_- \left[f(CR_x(\phi+b)) - f(CR_x(\phi-b))\right]
where :math:`f` is an expectation value depending on :math:`CR_x(\phi)`, and
- :math:`a = \pi/2`
- :math:`b = 3\pi/2`
- :math:`c_{\pm} = (\sqrt{2} \pm 1)/{4\sqrt{2}}`
Args:
phi (float): rotation angle :math:`\phi`
wires (Sequence[int]): the wire the operation acts on
id (str or None): String representing the operation (optional)
"""
num_wires = 2
"""int: Number of wires that the operation acts on."""
num_params = 1
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = (0,)
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
name = "CRX"
parameter_frequencies = [(0.5, 1.0)]
def __init__(self, phi, wires, id=None):
super().__init__(qml.RX(phi, wires=wires[1:]), control_wires=wires[0], id=id)
def __repr__(self):
return f"CRX({self.data[0]}, wires={self.wires.tolist()})"
@classmethod
def _unflatten(cls, data, metadata):
base = data[0]
control_wires = metadata[0]
return cls(*base.data, wires=control_wires + base.wires)
[docs] @staticmethod
def compute_matrix(theta): # pylint: disable=arguments-differ
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.CRX.matrix`
Args:
theta (tensor_like or float): rotation angle
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.CRX.compute_matrix(torch.tensor(0.5))
tensor([[1.0+0.0j, 0.0+0.0j, 0.0+0.0j, 0.0+0.0j],
[0.0+0.0j, 1.0+0.0j, 0.0+0.0j, 0.0+0.0j],
[0.0+0.0j, 0.0+0.0j, 0.9689+0.0j, 0.0-0.2474j],
[0.0+0.0j, 0.0+0.0j, 0.0-0.2474j, 0.9689+0.0j]])
"""
interface = qml.math.get_interface(theta)
c = qml.math.cos(theta / 2)
s = qml.math.sin(theta / 2)
if interface == "tensorflow":
c = qml.math.cast_like(c, 1j)
s = qml.math.cast_like(s, 1j)
# The following avoids casting an imaginary quantity to reals when back propagating
c = (1 + 0j) * c
js = -1j * s
ones = qml.math.ones_like(js)
zeros = qml.math.zeros_like(js)
matrix = [
[ones, zeros, zeros, zeros],
[zeros, ones, zeros, zeros],
[zeros, zeros, c, js],
[zeros, zeros, js, c],
]
return qml.math.stack([stack_last(row) for row in matrix], axis=-2)
[docs] @staticmethod
def compute_decomposition(phi, wires): # pylint: disable=arguments-differ
r"""Representation of the operator as a product of other operators (static method). :
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.CRot.decomposition`.
Args:
phi (float): rotation angle :math:`\phi`
wires (Iterable, Wires): the wires the operation acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> qml.CRX.compute_decomposition(1.2, wires=(0,1))
[RZ(1.5707963267948966, wires=[1]),
RY(0.6, wires=[1]),
CNOT(wires=[0, 1]),
RY(-0.6, wires=[1]),
CNOT(wires=[0, 1]),
RZ(-1.5707963267948966, wires=[1])]
"""
pi_half = qml.math.ones_like(phi) * (np.pi / 2)
return [
qml.RZ(pi_half, wires=wires[1]),
qml.RY(phi / 2, wires=wires[1]),
qml.CNOT(wires=wires),
qml.RY(-phi / 2, wires=wires[1]),
qml.CNOT(wires=wires),
qml.RZ(-pi_half, wires=wires[1]),
]
[docs]class CRY(ControlledOp):
r"""The controlled-RY operator
.. math::
\begin{align}
CR_y(\phi) &=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0\\
0 & 0 & \cos(\phi/2) & -\sin(\phi/2)\\
0 & 0 & \sin(\phi/2) & \cos(\phi/2)
\end{bmatrix}.
\end{align}
**Details:**
* Number of wires: 2
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
* Gradient recipe: The controlled-RY operator satisfies a four-term parameter-shift rule
(see Appendix F, https://doi.org/10.1088/1367-2630/ac2cb3):
.. math::
\frac{d}{d\phi}f(CR_y(\phi)) = c_+ \left[f(CR_y(\phi+a)) - f(CR_y(\phi-a))\right] - c_- \left[f(CR_y(\phi+b)) - f(CR_y(\phi-b))\right]
where :math:`f` is an expectation value depending on :math:`CR_y(\phi)`, and
- :math:`a = \pi/2`
- :math:`b = 3\pi/2`
- :math:`c_{\pm} = (\sqrt{2} \pm 1)/{4\sqrt{2}}`
Args:
phi (float): rotation angle :math:`\phi`
wires (Sequence[int]): the wire the operation acts on
id (str or None): String representing the operation (optional)
"""
num_wires = 2
"""int: Number of wires that the operation acts on."""
num_params = 1
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = (0,)
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
name = "CRY"
parameter_frequencies = [(0.5, 1.0)]
def __init__(self, phi, wires, id=None):
super().__init__(qml.RY(phi, wires=wires[1:]), control_wires=wires[0], id=id)
def __repr__(self):
return f"CRY({self.data[0]}, wires={self.wires.tolist()}))"
@classmethod
def _unflatten(cls, data, metadata):
base = data[0]
control_wires = metadata[0]
return cls(*base.data, wires=control_wires + base.wires)
[docs] @staticmethod
def compute_matrix(theta): # pylint: disable=arguments-differ
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.CRY.matrix`
Args:
theta (tensor_like or float): rotation angle
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.CRY.compute_matrix(torch.tensor(0.5))
tensor([[ 1.0000+0.j, 0.0000+0.j, 0.0000+0.j, 0.0000+0.j],
[ 0.0000+0.j, 1.0000+0.j, 0.0000+0.j, 0.0000+0.j],
[ 0.0000+0.j, 0.0000+0.j, 0.9689+0.j, -0.2474-0.j],
[ 0.0000+0.j, 0.0000+0.j, 0.2474+0.j, 0.9689+0.j]])
"""
interface = qml.math.get_interface(theta)
c = qml.math.cos(theta / 2)
s = qml.math.sin(theta / 2)
if interface == "tensorflow":
c = qml.math.cast_like(c, 1j)
s = qml.math.cast_like(s, 1j)
# The following avoids casting an imaginary quantity to reals when back propagating
c = (1 + 0j) * c
s = (1 + 0j) * s
ones = qml.math.ones_like(s)
zeros = qml.math.zeros_like(s)
matrix = [
[ones, zeros, zeros, zeros],
[zeros, ones, zeros, zeros],
[zeros, zeros, c, -s],
[zeros, zeros, s, c],
]
return qml.math.stack([stack_last(row) for row in matrix], axis=-2)
[docs] @staticmethod
def compute_decomposition(phi, wires): # pylint: disable=arguments-differ
r"""Representation of the operator as a product of other operators (static method). :
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.CRY.decomposition`.
Args:
phi (float): rotation angle :math:`\phi`
wires (Iterable, Wires): wires that the operator acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> qml.CRY.compute_decomposition(1.2, wires=(0,1))
[RY(0.6, wires=[1]),
CNOT(wires=[0, 1]),
RY(-0.6, wires=[1]),
CNOT(wires=[0, 1])]
"""
return [
qml.RY(phi / 2, wires=wires[1]),
qml.CNOT(wires=wires),
qml.RY(-phi / 2, wires=wires[1]),
qml.CNOT(wires=wires),
]
[docs]class CRZ(ControlledOp):
r"""The controlled-RZ operator
.. math::
\begin{align}
CR_z(\phi) &=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0\\
0 & 0 & e^{-i\phi/2} & 0\\
0 & 0 & 0 & e^{i\phi/2}
\end{bmatrix}.
\end{align}
.. note:: The subscripts of the operations in the formula refer to the wires they act on, e.g. 1 corresponds
to the first element in ``wires`` that is the **control qubit**.
**Details:**
* Number of wires: 2
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
* Gradient recipe: The controlled-RZ operator satisfies a four-term parameter-shift rule
(see Appendix F, https://doi.org/10.1088/1367-2630/ac2cb3):
.. math::
\frac{d}{d\phi}f(CR_z(\phi)) = c_+ \left[f(CR_z(\phi+a)) - f(CR_z(\phi-a))\right] - c_- \left[f(CR_z(\phi+b)) - f(CR_z(\phi-b))\right]
where :math:`f` is an expectation value depending on :math:`CR_z(\phi)`, and
- :math:`a = \pi/2`
- :math:`b = 3\pi/2`
- :math:`c_{\pm} = (\sqrt{2} \pm 1)/{4\sqrt{2}}`
Args:
phi (float): rotation angle :math:`\phi`
wires (Sequence[int]): the wire the operation acts on
id (str or None): String representing the operation (optional)
"""
num_wires = 2
"""int: Number of wires that the operation acts on."""
num_params = 1
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = (0,)
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
name = "CRZ"
parameter_frequencies = [(0.5, 1.0)]
def __init__(self, phi, wires, id=None):
super().__init__(qml.RZ(phi, wires=wires[1:]), control_wires=wires[0], id=id)
def __repr__(self):
return f"CRZ({self.data[0]}, wires={self.wires})"
@classmethod
def _unflatten(cls, data, metadata):
base = data[0]
control_wires = metadata[0]
return cls(*base.data, wires=control_wires + base.wires)
[docs] @staticmethod
def compute_matrix(theta): # pylint: disable=arguments-differ
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.CRZ.matrix`
Args:
theta (tensor_like or float): rotation angle
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.CRZ.compute_matrix(torch.tensor(0.5))
tensor([[1.0+0.0j, 0.0+0.0j, 0.0+0.0j, 0.0+0.0j],
[0.0+0.0j, 1.0+0.0j, 0.0+0.0j, 0.0+0.0j],
[0.0+0.0j, 0.0+0.0j, 0.9689-0.2474j, 0.0+0.0j],
[0.0+0.0j, 0.0+0.0j, 0.0+0.0j, 0.9689+0.2474j]])
"""
if qml.math.get_interface(theta) == "tensorflow":
p = qml.math.exp(-0.5j * qml.math.cast_like(theta, 1j))
if qml.math.ndim(p) == 0:
return qml.math.diag([1, 1, p, qml.math.conj(p)])
ones = qml.math.ones_like(p)
diags = stack_last([ones, ones, p, qml.math.conj(p)])
return diags[:, :, np.newaxis] * qml.math.cast_like(qml.math.eye(4, like=diags), diags)
signs = qml.math.array([0, 0, 1, -1], like=theta)
arg = -0.5j * theta
if qml.math.ndim(arg) == 0:
return qml.math.diag(qml.math.exp(arg * signs))
diags = qml.math.exp(qml.math.outer(arg, signs))
return diags[:, :, np.newaxis] * qml.math.cast_like(qml.math.eye(4, like=diags), diags)
[docs] @staticmethod
def compute_eigvals(theta, **_): # pylint: disable=arguments-differ
r"""Eigenvalues of the operator in the computational basis (static method).
If :attr:`diagonalizing_gates` are specified and implement a unitary :math:`U^{\dagger}`,
the operator can be reconstructed as
.. math:: O = U \Sigma U^{\dagger},
where :math:`\Sigma` is the diagonal matrix containing the eigenvalues.
Otherwise, no particular order for the eigenvalues is guaranteed.
.. seealso:: :meth:`~.CRZ.eigvals`
Args:
theta (tensor_like or float): rotation angle
Returns:
tensor_like: eigenvalues
**Example**
>>> qml.CRZ.compute_eigvals(torch.tensor(0.5))
tensor([1.0000+0.0000j, 1.0000+0.0000j, 0.9689-0.2474j, 0.9689+0.2474j])
"""
if qml.math.get_interface(theta) == "tensorflow":
phase = qml.math.exp(-0.5j * qml.math.cast_like(theta, 1j))
ones = qml.math.ones_like(phase)
return stack_last([ones, ones, phase, qml.math.conj(phase)])
prefactors = qml.math.array([0, 0, -0.5j, 0.5j], like=theta)
if qml.math.ndim(theta) == 0:
product = theta * prefactors
else:
product = qml.math.outer(theta, prefactors)
return qml.math.exp(product)
[docs] @staticmethod
def compute_decomposition(phi, wires): # pylint: disable=arguments-differ
r"""Representation of the operator as a product of other operators (static method). :
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.CRZ.decomposition`.
Args:
phi (float): rotation angle :math:`\phi`
wires (Iterable, Wires): wires that the operator acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> qml.CRZ.compute_decomposition(1.2, wires=(0,1))
[PhaseShift(0.6, wires=[1]),
CNOT(wires=[0, 1]),
PhaseShift(-0.6, wires=[1]),
CNOT(wires=[0, 1])]
"""
return [
qml.PhaseShift(phi / 2, wires=wires[1]),
qml.CNOT(wires=wires),
qml.PhaseShift(-phi / 2, wires=wires[1]),
qml.CNOT(wires=wires),
]
[docs]class CRot(ControlledOp):
r"""The controlled-Rot operator
.. math:: CR(\phi, \theta, \omega) = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0\\
0 & 0 & e^{-i(\phi+\omega)/2}\cos(\theta/2) & -e^{i(\phi-\omega)/2}\sin(\theta/2)\\
0 & 0 & e^{-i(\phi-\omega)/2}\sin(\theta/2) & e^{i(\phi+\omega)/2}\cos(\theta/2)
\end{bmatrix}.
.. note:: The first wire provided corresponds to the **control qubit**.
**Details:**
* Number of wires: 2
* Number of parameters: 3
* Number of dimensions per parameter: (0, 0, 0)
* Gradient recipe: The controlled-Rot operator satisfies a four-term parameter-shift rule
(see Appendix F, https://doi.org/10.1088/1367-2630/ac2cb3):
.. math::
\frac{d}{d\mathbf{x}_i}f(CR(\mathbf{x}_i)) = c_+ \left[f(CR(\mathbf{x}_i+a)) - f(CR(\mathbf{x}_i-a))\right] - c_- \left[f(CR(\mathbf{x}_i+b)) - f(CR(\mathbf{x}_i-b))\right]
where :math:`f` is an expectation value depending on :math:`CR(\mathbf{x}_i)`, and
- :math:`\mathbf{x} = (\phi, \theta, \omega)` and `i` is an index to :math:`\mathbf{x}`
- :math:`a = \pi/2`
- :math:`b = 3\pi/2`
- :math:`c_{\pm} = (\sqrt{2} \pm 1)/{4\sqrt{2}}`
Args:
phi (float): rotation angle :math:`\phi`
theta (float): rotation angle :math:`\theta`
omega (float): rotation angle :math:`\omega`
wires (Sequence[int]): the wire the operation acts on
id (str or None): String representing the operation (optional)
"""
num_wires = 2
"""int: Number of wires this operator acts on."""
num_params = 3
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = (0, 0, 0)
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
name = "CRot"
parameter_frequencies = [(0.5, 1.0), (0.5, 1.0), (0.5, 1.0)]
def __init__(self, phi, theta, omega, wires, id=None): # pylint: disable=too-many-arguments
super().__init__(qml.Rot(phi, theta, omega, wires=wires[1]), control_wires=wires[0], id=id)
def __repr__(self):
params = ", ".join([repr(p) for p in self.parameters])
return f"CRot({params}, wires={self.wires})"
@classmethod
def _unflatten(cls, data, metadata):
base = data[0]
control_wires = metadata[0]
return cls(*base.data, wires=control_wires + base.wires)
[docs] @staticmethod
def compute_matrix(phi, theta, omega): # pylint: disable=arguments-differ
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.CRot.matrix`
Args:
phi(tensor_like or float): first rotation angle
theta (tensor_like or float): second rotation angle
omega (tensor_like or float): third rotation angle
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.CRot.compute_matrix(torch.tensor(0.1), torch.tensor(0.2), torch.tensor(0.3))
tensor([[ 1.0+0.0j, 0.0+0.0j, 0.0+0.0j, 0.0+0.0j],
[ 0.0+0.0j, 1.0+0.0j, 0.0+0.0j, 0.0+0.0j],
[ 0.0+0.0j, 0.0+0.0j, 0.9752-0.1977j, -0.0993+0.0100j],
[ 0.0+0.0j, 0.0+0.0j, 0.0993+0.0100j, 0.9752+0.1977j]])
"""
# It might be that they are in different interfaces, e.g.,
# CRot(0.2, 0.3, tf.Variable(0.5), wires=[0, 1])
# So we need to make sure the matrix comes out having the right type
interface = qml.math.get_interface(phi, theta, omega)
c = qml.math.cos(theta / 2)
s = qml.math.sin(theta / 2)
# If anything is not tensorflow, it has to be casted
if interface == "tensorflow":
phi = qml.math.cast_like(qml.math.asarray(phi, like=interface), 1j)
omega = qml.math.cast_like(qml.math.asarray(omega, like=interface), 1j)
c = qml.math.cast_like(qml.math.asarray(c, like=interface), 1j)
s = qml.math.cast_like(qml.math.asarray(s, like=interface), 1j)
# The following variable is used to assert the all terms to be stacked have same shape
one = qml.math.ones_like(phi) * qml.math.ones_like(omega)
c = c * one
s = s * one
o = qml.math.ones_like(c)
z = qml.math.zeros_like(c)
mat = [
[o, z, z, z],
[z, o, z, z],
[
z,
z,
qml.math.exp(-0.5j * (phi + omega)) * c,
-qml.math.exp(0.5j * (phi - omega)) * s,
],
[
z,
z,
qml.math.exp(-0.5j * (phi - omega)) * s,
qml.math.exp(0.5j * (phi + omega)) * c,
],
]
return qml.math.stack([stack_last(row) for row in mat], axis=-2)
[docs] @staticmethod
def compute_decomposition(phi, theta, omega, wires): # pylint: disable=arguments-differ
r"""Representation of the operator as a product of other operators (static method). :
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.CRot.decomposition`.
Args:
phi (float): rotation angle :math:`\phi`
theta (float): rotation angle :math:`\theta`
omega (float): rotation angle :math:`\omega`
wires (Iterable, Wires): the wires the operation acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> qml.CRot.compute_decomposition(1.234, 2.34, 3.45, wires=[0, 1])
[RZ(-1.108, wires=[1]),
CNOT(wires=[0, 1]),
RZ(-2.342, wires=[1]),
RY(-1.17, wires=[1]),
CNOT(wires=[0, 1]),
RY(1.17, wires=[1]),
RZ(3.45, wires=[1])]
"""
return [
qml.RZ((phi - omega) / 2, wires=wires[1]),
qml.CNOT(wires=wires),
qml.RZ(-(phi + omega) / 2, wires=wires[1]),
qml.RY(-theta / 2, wires=wires[1]),
qml.CNOT(wires=wires),
qml.RY(theta / 2, wires=wires[1]),
qml.RZ(omega, wires=wires[1]),
]
[docs]class ControlledPhaseShift(ControlledOp):
r"""A qubit controlled phase shift.
.. math:: CR_\phi(\phi) = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & e^{i\phi}
\end{bmatrix}.
.. note:: The first wire provided corresponds to the **control qubit**.
**Details:**
* Number of wires: 2
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
* Gradient recipe: :math:`\frac{d}{d\phi}f(CR_\phi(\phi)) = \frac{1}{2}\left[f(CR_\phi(\phi+\pi/2)) - f(CR_\phi(\phi-\pi/2))\right]`
where :math:`f` is an expectation value depending on :math:`CR_{\phi}(\phi)`.
Args:
phi (float): rotation angle :math:`\phi`
wires (Sequence[int]): the wire the operation acts on
id (str or None): String representing the operation (optional)
"""
num_wires = 2
"""int: Number of wires the operator acts on."""
num_params = 1
"""int: Number of trainable parameters that the operator depends on."""
ndim_params = (0,)
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
name = "ControlledPhaseShift"
parameter_frequencies = [(1,)]
def __init__(self, phi, wires, id=None):
super().__init__(qml.PhaseShift(phi, wires=wires[1:]), control_wires=wires[0], id=id)
def __repr__(self):
return f"ControlledPhaseShift({self.data[0]}, wires={self.wires})"
@classmethod
def _unflatten(cls, data, metadata):
base = data[0]
control_wires = metadata[0]
return cls(*base.data, wires=control_wires + base.wires)
[docs] @staticmethod
def compute_matrix(phi): # pylint: disable=arguments-differ
r"""Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires.
Implicitly, this assumes that the wires of the operator correspond to the global wire order.
.. seealso:: :meth:`~.ControlledPhaseShift.matrix`
Args:
phi (tensor_like or float): phase shift
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.ControlledPhaseShift.compute_matrix(torch.tensor(0.5))
tensor([[1.0+0.0j, 0.0+0.0j, 0.0+0.0j, 0.0000+0.0000j],
[0.0+0.0j, 1.0+0.0j, 0.0+0.0j, 0.0000+0.0000j],
[0.0+0.0j, 0.0+0.0j, 1.0+0.0j, 0.0000+0.0000j],
[0.0+0.0j, 0.0+0.0j, 0.0+0.0j, 0.8776+0.4794j]])
"""
if qml.math.get_interface(phi) == "tensorflow":
p = qml.math.exp(1j * qml.math.cast_like(phi, 1j))
if qml.math.ndim(p) == 0:
return qml.math.diag([1, 1, 1, p])
ones = qml.math.ones_like(p)
diags = stack_last([ones, ones, ones, p])
return diags[:, :, np.newaxis] * qml.math.cast_like(qml.math.eye(4, like=diags), diags)
signs = qml.math.array([0, 0, 0, 1], like=phi)
arg = 1j * phi
if qml.math.ndim(arg) == 0:
return qml.math.diag(qml.math.exp(arg * signs))
diags = qml.math.exp(qml.math.outer(arg, signs))
return diags[:, :, np.newaxis] * qml.math.cast_like(qml.math.eye(4, like=diags), diags)
[docs] @staticmethod
def compute_eigvals(phi, **_): # pylint: disable=arguments-differ
r"""Eigenvalues of the operator in the computational basis (static method).
If :attr:`diagonalizing_gates` are specified and implement a unitary :math:`U^{\dagger}`,
the operator can be reconstructed as
.. math:: O = U \Sigma U^{\dagger},
where :math:`\Sigma` is the diagonal matrix containing the eigenvalues.
Otherwise, no particular order for the eigenvalues is guaranteed.
.. seealso:: :meth:`~.ControlledPhaseShift.eigvals`
Args:
phi (tensor_like or float): phase shift
Returns:
tensor_like: eigenvalues
**Example**
>>> qml.ControlledPhaseShift.compute_eigvals(torch.tensor(0.5))
tensor([1.0000+0.0000j, 1.0000+0.0000j, 1.0000+0.0000j, 0.8776+0.4794j])
"""
if qml.math.get_interface(phi) == "tensorflow":
phase = qml.math.exp(1j * qml.math.cast_like(phi, 1j))
ones = qml.math.ones_like(phase)
return stack_last([ones, ones, ones, phase])
prefactors = qml.math.array([0, 0, 0, 1j], like=phi)
if qml.math.ndim(phi) == 0:
product = phi * prefactors
else:
product = qml.math.outer(phi, prefactors)
return qml.math.exp(product)
[docs] @staticmethod
def compute_decomposition(phi, wires): # pylint: disable=arguments-differ
r"""Representation of the operator as a product of other operators (static method). :
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.ControlledPhaseShift.decomposition`.
Args:
phi (float): rotation angle :math:`\phi`
wires (Iterable, Wires): wires that the operator acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> qml.ControlledPhaseShift.compute_decomposition(1.234, wires=(0,1))
[PhaseShift(0.617, wires=[0]),
CNOT(wires=[0, 1]),
PhaseShift(-0.617, wires=[1]),
CNOT(wires=[0, 1]),
PhaseShift(0.617, wires=[1])]
"""
return [
qml.PhaseShift(phi / 2, wires=wires[0]),
qml.CNOT(wires=wires),
qml.PhaseShift(-phi / 2, wires=wires[1]),
qml.CNOT(wires=wires),
qml.PhaseShift(phi / 2, wires=wires[1]),
]
CPhase = ControlledPhaseShift
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