Source code for pennylane.ops.qubit.parametric_ops_multi_qubit
# Copyright 2018-2023 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.
# pylint: disable=too-many-arguments
"""
This submodule contains the discrete-variable quantum operations that are the
core parameterized gates.
"""
# pylint:disable=abstract-method,arguments-differ,protected-access,invalid-overridden-method
import functools
from operator import matmul
import numpy as np
import pennylane as qml
from pennylane.math import expand_matrix
from pennylane.operation import AnyWires, Operation
from pennylane.utils import pauli_eigs
from pennylane.wires import Wires
from .non_parametric_ops import Hadamard, PauliX, PauliY, PauliZ
from .parametric_ops_single_qubit import _can_replace, stack_last, RX, RY, RZ, PhaseShift
[docs]class MultiRZ(Operation):
r"""
Arbitrary multi Z rotation.
.. math::
MultiRZ(\theta) = \exp(-i \frac{\theta}{2} Z^{\otimes n})
**Details:**
* Number of wires: Any
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
* Gradient recipe: :math:`\frac{d}{d\theta}f(MultiRZ(\theta)) = \frac{1}{2}\left[f(MultiRZ(\theta +\pi/2)) - f(MultiRZ(\theta-\pi/2))\right]`
where :math:`f` is an expectation value depending on :math:`MultiRZ(\theta)`.
.. note::
If the ``MultiRZ`` gate is not supported on the targeted device, PennyLane
will decompose the gate using :class:`~.RZ` and :class:`~.CNOT` gates.
Args:
theta (tensor_like or float): rotation angle :math:`\theta`
wires (Sequence[int] or int): the wires the operation acts on
id (str or None): String representing the operation (optional)
"""
num_wires = AnyWires
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."""
grad_method = "A"
parameter_frequencies = [(1,)]
def _flatten(self):
return self.data, (self.wires, tuple())
def __init__(self, theta, wires=None, id=None):
wires = Wires(wires)
self.hyperparameters["num_wires"] = len(wires)
super().__init__(theta, wires=wires, id=id)
[docs] @staticmethod
def compute_matrix(theta, num_wires): # 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:`~.MultiRZ.matrix`
Args:
theta (tensor_like or float): rotation angle
num_wires (int): number of wires the rotation acts on
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.MultiRZ.compute_matrix(torch.tensor(0.1), 2)
tensor([[0.9988-0.0500j, 0.0000+0.0000j, 0.0000+0.0000j, 0.0000+0.0000j],
[0.0000+0.0000j, 0.9988+0.0500j, 0.0000+0.0000j, 0.0000+0.0000j],
[0.0000+0.0000j, 0.0000+0.0000j, 0.9988+0.0500j, 0.0000+0.0000j],
[0.0000+0.0000j, 0.0000+0.0000j, 0.0000+0.0000j, 0.9988-0.0500j]])
"""
eigs = qml.math.convert_like(pauli_eigs(num_wires), theta)
if qml.math.get_interface(theta) == "tensorflow":
theta = qml.math.cast_like(theta, 1j)
eigs = qml.math.cast_like(eigs, 1j)
if qml.math.ndim(theta) == 0:
return qml.math.diag(qml.math.exp(-0.5j * theta * eigs))
diags = qml.math.exp(qml.math.outer(-0.5j * theta, eigs))
return diags[:, :, np.newaxis] * qml.math.cast_like(
qml.math.eye(2**num_wires, like=diags), diags
)
[docs] @staticmethod
def compute_eigvals(theta, num_wires): # 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:`~.MultiRZ.eigvals`
Args:
theta (tensor_like or float): rotation angle
num_wires (int): number of wires the rotation acts on
Returns:
tensor_like: eigenvalues
**Example**
>>> qml.MultiRZ.compute_eigvals(torch.tensor(0.5), 3)
tensor([0.9689-0.2474j, 0.9689+0.2474j, 0.9689+0.2474j, 0.9689-0.2474j,
0.9689+0.2474j, 0.9689-0.2474j, 0.9689-0.2474j, 0.9689+0.2474j])
"""
eigs = qml.math.convert_like(pauli_eigs(num_wires), theta)
if qml.math.get_interface(theta) == "tensorflow":
theta = qml.math.cast_like(theta, 1j)
eigs = qml.math.cast_like(eigs, 1j)
if qml.math.ndim(theta) == 0:
return qml.math.exp(-0.5j * theta * eigs)
return qml.math.exp(qml.math.outer(-0.5j * theta, eigs))
[docs] @staticmethod
def compute_decomposition(
theta, wires, **kwargs
): # pylint: disable=arguments-differ,unused-argument
r"""Representation of the operator as a product of other operators (static method). :
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.MultiRZ.decomposition`.
Args:
theta (float): rotation angle :math:`\theta`
wires (Iterable, Wires): the wires the operation acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> qml.MultiRZ.compute_decomposition(1.2, wires=(0,1))
[CNOT(wires=[1, 0]), RZ(1.2, wires=[0]), CNOT(wires=[1, 0])]
"""
ops = [qml.CNOT(wires=(w0, w1)) for w0, w1 in zip(wires[~0:0:-1], wires[~1::-1])]
ops.append(RZ(theta, wires=wires[0]))
ops += [qml.CNOT(wires=(w0, w1)) for w0, w1 in zip(wires[1:], wires[:~0])]
return ops
[docs] def simplify(self):
theta = self.data[0] % (4 * np.pi)
if _can_replace(theta, 0):
return qml.Identity(wires=self.wires[0])
return MultiRZ(theta, wires=self.wires)
[docs]class PauliRot(Operation):
r"""
Arbitrary Pauli word rotation.
.. math::
RP(\theta, P) = \exp(-i \frac{\theta}{2} P)
**Details:**
* Number of wires: Any
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
* Gradient recipe: :math:`\frac{d}{d\theta}f(RP(\theta)) = \frac{1}{2}\left[f(RP(\theta +\pi/2)) - f(RP(\theta-\pi/2))\right]`
where :math:`f` is an expectation value depending on :math:`RP(\theta)`.
.. note::
If the ``PauliRot`` gate is not supported on the targeted device, PennyLane
will decompose the gate using :class:`~.RX`, :class:`~.Hadamard`, :class:`~.RZ`
and :class:`~.CNOT` gates.
Args:
theta (float): rotation angle :math:`\theta`
pauli_word (string): the Pauli word defining the rotation
wires (Sequence[int] or int): the wire the operation acts on
id (str or None): String representing the operation (optional)
**Example**
>>> dev = qml.device('default.qubit', wires=1)
>>> @qml.qnode(dev)
... def example_circuit():
... qml.PauliRot(0.5, 'X', wires=0)
... return qml.expval(qml.Z(0))
>>> print(example_circuit())
0.8775825618903724
"""
num_wires = AnyWires
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."""
do_check_domain = False
grad_method = "A"
parameter_frequencies = [(1,)]
_ALLOWED_CHARACTERS = "IXYZ"
_PAULI_CONJUGATION_MATRICES = {
"X": Hadamard.compute_matrix(),
"Y": RX.compute_matrix(np.pi / 2),
"Z": np.array([[1, 0], [0, 1]]),
}
def __init__(self, theta, pauli_word, wires=None, id=None):
super().__init__(theta, wires=wires, id=id)
self.hyperparameters["pauli_word"] = pauli_word
if not PauliRot._check_pauli_word(pauli_word):
raise ValueError(
f'The given Pauli word "{pauli_word}" contains characters that are not allowed. '
"Allowed characters are I, X, Y and Z"
)
num_wires = 1 if isinstance(wires, int) else len(wires)
if not len(pauli_word) == num_wires:
raise ValueError(
f"The given Pauli word has length {len(pauli_word)}, length "
f"{num_wires} was expected for wires {wires}"
)
def __repr__(self):
return f"PauliRot({self.data[0]}, {self.hyperparameters['pauli_word']}, wires={self.wires.tolist()})"
[docs] def label(self, decimals=None, base_label=None, cache=None):
r"""A customizable string representation of the operator.
Args:
decimals=None (int): If ``None``, no parameters are included. Else,
specifies how to round the parameters.
base_label=None (str): overwrite the non-parameter component of the label
cache=None (dict): dictionary that caries information between label calls
in the same drawing
Returns:
str: label to use in drawings
**Example:**
>>> op = qml.PauliRot(0.1, "XYY", wires=(0,1,2))
>>> op.label()
'RXYY'
>>> op.label(decimals=2)
'RXYY\n(0.10)'
>>> op.label(base_label="PauliRot")
'PauliRot\n(0.10)'
"""
pauli_word = self.hyperparameters["pauli_word"]
op_label = base_label or ("R" + pauli_word)
# TODO[dwierichs]: Implement a proper label for parameter-broadcasted operators
if decimals is not None and self.batch_size is None:
param_string = f"\n({qml.math.asarray(self.parameters[0]):.{decimals}f})"
op_label += param_string
return op_label
@staticmethod
def _check_pauli_word(pauli_word):
"""Check that the given Pauli word has correct structure.
Args:
pauli_word (str): Pauli word to be checked
Returns:
bool: Whether the Pauli word has correct structure.
"""
return all(pauli in PauliRot._ALLOWED_CHARACTERS for pauli in set(pauli_word))
[docs] @staticmethod
def compute_matrix(theta, pauli_word): # 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:`~.PauliRot.matrix`
Args:
theta (tensor_like or float): rotation angle
pauli_word (str): string representation of Pauli word
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.PauliRot.compute_matrix(0.5, 'X')
[[9.6891e-01+4.9796e-18j 2.7357e-17-2.4740e-01j]
[2.7357e-17-2.4740e-01j 9.6891e-01+4.9796e-18j]]
"""
if not PauliRot._check_pauli_word(pauli_word):
raise ValueError(
f'The given Pauli word "{pauli_word}" contains characters that are not allowed. '
"Allowed characters are I, X, Y and Z"
)
interface = qml.math.get_interface(theta)
if interface == "tensorflow":
theta = qml.math.cast_like(theta, 1j)
# Simplest case is if the Pauli is the identity matrix
if set(pauli_word) == {"I"}:
exp = qml.math.exp(-0.5j * theta)
iden = qml.math.eye(2 ** len(pauli_word), like=theta)
if qml.math.get_interface(theta) == "tensorflow":
iden = qml.math.cast_like(iden, 1j)
if qml.math.get_interface(theta) == "torch":
td = exp.device
iden = iden.to(td)
if qml.math.ndim(theta) == 0:
return exp * iden
return qml.math.stack([e * iden for e in exp])
# We first generate the matrix excluding the identity parts and expand it afterwards.
# To this end, we have to store on which wires the non-identity parts act
non_identity_wires, non_identity_gates = zip(
*[(wire, gate) for wire, gate in enumerate(pauli_word) if gate != "I"]
)
multi_Z_rot_matrix = MultiRZ.compute_matrix(theta, len(non_identity_gates))
# now we conjugate with Hadamard and RX to create the Pauli string
conjugation_matrix = functools.reduce(
qml.math.kron,
[PauliRot._PAULI_CONJUGATION_MATRICES[gate] for gate in non_identity_gates],
)
if interface == "tensorflow":
conjugation_matrix = qml.math.cast_like(conjugation_matrix, 1j)
# Note: we use einsum with reverse arguments here because it is not multi-dispatched
# and the tensordot containing multi_Z_rot_matrix should decide about the interface
return expand_matrix(
qml.math.einsum(
"...jk,ij->...ik",
qml.math.tensordot(multi_Z_rot_matrix, conjugation_matrix, axes=[[-1], [0]]),
qml.math.conj(conjugation_matrix),
),
non_identity_wires,
list(range(len(pauli_word))),
)
[docs] def generator(self):
pauli_word = self.hyperparameters["pauli_word"]
wire_map = {w: i for i, w in enumerate(self.wires)}
return -0.5 * qml.pauli.string_to_pauli_word(pauli_word, wire_map=wire_map)
[docs] @staticmethod
def compute_eigvals(theta, pauli_word): # 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:`~.PauliRot.eigvals`
Returns:
tensor_like: eigenvalues
**Example**
>>> qml.PauliRot.compute_eigvals(torch.tensor(0.5), "X")
tensor([0.9689-0.2474j, 0.9689+0.2474j])
"""
if qml.math.get_interface(theta) == "tensorflow":
theta = qml.math.cast_like(theta, 1j)
# Identity must be treated specially because its eigenvalues are all the same
if set(pauli_word) == {"I"}:
exp = qml.math.exp(-0.5j * theta)
ones = qml.math.ones(2 ** len(pauli_word), like=theta)
if qml.math.get_interface(theta) == "tensorflow":
ones = qml.math.cast_like(ones, 1j)
if qml.math.ndim(theta) == 0:
return exp * ones
return qml.math.tensordot(exp, ones, axes=0)
return MultiRZ.compute_eigvals(theta, len(pauli_word))
[docs] @staticmethod
def compute_decomposition(theta, wires, pauli_word):
r"""Representation of the operator as a product of other operators (static method). :
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.PauliRot.decomposition`.
Args:
theta (float): rotation angle :math:`\theta`
wires (Iterable, Wires): the wires the operation acts on
pauli_word (string): the Pauli word defining the rotation
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> qml.PauliRot.compute_decomposition(1.2, "XY", wires=(0,1))
[Hadamard(wires=[0]),
RX(1.5707963267948966, wires=[1]),
MultiRZ(1.2, wires=[0, 1]),
Hadamard(wires=[0]),
RX(-1.5707963267948966, wires=[1])]
"""
if isinstance(wires, int): # Catch cases when the wire is passed as a single int.
wires = [wires]
# Check for identity and do nothing
if set(pauli_word) == {"I"}:
return []
active_wires, active_gates = zip(
*[(wire, gate) for wire, gate in zip(wires, pauli_word) if gate != "I"]
)
ops = []
for wire, gate in zip(active_wires, active_gates):
if gate == "X":
ops.append(Hadamard(wires=[wire]))
elif gate == "Y":
ops.append(RX(np.pi / 2, wires=[wire]))
ops.append(MultiRZ(theta, wires=list(active_wires)))
for wire, gate in zip(active_wires, active_gates):
if gate == "X":
ops.append(Hadamard(wires=[wire]))
elif gate == "Y":
ops.append(RX(-np.pi / 2, wires=[wire]))
return ops
[docs] def adjoint(self):
return PauliRot(-self.parameters[0], self.hyperparameters["pauli_word"], wires=self.wires)
[docs] def pow(self, z):
return [PauliRot(self.data[0] * z, self.hyperparameters["pauli_word"], wires=self.wires)]
[docs]class PCPhase(Operation):
r"""PCPhase(phi, dim, wires)
A projector-controlled phase gate.
This gate applies a complex phase :math:`e^{i\phi}` to the first :math:`dim`
basis vectors of the input state while applying a complex phase :math:`e^{-i \phi}`
to the remaining basis vectors. For example, consider the 2-qubit case where :math:`dim = 3`:
.. math:: \Pi(\phi) = \begin{bmatrix}
e^{i\phi} & 0 & 0 & 0 \\
0 & e^{i\phi} & 0 & 0 \\
0 & 0 & e^{i\phi} & 0 \\
0 & 0 & 0 & e^{-i\phi}
\end{bmatrix}.
**Details:**
* Number of wires: Any (the operation can act on any number of wires)
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
Args:
phi (float): rotation angle :math:`\phi`
dim (int): the dimension of the subspace
wires (Iterable[int, str], Wires): the wires the operation acts on
id (str or None): String representing the operation (optional)
**Example:**
We can define a circuit using :class:`~.PCPhase` as follows:
>>> dev = qml.device('default.qubit', wires=2)
>>> @qml.qnode(dev)
>>> def example_circuit():
... qml.PCPhase(0.27, dim = 2, wires=range(2))
... return qml.state()
The resulting operation applies a complex phase :math:`e^{0.27i}` to the first :math:`dim = 2`
basis vectors and :math:`e^{-0.27i}` to the remaining basis vectors.
>>> print(np.round(qml.matrix(example_circuit)(),2))
[[0.96+0.27j 0. +0.j 0. +0.j 0. +0.j ]
[0. +0.j 0.96+0.27j 0. +0.j 0. +0.j ]
[0. +0.j 0. +0.j 0.96-0.27j 0. +0.j ]
[0. +0.j 0. +0.j 0. +0.j 0.96-0.27j]]
We can also choose a different :math:`dim` value to apply the phase shift to a different set of
basis vectors as follows:
>>> pc_op = qml.PCPhase(1.23, dim=3, wires=[1, 2])
>>> print(np.round(qml.matrix(pc_op),2))
[[0.33+0.94j 0. +0.j 0. +0.j 0. +0.j ]
[0. +0.j 0.33+0.94j 0. +0.j 0. +0.j ]
[0. +0.j 0. +0.j 0.33+0.94j 0. +0.j ]
[0. +0.j 0. +0.j 0. +0.j 0.33-0.94j]]
"""
num_wires = AnyWires
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."""
basis = "Z"
grad_method = "A"
parameter_frequencies = [(2,)]
[docs] def generator(self):
dim, shape = self.hyperparameters["dimension"]
mat = np.diag([1 if index < dim else -1 for index in range(shape)])
return qml.Hermitian(mat, wires=self.wires)
def _flatten(self):
hyperparameter = (("dim", self.hyperparameters["dimension"][0]),)
return tuple(self.data), (self.wires, hyperparameter)
def __init__(self, phi, dim, wires, id=None):
wires = wires if isinstance(wires, Wires) else Wires(wires)
if not (isinstance(dim, int) and (dim <= 2 ** len(wires))):
raise ValueError(
f"The projected dimension {dim} must be an integer that is less than or equal to "
f"the max size of the matrix {2 ** len(wires)}. Try adding more wires."
)
super().__init__(phi, wires=wires, id=id)
self.hyperparameters["dimension"] = (dim, 2 ** len(wires))
[docs] @staticmethod
def compute_matrix(*params, **hyperparams):
"""Get the matrix representation of Pi-controlled phase unitary."""
phi = params[0]
d, t = hyperparams["dimension"]
if qml.math.get_interface(phi) == "tensorflow":
p = qml.math.exp(1j * qml.math.cast_like(phi, 1j))
minus_p = qml.math.exp(-1j * qml.math.cast_like(phi, 1j))
zeros = qml.math.zeros_like(p)
columns = []
for i in range(t):
columns.append(
[p if j == i else zeros for j in range(t)]
if i < d
else [minus_p if j == i else zeros for j in range(t)]
)
r = qml.math.stack(columns, like="tensorflow", axis=-2)
return r
arg = 1j * phi
prefactors = qml.math.array([1 if index < d else -1 for index in range(t)], like=phi)
if qml.math.ndim(arg) == 0:
return qml.math.diag(qml.math.exp(arg * prefactors))
diags = qml.math.exp(qml.math.outer(arg, prefactors))
return qml.math.stack([qml.math.diag(d) for d in diags])
[docs] @staticmethod
def compute_eigvals(*params, **hyperparams):
"""Get the eigvals for the Pi-controlled phase unitary."""
phi = params[0]
d, t = hyperparams["dimension"]
if qml.math.get_interface(phi) == "tensorflow":
phase = qml.math.exp(1j * qml.math.cast_like(phi, 1j))
minus_phase = qml.math.exp(-1j * qml.math.cast_like(phi, 1j))
return stack_last([phase if index < d else minus_phase for index in range(t)])
arg = 1j * phi
prefactors = qml.math.array([1 if index < d else -1 for index in range(t)], like=phi)
if qml.math.ndim(phi) == 0:
product = arg * prefactors
else:
product = qml.math.outer(arg, prefactors)
return qml.math.exp(product)
[docs] @staticmethod
def compute_decomposition(*params, wires=None, **hyperparams):
r"""Representation of the operator as a product of other operators (static method).
.. math:: O = O_1 O_2 \dots O_n.
.. note::
Operations making up the decomposition should be queued within the
``compute_decomposition`` method.
.. seealso:: :meth:`~.Operator.decomposition`.
Args:
*params (list): trainable parameters of the operator, as stored in the ``parameters`` attribute
wires (Iterable[Any], Wires): wires that the operator acts on
**hyperparams (dict): non-trainable hyper-parameters of the operator, as stored in the ``hyperparameters`` attribute
Returns:
list[Operator]: decomposition of the operator
"""
phi = params[0]
k, n = hyperparams["dimension"]
def _get_op_from_binary_rep(binary_rep, theta, wires):
if len(binary_rep) == 1:
op = (
PhaseShift(theta, wires[0])
if int(binary_rep)
else PauliX(wires[0]) @ PhaseShift(theta, wires[0]) @ PauliX(wires[0])
)
else:
base_op = (
PhaseShift(theta, wires[-1])
if int(binary_rep[-1])
else PauliX(wires[-1]) @ PhaseShift(theta, wires[-1]) @ PauliX(wires[-1])
)
op = qml.ctrl(
base_op, control=wires[:-1], control_values=[int(i) for i in binary_rep[:-1]]
)
return op
n_log2 = int(np.log2(n))
positive_binary_reps = [bin(_k)[2:].zfill(n_log2) for _k in range(k)]
negative_binary_reps = [bin(_k)[2:].zfill(n_log2) for _k in range(k, n)]
positive_ops = [
_get_op_from_binary_rep(br, phi, wires=wires) for br in positive_binary_reps
]
negative_ops = [
_get_op_from_binary_rep(br, -1 * phi, wires=wires) for br in negative_binary_reps
]
return positive_ops + negative_ops
[docs] def adjoint(self):
"""Computes the adjoint of the operator."""
phi = self.parameters[0]
dim, _ = self.hyperparameters["dimension"]
return PCPhase(-1 * phi, dim=dim, wires=self.wires)
[docs] def pow(self, z):
"""Computes the operator raised to z."""
phi = self.parameters[0]
dim, _ = self.hyperparameters["dimension"]
return [PCPhase(phi * z, dim=dim, wires=self.wires)]
[docs] def simplify(self):
"""Simplifies the operator if possible."""
phi = self.parameters[0] % (2 * np.pi)
dim, _ = self.hyperparameters["dimension"]
if _can_replace(phi, 0):
return qml.Identity(wires=self.wires[0])
return PCPhase(phi, dim=dim, wires=self.wires)
[docs] def label(self, decimals=None, base_label=None, cache=None):
"""The label of the operator when displayed in a circuit."""
return super().label(decimals=decimals, base_label=base_label or "∏_ϕ", cache=cache)
[docs]class IsingXX(Operation):
r"""
Ising XX coupling gate
.. math:: XX(\phi) = \exp(-i \frac{\phi}{2} (X \otimes X)) =
\begin{bmatrix} =
\cos(\phi / 2) & 0 & 0 & -i \sin(\phi / 2) \\
0 & \cos(\phi / 2) & -i \sin(\phi / 2) & 0 \\
0 & -i \sin(\phi / 2) & \cos(\phi / 2) & 0 \\
-i \sin(\phi / 2) & 0 & 0 & \cos(\phi / 2)
\end{bmatrix}.
.. note::
Special cases of using the :math:`XX` operator include:
* :math:`XX(0) = I`;
* :math:`XX(\pi) = i (X \otimes X)`.
**Details:**
* Number of wires: 2
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
* Gradient recipe: :math:`\frac{d}{d\phi}f(XX(\phi)) = \frac{1}{2}\left[f(XX(\phi +\pi/2)) - f(XX(\phi-\pi/2))\right]`
where :math:`f` is an expectation value depending on :math:`XX(\phi)`.
Args:
phi (float): the phase angle
wires (int): the subsystem the gate acts on
id (str or None): String representing the operation (optional)
"""
num_wires = 2
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."""
grad_method = "A"
parameter_frequencies = [(1,)]
def __init__(self, phi, wires, id=None):
super().__init__(phi, wires=wires, id=id)
[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.
.. seealso:: :meth:`~.IsingXX.matrix`
Args:
phi (tensor_like or float): phase angle
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.IsingXX.compute_matrix(torch.tensor(0.5))
tensor([[0.9689+0.0000j, 0.0000+0.0000j, 0.0000+0.0000j, 0.0000-0.2474j],
[0.0000+0.0000j, 0.9689+0.0000j, 0.0000-0.2474j, 0.0000+0.0000j],
[0.0000+0.0000j, 0.0000-0.2474j, 0.9689+0.0000j, 0.0000+0.0000j],
[0.0000-0.2474j, 0.0000+0.0000j, 0.0000+0.0000j, 0.9689+0.0000j]],
dtype=torch.complex128)
"""
c = qml.math.cos(phi / 2)
s = qml.math.sin(phi / 2)
eye = qml.math.eye(4, like=phi)
rev_eye = qml.math.convert_like(np.eye(4)[::-1].copy(), phi)
if qml.math.get_interface(phi) == "tensorflow":
c = qml.math.cast_like(c, 1j)
s = qml.math.cast_like(s, 1j)
eye = qml.math.cast_like(eye, 1j)
rev_eye = qml.math.cast_like(rev_eye, 1j)
# The following avoids casting an imaginary quantity to reals when backpropagating
js = -1j * s
if qml.math.ndim(phi) == 0:
return c * eye + js * rev_eye
return qml.math.tensordot(c, eye, axes=0) + qml.math.tensordot(js, rev_eye, axes=0)
[docs] @staticmethod
def compute_decomposition(phi, wires):
r"""Representation of the operator as a product of other operators (static method). :
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.IsingXX.decomposition`.
Args:
phi (float): the phase angle
wires (Iterable, Wires): the subsystem the gate acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> qml.IsingXX.compute_decomposition(1.23, wires=(0,1))
[CNOT(wires=[0, 1]), RX(1.23, wires=[0]), CNOT(wires=[0, 1]]
"""
decomp_ops = [
qml.CNOT(wires=wires),
RX(phi, wires=[wires[0]]),
qml.CNOT(wires=wires),
]
return decomp_ops
[docs] def simplify(self):
phi = self.data[0] % (4 * np.pi)
if _can_replace(phi, 0):
return qml.Identity(wires=self.wires[0])
return IsingXX(phi, wires=self.wires)
[docs]class IsingYY(Operation):
r"""
Ising YY coupling gate
.. math:: \mathtt{YY}(\phi) = \exp(-i \frac{\phi}{2} (Y \otimes Y)) =
\begin{bmatrix}
\cos(\phi / 2) & 0 & 0 & i \sin(\phi / 2) \\
0 & \cos(\phi / 2) & -i \sin(\phi / 2) & 0 \\
0 & -i \sin(\phi / 2) & \cos(\phi / 2) & 0 \\
i \sin(\phi / 2) & 0 & 0 & \cos(\phi / 2)
\end{bmatrix}.
.. note::
Special cases of using the :math:`YY` operator include:
* :math:`YY(0) = I`;
* :math:`YY(\pi) = i (Y \otimes Y)`.
**Details:**
* Number of wires: 2
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
* Gradient recipe: :math:`\frac{d}{d\phi}f(YY(\phi)) = \frac{1}{2}\left[f(YY(\phi +\pi/2)) - f(YY(\phi-\pi/2))\right]`
where :math:`f` is an expectation value depending on :math:`YY(\phi)`.
Args:
phi (float): the phase angle
wires (int): the subsystem the gate acts on
id (str or None): String representing the operation (optional)
"""
num_wires = 2
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."""
grad_method = "A"
parameter_frequencies = [(1,)]
def __init__(self, phi, wires, id=None):
super().__init__(phi, wires=wires, id=id)
[docs] @staticmethod
def compute_decomposition(phi, wires):
r"""Representation of the operator as a product of other operators (static method). :
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.IsingYY.decomposition`.
Args:
phi (float): the phase angle
wires (Iterable, Wires): the subsystem the gate acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> qml.IsingYY.compute_decomposition(1.23, wires=(0,1))
[CY(wires=[0, 1]), RY(1.23, wires=[0]), CY(wires=[0, 1])]
"""
return [
qml.CY(wires=wires),
RY(phi, wires=[wires[0]]),
qml.CY(wires=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:`~.IsingYY.matrix`
Args:
phi (tensor_like or float): phase angle
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.IsingYY.compute_matrix(torch.tensor(0.5))
tensor([[0.9689+0.0000j, 0.0000+0.0000j, 0.0000+0.0000j, 0.0000+0.2474j],
[0.0000+0.0000j, 0.9689+0.0000j, 0.0000-0.2474j, 0.0000+0.0000j],
[0.0000+0.0000j, 0.0000-0.2474j, 0.9689+0.0000j, 0.0000+0.0000j],
[0.0000+0.2474j, 0.0000+0.0000j, 0.0000+0.0000j, 0.9689+0.0000j]])
"""
c = qml.math.cos(phi / 2)
s = qml.math.sin(phi / 2)
if qml.math.get_interface(phi) == "tensorflow":
c = qml.math.cast_like(c, 1j)
s = qml.math.cast_like(s, 1j)
js = 1j * s
r_term = qml.math.cast_like(
qml.math.array(
[
[0.0, 0.0, 0.0, 1.0],
[0.0, 0.0, -1.0, 0.0],
[0.0, -1.0, 0.0, 0.0],
[1.0, 0.0, 0.0, 0.0],
],
like=js,
),
1j,
)
if qml.math.ndim(phi) == 0:
return c * qml.math.cast_like(qml.math.eye(4, like=c), c) + js * r_term
return qml.math.tensordot(c, np.eye(4), axes=0) + qml.math.tensordot(js, r_term, axes=0)
[docs] def simplify(self):
phi = self.data[0] % (4 * np.pi)
if _can_replace(phi, 0):
return qml.Identity(wires=self.wires[0])
return IsingYY(phi, wires=self.wires)
[docs]class IsingZZ(Operation):
r"""
Ising ZZ coupling gate
.. math:: ZZ(\phi) = \exp(-i \frac{\phi}{2} (Z \otimes Z)) =
\begin{bmatrix}
e^{-i \phi / 2} & 0 & 0 & 0 \\
0 & e^{i \phi / 2} & 0 & 0 \\
0 & 0 & e^{i \phi / 2} & 0 \\
0 & 0 & 0 & e^{-i \phi / 2}
\end{bmatrix}.
.. note::
Special cases of using the :math:`ZZ` operator include:
* :math:`ZZ(0) = I`;
* :math:`ZZ(\pi) = - (Z \otimes Z)`;
* :math:`ZZ(2\pi) = - I`;
**Details:**
* Number of wires: 2
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
* Gradient recipe: :math:`\frac{d}{d\phi}f(ZZ(\phi)) = \frac{1}{2}\left[f(ZZ(\phi +\pi/2)) - f(ZZ(\phi-\pi/2))\right]`
where :math:`f` is an expectation value depending on :math:`ZZ(\theta)`.
Args:
phi (float): the phase angle
wires (int): the subsystem the gate acts on
id (str or None): String representing the operation (optional)
"""
num_wires = 2
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."""
grad_method = "A"
parameter_frequencies = [(1,)]
def __init__(self, phi, wires, id=None):
super().__init__(phi, wires=wires, id=id)
[docs] @staticmethod
def compute_decomposition(phi, wires):
r"""Representation of the operator as a product of other operators (static method). :
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.IsingZZ.decomposition`.
Args:
phi (float): the phase angle
wires (Iterable, Wires): the subsystem the gate acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> qml.IsingZZ.compute_decomposition(1.23, wires=[0, 1])
[CNOT(wires=[0, 1]), RZ(1.23, wires=[1]), CNOT(wires=[0, 1])]
"""
return [
qml.CNOT(wires=wires),
RZ(phi, wires=[wires[1]]),
qml.CNOT(wires=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:`~.IsingZZ.matrix`
Args:
phi (tensor_like or float): phase angle
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.IsingZZ.compute_matrix(torch.tensor(0.5))
tensor([[0.9689-0.2474j, 0.0000+0.0000j, 0.0000+0.0000j, 0.0000+0.0000j],
[0.0000+0.0000j, 0.9689+0.2474j, 0.0000+0.0000j, 0.0000+0.0000j],
[0.0000+0.0000j, 0.0000+0.0000j, 0.9689+0.2474j, 0.0000+0.0000j],
[0.0000+0.0000j, 0.0000+0.0000j, 0.0000+0.0000j, 0.9689-0.2474j]])
"""
if qml.math.get_interface(phi) == "tensorflow":
p = qml.math.exp(-0.5j * qml.math.cast_like(phi, 1j))
if qml.math.ndim(p) == 0:
return qml.math.diag([p, qml.math.conj(p), qml.math.conj(p), p])
diags = stack_last([p, qml.math.conj(p), qml.math.conj(p), p])
return diags[:, :, np.newaxis] * qml.math.cast_like(qml.math.eye(4, like=diags), diags)
signs = qml.math.array([1, -1, -1, 1], like=phi)
arg = -0.5j * 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:`~.IsingZZ.eigvals`
Args:
phi (tensor_like or float): phase angle
Returns:
tensor_like: eigenvalues
**Example**
>>> qml.IsingZZ.compute_eigvals(torch.tensor(0.5))
tensor([0.9689-0.2474j, 0.9689+0.2474j, 0.9689+0.2474j, 0.9689-0.2474j])
"""
if qml.math.get_interface(phi) == "tensorflow":
phase = qml.math.exp(-0.5j * qml.math.cast_like(phi, 1j))
return stack_last([phase, qml.math.conj(phase), qml.math.conj(phase), phase])
prefactors = qml.math.array([-0.5j, 0.5j, 0.5j, -0.5j], like=phi)
if qml.math.ndim(phi) == 0:
product = phi * prefactors
else:
product = qml.math.outer(phi, prefactors)
return qml.math.exp(product)
[docs] def simplify(self):
phi = self.data[0] % (4 * np.pi)
if _can_replace(phi, 0):
return qml.Identity(wires=self.wires[0])
return IsingZZ(phi, wires=self.wires)
[docs]class IsingXY(Operation):
r"""
Ising (XX + YY) coupling gate
.. math:: \mathtt{XY}(\phi) = \exp(i \frac{\theta}{4} (X \otimes X + Y \otimes Y)) =
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & \cos(\phi / 2) & i \sin(\phi / 2) & 0 \\
0 & i \sin(\phi / 2) & \cos(\phi / 2) & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}.
.. note::
Special cases of using the :math:`XY` operator include:
* :math:`XY(0) = I`;
* :math:`XY(\frac{\pi}{2}) = \sqrt{iSWAP}`;
* :math:`XY(\pi) = iSWAP`;
**Details:**
* Number of wires: 2
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
* Gradient recipe: The XY operator satisfies a four-term parameter-shift rule
.. math::
\frac{d}{d \phi} f(XY(\phi))
= c_+ \left[ f(XY(\phi + a)) - f(XY(\phi - a)) \right]
- c_- \left[ f(XY(\phi + b)) - f(XY(\phi - b)) \right]
where :math:`f` is an expectation value depending on :math:`XY(\phi)`, and
- :math:`a = \pi / 2`
- :math:`b = 3 \pi / 2`
- :math:`c_{\pm} = (\sqrt{2} \pm 1)/{4 \sqrt{2}}`
Args:
phi (float): the phase angle
wires (int): the subsystem the gate acts on
id (str or None): String representing the operation (optional)
"""
num_wires = 2
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."""
grad_method = "A"
parameter_frequencies = [(0.5, 1.0)]
[docs] def generator(self):
return 0.25 * (
PauliX(wires=self.wires[0]) @ PauliX(wires=self.wires[1])
+ PauliY(wires=self.wires[0]) @ PauliY(wires=self.wires[1])
)
def __init__(self, phi, wires, id=None):
super().__init__(phi, wires=wires, id=id)
[docs] @staticmethod
def compute_decomposition(phi, wires):
r"""Representation of the operator as a product of other operators (static method). :
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.IsingXY.decomposition`.
Args:
phi (float): the phase angle
wires (Iterable, Wires): the subsystem the gate acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> qml.IsingXY.compute_decomposition(1.23, wires=(0,1))
[Hadamard(wires=[0]), CY(wires=[0, 1]), RY(0.615, wires=[0]), RX(-0.615, wires=[1]), CY(wires=[0, 1]), Hadamard(wires=[0])]
"""
return [
Hadamard(wires=[wires[0]]),
qml.CY(wires=wires),
RY(phi / 2, wires=[wires[0]]),
RX(-phi / 2, wires=[wires[1]]),
qml.CY(wires=wires),
Hadamard(wires=[wires[0]]),
]
[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:`~.IsingXY.matrix`
Args:
phi (tensor_like or float): phase angle
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.IsingXY.compute_matrix(0.5)
array([[1. +0.j , 0. +0.j , 0. +0.j , 0. +0.j ],
[0. +0.j , 0.96891242+0.j , 0. +0.24740396j, 0. +0.j ],
[0. +0.j , 0. +0.24740396j, 0.96891242+0.j , 0. +0.j ],
[0. +0.j , 0. +0.j , 0. +0.j , 1. +0.j ]])
"""
c = qml.math.cos(phi / 2)
s = qml.math.sin(phi / 2)
if qml.math.get_interface(phi) == "tensorflow":
c = qml.math.cast_like(c, 1j)
s = qml.math.cast_like(s, 1j)
js = 1j * s
off_diag = qml.math.cast_like(
qml.math.array(
[
[0.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 1.0, 0.0],
[0.0, 1.0, 0.0, 0.0],
[0.0, 0.0, 0.0, 0.0],
],
like=js,
),
1j,
)
if qml.math.ndim(phi) == 0:
return qml.math.diag([1, c, c, 1]) + js * off_diag
ones = qml.math.ones_like(c)
diags = stack_last([ones, c, c, ones])[:, :, np.newaxis]
return diags * np.eye(4) + qml.math.tensordot(js, off_diag, axes=0)
[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:`~.IsingXY.eigvals`
Args:
phi (tensor_like or float): phase angle
Returns:
tensor_like: eigenvalues
**Example**
>>> qml.IsingXY.compute_eigvals(0.5)
array([0.96891242+0.24740396j, 0.96891242-0.24740396j, 1. +0.j , 1. +0.j ])
"""
if qml.math.get_interface(phi) == "tensorflow":
phi = qml.math.cast_like(phi, 1j)
signs = np.array([1, -1, 0, 0])
if qml.math.ndim(phi) == 0:
return qml.math.exp(0.5j * phi * signs)
return qml.math.exp(qml.math.tensordot(0.5j * phi, signs, axes=0))
[docs] def simplify(self):
phi = self.data[0] % (4 * np.pi)
if _can_replace(phi, 0):
return qml.Identity(wires=self.wires[0])
return IsingXY(phi, wires=self.wires)
[docs]class PSWAP(Operation):
r"""Phase SWAP gate
.. math:: PSWAP(\phi) = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & e^{i \phi} & 0 \\
0 & e^{i \phi} & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}.
**Details:**
* Number of wires: 2
* Number of parameters: 1
* Gradient recipe:
.. math::
\frac{d}{d \phi} PSWAP(\phi)
= \frac{1}{2} \left[ PSWAP(\phi + \pi / 2) - PSWAP(\phi - \pi / 2) \right]
Args:
phi (float): the phase angle
wires (int): the subsystem the gate acts on
id (str or None): String representing the operation (optional)
"""
num_wires = 2
num_params = 1
"""int: Number of trainable parameters that the operator depends on."""
grad_method = "A"
grad_recipe = ([[0.5, 1, np.pi / 2], [-0.5, 1, -np.pi / 2]],)
def __init__(self, phi, wires, id=None):
super().__init__(phi, wires=wires, id=id)
[docs] @staticmethod
def compute_decomposition(phi, wires):
r"""Representation of the operator as a product of other operators (static method). :
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.PSWAP.decomposition`.
Args:
phi (float): the phase angle
wires (Iterable, Wires): the subsystem the gate acts on
Returns:
list[Operator]: decomposition into lower level operations
**Example:**
>>> qml.PSWAP.compute_decomposition(1.23, wires=(0,1))
[SWAP(wires=[0, 1]), CNOT(wires=[0, 1]), PhaseShift(1.23, wires=[1]), CNOT(wires=[0, 1])]
"""
return [
qml.SWAP(wires=wires),
qml.CNOT(wires=wires),
PhaseShift(phi, wires=[wires[1]]),
qml.CNOT(wires=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:`~.PSWAP.matrix`
Args:
phi (tensor_like or float): phase angle
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.PSWAP.compute_matrix(0.5)
array([[1. +0.j, 0. +0.j , 0. +0.j , 0. +0.j],
[0. +0.j, 0. +0.j , 0.87758256+0.47942554j, 0. +0.j],
[0. +0.j, 0.87758256+0.47942554j, 0. +0.j , 0. +0.j],
[0. +0.j, 0. +0.j , 0. +0.j , 1. +0.j]])
"""
if qml.math.get_interface(phi) == "tensorflow":
phi = qml.math.cast_like(phi, 1j)
e = qml.math.exp(1j * phi)
return qml.math.stack(
[
stack_last([1, 0, 0, 0]),
stack_last([0, 0, e, 0]),
stack_last([0, e, 0, 0]),
stack_last([0, 0, 0, 1]),
],
axis=-2,
)
[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:`~.PSWAP.eigvals`
Args:
phi (tensor_like or float): phase angle
Returns:
tensor_like: eigenvalues
**Example**
>>> qml.PSWAP.compute_eigvals(0.5)
array([ 1. +0.j , 1. +0.j, -0.87758256-0.47942554j, 0.87758256+0.47942554j])
"""
if qml.math.get_interface(phi) == "tensorflow":
phi = qml.math.cast_like(phi, 1j)
return qml.math.stack([1, 1, -qml.math.exp(1j * phi), qml.math.exp(1j * phi)])
[docs] def simplify(self):
phi = self.data[0] % (2 * np.pi)
if _can_replace(phi, 0):
return qml.SWAP(wires=self.wires)
return PSWAP(phi, wires=self.wires)
[docs]class CPhaseShift00(Operation):
r"""
A qubit controlled phase shift.
.. math:: CR_{00}(\phi) = \begin{bmatrix}
e^{i\phi} & 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** and controls
on the zero state :math:`|0\rangle`.
**Details:**
* Number of wires: 2
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
* Gradient recipe:
.. math::
\frac{d}{d \phi} CR_{00}(\phi)
= \frac{1}{2} \left[ CR_{00}(\phi + \pi / 2)
- CR_{00}(\phi - \pi / 2) \right]
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
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."""
grad_method = "A"
parameter_frequencies = [(1,)]
def __init__(self, phi, wires, id=None):
super().__init__(phi, wires=wires, id=id)
[docs] def label(self, decimals=None, base_label=None, cache=None):
return super().label(decimals=decimals, base_label="Rϕ(00)", cache=cache)
[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:`~.CPhaseShift00.matrix`
Args:
phi (tensor_like or float): phase shift
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.CPhaseShift00.compute_matrix(torch.tensor(0.5))
tensor([[0.8776+0.4794j, 0.0+0.0j, 0.0+0.0j, 0.0+0.0j],
[0.0000+0.0000j, 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]])
"""
if qml.math.get_interface(phi) == "tensorflow":
phi = qml.math.cast_like(phi, 1j)
exp_part = qml.math.exp(1j * phi)
if qml.math.ndim(phi) > 0:
ones = qml.math.ones_like(exp_part)
zeros = qml.math.zeros_like(exp_part)
matrix = [
[exp_part, zeros, zeros, zeros],
[zeros, ones, zeros, zeros],
[zeros, zeros, ones, zeros],
[zeros, zeros, zeros, ones],
]
return qml.math.stack([stack_last(row) for row in matrix], axis=-2)
return qml.math.diag([exp_part, 1, 1, 1])
[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:`~.CPhaseShift00.eigvals`
Args:
phi (tensor_like or float): phase shift
Returns:
tensor_like: eigenvalues
**Example**
>>> qml.CPhaseShift00.compute_eigvals(torch.tensor(0.5))
tensor([0.8776+0.4794j, 1.0000+0.0000j, 1.0000+0.0000j, 1.0000+0.0000j])
"""
if qml.math.get_interface(phi) == "tensorflow":
phi = qml.math.cast_like(phi, 1j)
exp_part = qml.math.exp(1j * phi)
ones = qml.math.ones_like(exp_part)
return stack_last([exp_part, ones, ones, ones])
[docs] @staticmethod
def compute_decomposition(phi, wires):
r"""Representation of the operator as a product of other operators (static method). :
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.CPhaseShift00.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.CPhaseShift00.compute_decomposition(1.234, wires=(0,1))
[X(0),
X(1),
PhaseShift(0.617, wires=[0]),
PhaseShift(0.617, wires=[1]),
CNOT(wires=[0, 1]),
PhaseShift(-0.617, wires=[1]),
CNOT(wires=[0, 1]),
X(1),
X(0)]
"""
decomp_ops = [
PauliX(wires[0]),
PauliX(wires[1]),
PhaseShift(phi / 2, wires=[wires[0]]),
PhaseShift(phi / 2, wires=[wires[1]]),
qml.CNOT(wires=wires),
PhaseShift(-phi / 2, wires=[wires[1]]),
qml.CNOT(wires=wires),
PauliX(wires[1]),
PauliX(wires[0]),
]
return decomp_ops
@property
def control_values(self):
"""str: The control values of the operation"""
return "0"
@property
def control_wires(self):
return self.wires[0:1]
[docs]class CPhaseShift01(Operation):
r"""
A qubit controlled phase shift.
.. math:: CR_{01\phi}(\phi) = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & e^{i\phi} & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}.
.. note:: The first wire provided corresponds to the **control qubit** and controls
on the zero state :math:`|0\rangle`.
**Details:**
* Number of wires: 2
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
* Gradient recipe:
.. math::
\frac{d}{d \phi} CR_{01}(\phi)
= \frac{1}{2} \left[ CR_{01}(\phi + \pi / 2)
- CR_{01}(\phi - \pi / 2) \right]
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
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."""
grad_method = "A"
parameter_frequencies = [(1,)]
def __init__(self, phi, wires, id=None):
super().__init__(phi, wires=wires, id=id)
[docs] def label(self, decimals=None, base_label=None, cache=None):
return super().label(decimals=decimals, base_label="Rϕ(01)", cache=cache)
[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:`~.CPhaseShift01.matrix`
Args:
phi (tensor_like or float): phase shift
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.CPhaseShift01.compute_matrix(torch.tensor(0.5))
tensor([[1.0+0.0j, 0.0000+0.0000j, 0.0+0.0j, 0.0+0.0j],
[0.0+0.0j, 0.8776+0.4794j, 0.0+0.0j, 0.0+0.0j],
[0.0+0.0j, 0.0000+0.0000j, 1.0+0.0j, 0.0+0.0j],
[0.0+0.0j, 0.0000+0.0000j, 0.0+0.0j, 1.0+0.0j]])
"""
if qml.math.get_interface(phi) == "tensorflow":
phi = qml.math.cast_like(phi, 1j)
exp_part = qml.math.exp(1j * phi)
if qml.math.ndim(phi) > 0:
ones = qml.math.ones_like(exp_part)
zeros = qml.math.zeros_like(exp_part)
matrix = [
[ones, zeros, zeros, zeros],
[zeros, exp_part, zeros, zeros],
[zeros, zeros, ones, zeros],
[zeros, zeros, zeros, ones],
]
return qml.math.stack([stack_last(row) for row in matrix], axis=-2)
return qml.math.diag([1, exp_part, 1, 1])
[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:`~.CPhaseShift01.eigvals`
Args:
phi (tensor_like or float): phase shift
Returns:
tensor_like: eigenvalues
**Example**
>>> qml.CPhaseShift01.compute_eigvals(torch.tensor(0.5))
tensor([1.0000+0.0000j, 0.8776+0.4794j, 1.0000+0.0000j, 1.0000+0.0000j])
"""
if qml.math.get_interface(phi) == "tensorflow":
phi = qml.math.cast_like(phi, 1j)
exp_part = qml.math.exp(1j * phi)
ones = qml.math.ones_like(exp_part)
return stack_last([ones, exp_part, ones, ones])
[docs] @staticmethod
def compute_decomposition(phi, wires):
r"""Representation of the operator as a product of other operators (static method). :
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.CPhaseShift01.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.CPhaseShift01.compute_decomposition(1.234, wires=(0,1))
[X(0),
PhaseShift(0.617, wires=[0]),
PhaseShift(0.617, wires=[1]),
CNOT(wires=[0, 1]),
PhaseShift(-0.617, wires=[1]),
CNOT(wires=[0, 1]),
X(0)]
"""
decomp_ops = [
PauliX(wires[0]),
PhaseShift(phi / 2, wires=[wires[0]]),
PhaseShift(phi / 2, wires=[wires[1]]),
qml.CNOT(wires=wires),
PhaseShift(-phi / 2, wires=[wires[1]]),
qml.CNOT(wires=wires),
PauliX(wires[0]),
]
return decomp_ops
@property
def control_values(self):
"""str: The control values of the operation"""
return "0"
@property
def control_wires(self):
return self.wires[0:1]
[docs]class CPhaseShift10(Operation):
r"""
A qubit controlled phase shift.
.. math:: CR_{10\phi}(\phi) = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & e^{i\phi} & 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: 1
* Number of dimensions per parameter: (0,)
* Gradient recipe:
.. math::
\frac{d}{d \phi} CR_{10}(\phi)
= \frac{1}{2} \left[ CR_{10}(\phi + \pi / 2)
- CR_{10}(\phi - \pi / 2) \right]
Args:
phi (float): rotation angle :math:`\phi`
wires (Any, Wires): the wire the operation acts on
id (str or None): String representing the operation (optional)
"""
num_wires = 2
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."""
grad_method = "A"
parameter_frequencies = [(1,)]
def __init__(self, phi, wires, id=None):
super().__init__(phi, wires=wires, id=id)
[docs] def label(self, decimals=None, base_label=None, cache=None):
return super().label(decimals=decimals, base_label="Rϕ(10)", cache=cache)
[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:`~.CPhaseShift10.matrix`
Args:
phi (tensor_like or float): phase shift
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.CPhaseShift10.compute_matrix(torch.tensor(0.5))
tensor([[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, 0.0+0.0j],
[0.0+0.0j, 0.0+0.0j, 0.0000+0.0000j, 1.0+0.0j]])
"""
if qml.math.get_interface(phi) == "tensorflow":
phi = qml.math.cast_like(phi, 1j)
exp_part = qml.math.exp(1j * phi)
if qml.math.ndim(phi) > 0:
ones = qml.math.ones_like(exp_part)
zeros = qml.math.zeros_like(exp_part)
matrix = [
[ones, zeros, zeros, zeros],
[zeros, ones, zeros, zeros],
[zeros, zeros, exp_part, zeros],
[zeros, zeros, zeros, ones],
]
return qml.math.stack([stack_last(row) for row in matrix], axis=-2)
return qml.math.diag([1, 1, exp_part, 1])
[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:`~.CPhaseShift10.eigvals`
Args:
phi (tensor_like or float): phase shift
Returns:
tensor_like: eigenvalues
**Example**
>>> qml.CPhaseShift10.compute_eigvals(torch.tensor(0.5))
tensor([1.0000+0.0000j, 1.0000+0.0000j, 0.8776+0.4794j, 1.0000+0.0000j])
"""
if qml.math.get_interface(phi) == "tensorflow":
phi = qml.math.cast_like(phi, 1j)
exp_part = qml.math.exp(1j * phi)
ones = qml.math.ones_like(exp_part)
return stack_last([ones, ones, exp_part, ones])
[docs] @staticmethod
def compute_decomposition(phi, wires):
r"""Representation of the operator as a product of other operators (static method). :
.. math:: O = O_1 O_2 \dots O_n.
.. seealso:: :meth:`~.CPhaseShift10.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.CPhaseShift10.compute_decomposition(1.234, wires=(0,1))
[X(1),
PhaseShift(0.617, wires=[0]),
PhaseShift(0.617, wires=[1]),
CNOT(wires=[0, 1]),
PhaseShift(-0.617, wires=[1]),
CNOT(wires=[0, 1]),
X(1)]
"""
decomp_ops = [
PauliX(wires[1]),
PhaseShift(phi / 2, wires=[wires[0]]),
PhaseShift(phi / 2, wires=[wires[1]]),
qml.CNOT(wires=wires),
PhaseShift(-phi / 2, wires=[wires[1]]),
qml.CNOT(wires=wires),
PauliX(wires[1]),
]
return decomp_ops
@property
def control_wires(self):
return self.wires[0:1]
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