Source code for pennylane.ops.qutrit.parametric_ops
# Copyright 2018-2022 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 parametrized gates for qutrits.
"""
import functools
import numpy as np
import pennylane as qml
from pennylane.operation import Operation
stack_last = functools.partial(qml.math.stack, axis=-1)
def validate_subspace(subspace):
"""Validate the subspace for qutrit operations.
This method determines whether a given subspace for qutrit operations
is defined correctly or not. If not, a ``ValueError`` is thrown.
Args:
subspace (tuple[int]): Subspace to check for correctness
"""
if not hasattr(subspace, "__iter__") or len(subspace) != 2:
raise ValueError(
"The subspace must be a sequence with two unique elements from the set {0, 1, 2}."
)
if not all(s in {0, 1, 2} for s in subspace):
raise ValueError("Elements of the subspace must be 0, 1, or 2.")
if subspace[0] == subspace[1]:
raise ValueError("Elements of subspace list must be unique.")
return tuple(sorted(subspace))
[docs]class TRX(Operation):
r"""
The single qutrit X rotation
Performs the RX operation on the specified 2D subspace. The subspace is
given as a keyword argument and determines which two of three single-qutrit
basis states the operation applies to.
The construction of this operator is based on section 3 of
`Di et al. (2012) <https://arxiv.org/abs/1105.5485>`_.
.. math:: TRX^{jk}(\phi) = \exp(-i\phi\sigma_x^{jk}/2),
where :math:`\sigma_x^{jk} = |j\rangle\langle k| + |k\rangle\langle j|;`
:math:`j, k \in \{0, 1, 2\}, j < k`.
.. seealso:: :class:`~.RX`
**Details:**
* Number of wires: 1
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
Args:
phi (float): rotation angle :math:`\phi`
wires (Sequence[int] or int): the wire the operation acts on
subspace (Sequence[int]): the 2D subspace on which to apply operation
id (str or None): String representing the operation (optional)
**Example**
The specified subspace will determine which basis states the operation actually
applies to:
>>> qml.TRX(0.5, wires=0, subspace=(0, 1)).matrix()
array([[0.96891242+0.j , 0. -0.24740396j, 0. +0.j ],
[0. -0.24740396j, 0.96891242+0.j , 0. +0.j ],
[0. +0.j , 0. +0.j , 1. +0.j ]])
>>> qml.TRX(0.5, wires=0, subspace=(0, 2)).matrix()
array([[0.96891242+0.j , 0. +0.j , 0. -0.24740396j],
[0. +0.j , 1. +0.j , 0. +0.j ],
[0. -0.24740396j, 0. +0.j , 0.96891242+0.j ]])
>>> qml.TRX(0.5, wires=0, subspace=(1, 2)).matrix()
array([[1. +0.j , 0. +0.j , 0. +0.j ],
[0. +0.j , 0.96891242+0.j , 0. -0.24740396j],
[0. +0.j , 0. -0.24740396j, 0.96891242+0.j ]])
"""
num_wires = 1
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)]
# Internal dictionary to map subpsaces to Gell-Mann observable for the generator
_index_dict = {(0, 1): 1, (0, 2): 4, (1, 2): 6}
[docs] def generator(self):
# this generator returns SProd, even with the old op_math, because other options are not suitable
# to qudit operators (for example, they do not have a matrix defined as a Hamiltonian)
return qml.s_prod(-0.5, qml.GellMann(self.wires, index=self._index_dict[self.subspace]))
def __init__(self, phi, wires, subspace=(0, 1), id=None):
self._subspace = validate_subspace(subspace)
self._hyperparameters = {
"subspace": self._subspace,
}
super().__init__(phi, wires=wires, id=id)
@property
def subspace(self):
"""The single-qutrit basis states which the operator acts on
This subspace determines which two single-qutrit basis states the operator acts on.
The remaining basis state is not affected by the operator.
Returns:
tuple[int]: subspace on which operator acts
"""
return self._subspace
[docs] @staticmethod
def compute_matrix(
theta, subspace=(0, 1)
): # pylint: disable=arguments-differ,dangerous-default-value
r"""Representation of the operator as a canonical matrix in the computational basis.
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:`~.TRX.matrix`
Args:
theta (tensor_like or float): rotation angle
subspace (Sequence[int]): the 2D subspace on which to apply the operation
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.TRX.compute_matrix(torch.tensor(0.5), subspace=(0, 2))
tensor([[0.9689+0.0000j, 0.0000+0.0000j, 0.0000-0.2474j],
[0.0000+0.0000j, 1.0000+0.0000j, 0.0000+0.0000j],
[0.0000-0.2474j, 0.0000+0.0000j, 0.9689+0.0000j]])
"""
c = qml.math.cos(theta / 2)
s = qml.math.sin(theta / 2)
if qml.math.get_interface(theta) == "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 backpropagating
c = (1 + 0j) * c
js = -1j * s
one = qml.math.ones_like(c)
z = qml.math.zeros_like(c)
diags = [one, one, one]
diags[subspace[0]] = c
diags[subspace[1]] = c
off_diags = [z, z, z]
off_diags[qml.math.sum(subspace) - 1] = js
return qml.math.stack(
[
stack_last([diags[0], off_diags[0], off_diags[1]]),
stack_last([off_diags[0], diags[1], off_diags[2]]),
stack_last([off_diags[1], off_diags[2], diags[2]]),
],
axis=-2,
)
[docs]class TRY(Operation):
r"""
The single qutrit Y rotation
Performs the RY operation on the specified 2D subspace. The subspace is
given as a keyword argument and determines which two of three single-qutrit
basis states the operation applies to.
The construction of this operator is based on section 3 of
`Di et al. (2012) <https://arxiv.org/abs/1105.5485>`_.
.. math:: TRY^{jk}(\phi) = \exp(-i\phi\sigma_y^{jk}/2),
where :math:`\sigma_y^{jk} = -i |j\rangle\langle k| + i |k\rangle\langle j|;`
:math:`j, k \in \{0, 1, 2\}, j < k`.
**Details:**
* Number of wires: 1
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
Args:
phi (float): rotation angle :math:`\phi`
wires (Sequence[int] or int): the wire the operation acts on
subspace (Sequence[int]): the 2D subspace on which to apply operation
id (str or None): String representing the operation (optional)
**Example**
The specified subspace will determine which basis states the operation actually
applies to:
>>> qml.TRY(0.5, wires=0, subspace=(0, 1)).matrix()
array([[ 0.96891242+0.j, -0.24740396-0.j, 0. +0.j],
[ 0.24740396+0.j, 0.96891242+0.j, 0. +0.j],
[ 0. +0.j, 0. +0.j, 1. +0.j]])
>>> qml.TRY(0.5, wires=0, subspace=(0, 2)).matrix()
array([[ 0.96891242+0.j, 0. +0.j, -0.24740396-0.j],
[ 0. +0.j, 1. +0.j, 0. +0.j],
[ 0.24740396+0.j, 0. +0.j, 0.96891242+0.j]])
>>> qml.TRY(0.5, wires=0, subspace=(1, 2)).matrix()
array([[ 1. +0.j, 0. +0.j, 0. +0.j],
[ 0. +0.j, 0.96891242+0.j, -0.24740396-0.j],
[ 0. +0.j, 0.24740396+0.j, 0.96891242+0.j]])
"""
num_wires = 1
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)]
# Internal dictionary to map subpsaces to Gell-Mann observable for the generator
_index_dict = {(0, 1): 2, (0, 2): 5, (1, 2): 7}
[docs] def generator(self):
# this generator returns SProd, even with the old op_math, because other options are not suitable
# to qudit operators (for example, they do not have a matrix defined as a Hamiltonian)
return qml.s_prod(-0.5, qml.GellMann(self.wires, index=self._index_dict[self.subspace]))
def __init__(self, phi, wires, subspace=(0, 1), id=None):
self._subspace = validate_subspace(subspace)
self._hyperparameters = {
"subspace": self._subspace,
}
super().__init__(phi, wires=wires, id=id)
@property
def subspace(self):
"""The single-qutrit basis states which the operator acts on
This subspace determines which two single-qutrit basis states the operator acts on.
The remaining basis state is not affected by the operator.
Returns:
tuple[int]: subspace on which operator acts
"""
return self._subspace
[docs] @staticmethod
def compute_matrix(theta, subspace=(0, 1)): # 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:`~.TRY.matrix`
Args:
theta (tensor_like or float): rotation angle
subspace (Sequence[int]): the 2D subspace on which to apply operation
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.TRY.compute_matrix(torch.tensor(0.5), subspace=(0, 2))
tensor([[ 0.9689+0.j, 0.0000+0.j, -0.2474-0.j],
[ 0.0000+0.j, 1.0000+0.j, 0.0000+0.j],
[ 0.2474+0.j, 0.0000+0.j, 0.9689+0.j]])
"""
c = qml.math.cos(theta / 2)
s = qml.math.sin(theta / 2)
if qml.math.get_interface(theta) == "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 backpropagating
c = (1 + 0j) * c
s = (1 + 0j) * s
one = qml.math.ones_like(c)
z = qml.math.zeros_like(c)
diags = [one, one, one]
diags[subspace[0]] = c
diags[subspace[1]] = c
off_diags = [z, z, z]
off_diags[qml.math.sum(subspace) - 1] = s
return qml.math.stack(
[
stack_last([diags[0], -off_diags[0], -off_diags[1]]),
stack_last([off_diags[0], diags[1], -off_diags[2]]),
stack_last([off_diags[1], off_diags[2], diags[2]]),
],
axis=-2,
)
[docs]class TRZ(Operation):
r"""The single qutrit Z rotation
Performs the RZ operation on the specified 2D subspace. The subspace is
given as a keyword argument and determines which two of three single-qutrit
basis states the operation applies to.
The construction of this operator is based on section 3 of
`Di et al. (2012) <https://arxiv.org/abs/1105.5485>`_.
.. math:: TRZ^{jk}(\phi) = \exp(-i\phi\sigma_z^{jk}/2),
where :math:`\sigma_z^{jk} = |j\rangle\langle j| - |k\rangle\langle k|;`
:math:`j, k \in \{0, 1, 2\}, j < k`.
**Details:**
* Number of wires: 1
* Number of parameters: 1
* Number of dimensions per parameter: (0,)
Args:
phi (float): rotation angle :math:`\phi`
wires (Sequence[int] or int): the wire the operation acts on
subspace (Sequence[int]): the 2D subspace on which to apply operation
id (str or None): String representing the operation (optional)
**Example**
The specified subspace will determine which basis states the operation actually
applies to:
>>> qml.TRZ(0.5, wires=0, subspace=(0, 1)).matrix()
array([[0.96891242-0.24740396j, 0. +0.j , 0. +0.j ],
[0. +0.j , 0.96891242+0.24740396j, 0. +0.j ],
[0. +0.j , 0. +0.j , 1. +0.j ]])
>>> qml.TRZ(0.5, wires=0, subspace=(0, 2)).matrix()
array([[0.96891242-0.24740396j, 0. +0.j , 0. +0.j ],
[0. +0.j , 1. +0.j , 0. +0.j ],
[0. +0.j , 0. +0.j , 0.96891242+0.24740396j]])
>>> qml.TRZ(0.5, wires=0, subspace=(1, 2)).matrix()
array([[1. +0.j , 0. +0.j , 0. +0.j ],
[0. +0.j , 0.96891242-0.24740396j, 0. +0.j ],
[0. +0.j , 0. +0.j , 0.96891242+0.24740396j]])
"""
num_wires = 1
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)]
[docs] def generator(self):
# these generators return SProd and Sum, even with the old op_math, because other options are
# not suitable to qudit operators (for example, they do not have a matrix defined as a Hamiltonian)
if self.subspace == (0, 1):
return qml.s_prod(-0.5, qml.GellMann(wires=self.wires, index=3))
if self.subspace == (0, 2):
coeffs = [-0.25, -0.25 * np.sqrt(3)]
obs = [qml.GellMann(wires=self.wires, index=3), qml.GellMann(wires=self.wires, index=8)]
return qml.dot(coeffs, obs)
coeffs = [0.25, -0.25 * np.sqrt(3)]
obs = [qml.GellMann(wires=self.wires, index=3), qml.GellMann(wires=self.wires, index=8)]
return qml.dot(coeffs, obs)
def __init__(self, phi, wires, subspace=(0, 1), id=None):
self._subspace = validate_subspace(subspace)
self._hyperparameters = {
"subspace": self._subspace,
}
super().__init__(phi, wires=wires, id=id)
@property
def subspace(self):
"""The single-qutrit basis states which the operator acts on
This subspace determines which two single-qutrit basis states the operator acts on.
The remaining basis state is not affected by the operator.
Returns:
tuple[int]: subspace on which operator acts
"""
return self._subspace
[docs] @staticmethod
def compute_matrix(theta, subspace=(0, 1)): # 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:`~.TRZ.matrix`
Args:
theta (tensor_like or float): rotation angle
subspace (Sequence[int]): the 2D subspace on which to apply operation
Returns:
tensor_like: canonical matrix
**Example**
>>> qml.TRZ.compute_matrix(torch.tensor(0.5), subspace=(0, 2))
tensor([[0.9689-0.2474j, 0.0000+0.0000j, 0.0000+0.0000j],
[0.0000+0.0000j, 1.0000+0.0000j, 0.0000+0.0000j],
[0.0000+0.0000j, 0.0000+0.0000j, 0.9689+0.2474j]])
"""
if qml.math.get_interface(theta) == "tensorflow":
theta = qml.math.cast_like(theta, 1j)
p = qml.math.exp(-1j * theta / 2)
one = qml.math.ones_like(p)
z = qml.math.zeros_like(p)
diags = [one, one, one]
diags[subspace[0]] = p
diags[subspace[1]] = qml.math.conj(p)
return qml.math.stack(
[
stack_last([diags[0], z, z]),
stack_last([z, diags[1], z]),
stack_last([z, z, diags[2]]),
],
axis=-2,
)
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