Source code for pennylane.ops.qubit.parametric_ops_multi_qubit
# Copyright 2018-2025 Xanadu Quantum Technologies Inc.
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
# http://www.apache.org/licenses/LICENSE-2.0
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""
This submodule contains the discrete-variable quantum operations that are the
core parametrized gates.
"""
# pylint: disable=arguments-differ
import functools
from collections import Counter
from operator import matmul
from typing import Optional, Union
import numpy as np
import pennylane as qml
from pennylane import math, queuing
from pennylane.decomposition import add_decomps, controlled_resource_rep, register_resources
from pennylane.decomposition.symbolic_decomposition import adjoint_rotation, pow_rotation
from pennylane.math.decomposition import decomp_int_to_powers_of_two
from pennylane.operation import FlatPytree, Operation, Operator
from pennylane.typing import TensorLike
from pennylane.wires import Wires, WiresLike
from .non_parametric_ops import Hadamard, PauliX, PauliY, PauliZ
from .parametric_ops_single_qubit import RX, RY, RZ, PhaseShift, _can_replace, stack_last
[docs]
class MultiRZ(Operation):
r"""
Arbitrary multi Z rotation.
.. math::
MultiRZ(\theta) = \exp\left(-i \frac{\theta}{2} Z^{\otimes n}\right)
**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 (TensorLike): 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_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."""
resource_keys = {"num_wires"}
grad_method = "A"
parameter_frequencies = [(1,)]
def _flatten(self) -> FlatPytree:
return self.data, (self.wires, tuple())
def __init__(self, theta: TensorLike, wires: WiresLike, id: Optional[str] = None):
wires = Wires(wires)
self.hyperparameters["num_wires"] = len(wires)
super().__init__(theta, wires=wires, id=id)
if not self._wires:
raise ValueError(
f"{self.name}: wrong number of wires. At least one wire has to be provided."
)
[docs]
@staticmethod
def compute_matrix(theta: TensorLike, num_wires: int) -> TensorLike:
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 (TensorLike): rotation angle
num_wires (int): number of wires the rotation acts on
Returns:
TensorLike: 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 = math.convert_like(qml.pauli.pauli_eigs(num_wires), theta)
if math.get_interface(theta) == "tensorflow":
theta = math.cast_like(theta, 1j)
eigs = math.cast_like(eigs, 1j)
if math.ndim(theta) == 0:
return math.diag(math.exp(-0.5j * theta * eigs))
diags = math.exp(math.outer(-0.5j * theta, eigs))
return diags[:, :, np.newaxis] * math.cast_like(math.eye(2**num_wires, like=diags), diags)
[docs]
def generator(self) -> "qml.Hamiltonian":
return qml.Hamiltonian([-0.5], [functools.reduce(matmul, [PauliZ(w) for w in self.wires])])
[docs]
@staticmethod
def compute_eigvals(theta: TensorLike, num_wires: int) -> TensorLike:
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 (TensorLike): rotation angle
num_wires (int): number of wires the rotation acts on
Returns:
TensorLike: 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 = math.convert_like(qml.pauli.pauli_eigs(num_wires), theta)
if math.get_interface(theta) == "tensorflow":
theta = math.cast_like(theta, 1j)
eigs = math.cast_like(eigs, 1j)
if math.ndim(theta) == 0:
return math.exp(-0.5j * theta * eigs)
return math.exp(math.outer(-0.5j * theta, eigs))
[docs]
@staticmethod
def compute_decomposition( # pylint: disable=arguments-differ,unused-argument
theta: TensorLike, wires: WiresLike, **kwargs
) -> list[Operator]:
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 (TensorLike): 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
@property
def resource_params(self) -> dict:
return {"num_wires": self.hyperparameters["num_wires"]}
[docs]
def pow(self, z: Union[int, float]) -> list[Operator]:
return [MultiRZ(self.data[0] * z, wires=self.wires)]
[docs]
def simplify(self) -> "MultiRZ":
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)
def _multi_rz_decomposition_resources(num_wires):
return {qml.RZ: 1, qml.CNOT: 2 * (num_wires - 1)}
@register_resources(_multi_rz_decomposition_resources)
def _multi_rz_decomposition(theta: TensorLike, wires: WiresLike, **__):
@qml.for_loop(len(wires) - 1, 0, -1)
def _pre_cnot(i):
qml.CNOT(wires=(wires[i], wires[i - 1]))
@qml.for_loop(1, len(wires), 1)
def _post_cnot(i):
qml.CNOT(wires=(wires[i], wires[i - 1]))
_pre_cnot() # pylint: disable=no-value-for-parameter
qml.RZ(theta, wires=wires[0])
_post_cnot() # pylint: disable=no-value-for-parameter
add_decomps(MultiRZ, _multi_rz_decomposition)
add_decomps("Adjoint(MultiRZ)", adjoint_rotation)
add_decomps("Pow(MultiRZ)", pow_rotation)
[docs]
class PauliRot(Operation):
r"""
Arbitrary Pauli word rotation.
.. math::
RP(\theta, P) = \exp\left(-i \frac{\theta}{2} P\right)
**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_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,)]
resource_keys = {
"pauli_word",
}
_ALLOWED_CHARACTERS = "IXYZ"
_PAULI_CONJUGATION_MATRICES = {
"X": Hadamard.compute_matrix(),
"Y": RX.compute_matrix(np.pi / 2),
"Z": np.array([[1, 0], [0, 1]]),
}
@classmethod
def _primitive_bind_call(cls, theta, pauli_word, wires=None, id=None):
return super()._primitive_bind_call(theta, pauli_word=pauli_word, wires=wires, id=id)
def __init__(
self,
theta: TensorLike,
pauli_word: str,
wires: WiresLike,
id: Optional[str] = None,
):
super().__init__(theta, wires=wires, id=id)
if not self._wires:
raise ValueError(
f"{self.name}: wrong number of wires. At least one wire has to be provided."
)
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 number of wires must be equal to the length of the Pauli word. "
f"The Pauli word {pauli_word} has length {len(pauli_word)}, and "
f"{num_wires} wires were given {wires}."
)
def __repr__(self) -> str:
return f"PauliRot({self.data[0]}, {self.hyperparameters['pauli_word']}, wires={self.wires.tolist()})"
[docs]
def label(
self,
decimals: Optional[int] = None,
base_label: Optional[str] = None,
cache: Optional[dict] = None,
) -> str:
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({math.asarray(self.parameters[0]):.{decimals}f})"
op_label += param_string
return op_label
@property
def resource_params(self) -> dict:
return {"pauli_word": self.hyperparameters["pauli_word"]}
@staticmethod
def _check_pauli_word(pauli_word) -> bool:
"""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: TensorLike, pauli_word: str) -> TensorLike:
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 (TensorLike): rotation angle
pauli_word (str): string representation of Pauli word
Returns:
TensorLike: 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 = math.get_interface(theta)
if interface == "tensorflow":
theta = math.cast_like(theta, 1j)
# Simplest case is if the Pauli is the identity matrix
if set(pauli_word) == {"I"}:
return qml.GlobalPhase.compute_matrix(0.5 * theta, n_wires=len(pauli_word))
# 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(
math.kron,
[PauliRot._PAULI_CONJUGATION_MATRICES[gate] for gate in non_identity_gates],
)
if interface == "tensorflow":
conjugation_matrix = 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 math.expand_matrix(
math.einsum(
"...jk,ij->...ik",
math.tensordot(multi_Z_rot_matrix, conjugation_matrix, axes=[[-1], [0]]),
math.conj(conjugation_matrix),
),
non_identity_wires,
list(range(len(pauli_word))),
)
[docs]
def generator(self) -> "qml.Hamiltonian":
pauli_word = self.hyperparameters["pauli_word"]
wire_map = {w: i for i, w in enumerate(self.wires)}
return qml.Hamiltonian(
[-0.5], [qml.pauli.string_to_pauli_word(pauli_word, wire_map=wire_map)]
)
[docs]
@staticmethod
def compute_eigvals(theta: TensorLike, pauli_word: str) -> TensorLike:
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:
TensorLike: eigenvalues
**Example**
>>> qml.PauliRot.compute_eigvals(torch.tensor(0.5), "X")
tensor([0.9689-0.2474j, 0.9689+0.2474j])
"""
if math.get_interface(theta) == "tensorflow":
theta = math.cast_like(theta, 1j)
# Identity must be treated specially because its eigenvalues are all the same
if set(pauli_word) == {"I"}:
return qml.GlobalPhase.compute_eigvals(0.5 * theta, n_wires=len(pauli_word))
return MultiRZ.compute_eigvals(theta, len(pauli_word))
[docs]
@staticmethod
def compute_decomposition(
theta: TensorLike, wires: WiresLike, pauli_word: str
) -> list[Operator]:
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 (TensorLike): 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))
[H(0),
RX(1.5707963267948966, wires=[1]),
MultiRZ(1.2, wires=[0, 1]),
H(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 [qml.GlobalPhase(phi=theta / 2)]
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)]
def _pauli_rot_resources(pauli_word):
if set(pauli_word) == {"I"}:
return {qml.GlobalPhase: 1}
num_active_wires = len(pauli_word.replace("I", ""))
return {
qml.Hadamard: 2 * pauli_word.count("X"),
qml.RX: 2 * pauli_word.count("Y"),
qml.resource_rep(qml.MultiRZ, num_wires=num_active_wires): 1,
}
@register_resources(_pauli_rot_resources)
def _pauli_rot_decomposition(theta, pauli_word, wires, **__):
if set(pauli_word) == {"I"}:
qml.GlobalPhase(theta / 2)
return
active_wires, active_gates = zip(
*[(wire, gate) for wire, gate in zip(wires, pauli_word) if gate != "I"]
)
for wire, gate in zip(active_wires, active_gates):
if gate == "X":
qml.Hadamard(wires=[wire])
elif gate == "Y":
qml.RX(np.pi / 2, wires=[wire])
qml.MultiRZ(theta, wires=list(active_wires))
for wire, gate in zip(active_wires, active_gates):
if gate == "X":
qml.Hadamard(wires=[wire])
elif gate == "Y":
qml.RX(-np.pi / 2, wires=[wire])
add_decomps(PauliRot, _pauli_rot_decomposition)
add_decomps("Adjoint(PauliRot)", adjoint_rotation)
add_decomps("Pow(PauliRot)", pow_rotation)
[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 ``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}.
This can also be written as :math:`\Pi(\phi) = \exp(i\phi(2\Pi-\mathbb{I}_N))`, where
:math:`N=2^n` is the Hilbert space dimension for :math:`n` qubits and :math:`\Pi` is
the diagonal projector with ``dim`` ones and ``N-dim`` zeros.
**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:
>>> op_3 = qml.PCPhase(0.27, dim = 3, wires=range(3))
The resulting operation applies a complex phase :math:`e^{0.27i}` to the first :math:`dim = 3`
basis vectors and :math:`e^{-0.27i}` to the remaining basis vectors, as we can see from
the diagonal of the matrix for this circuit.
>>> print(np.round(np.diag(qml.matrix(op_3)),2))
[0.96+0.27j 0.96+0.27j 0.96+0.27j 0.96-0.27j 0.96-0.27j 0.96-0.27j
0.96-0.27j 0.96-0.27j]
We can also choose a different ``dim`` value to apply the phase shift to a different set of
basis vectors as follows:
>>> op_7 = qml.PCPhase(1.23, dim=7, wires=[1, 2, 3])
>>> print(np.round(np.diag(qml.matrix(op_7)),2))
[0.33+0.94j 0.33+0.94j 0.33+0.94j 0.33+0.94j 0.33+0.94j 0.33+0.94j
0.33+0.94j 0.33-0.94j]
``PCPhase`` operations are decomposed into (multi-)controlled :class:`~.PhaseShift`
operations which share the same control values on common control wires, and Pauli-X operations,
possibly complemented by a global phase.
>>> op_13 = qml.PCPhase(1.23, dim=13, wires=[1, 2, 3, 4])
>>> print(qml.draw(op_13.decomposition)())
1: ──GlobalPhase(-1.23)─╭●─────────╭●───────────┤
2: ──GlobalPhase(-1.23)─╰Rϕ(-2.46)─├●───────────┤
3: ──GlobalPhase(-1.23)────────────├○───────────┤
4: ──GlobalPhase(-1.23)──X─────────╰Rϕ(2.46)──X─┤
If ``dim`` is a power of two, a single (multi-controlled) ``PhaseShift`` gate is sufficient:
>>> op_16 = qml.PCPhase(1.23, dim=16, wires=range(6))
>>> print(qml.draw(op_16.decomposition, wire_order=range(6), show_all_wires=True)())
0: ──GlobalPhase(1.23)────╭○───────────┤
1: ──GlobalPhase(1.23)──X─╰Rϕ(2.46)──X─┤
2: ──GlobalPhase(1.23)─────────────────┤
3: ──GlobalPhase(1.23)─────────────────┤
4: ──GlobalPhase(1.23)─────────────────┤
5: ──GlobalPhase(1.23)─────────────────┤
"""
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,)]
resource_keys = {"num_wires", "dim"}
[docs]
def generator(self) -> "qml.Hermitian":
r"""Generator of the ``PCPhase`` operator, which is in single-parameter-form.
The operator reads
.. math:: \Pi(\phi) = e^{i\phi (2\Pi - \mathbb{I}_N)},
where :math:`\Pi` is the projector onto the first :math`d` (``dim``) computational basis
states and :math:`N=2^n` is the Hilbert space dimension for :math:`n` qubits.
Correspondingly, the generator is
:math:`2\Pi - \mathbb{I}_N=\text{diag}(\underset{d\text{ times}}{\underbrace{1, \dots, 1}},\underset{(N-d)\text{ times}}{\underbrace{-1, \dots, -1}})`:
>>> qml.PCPhase(0.5, dim=3, wires=[0, 1]).generator()
Hermitian(array([[ 1, 0, 0, 0],
[ 0, 1, 0, 0],
[ 0, 0, 1, 0],
[ 0, 0, 0, -1]]), wires=[0, 1])
"""
dim, N = self.hyperparameters["dimension"]
mat = np.diag([1] * dim + [-1] * (N - dim))
return qml.Hermitian(mat, wires=self.wires)
def _flatten(self) -> FlatPytree:
hyperparameter = (("dim", self.hyperparameters["dimension"][0]),)
return tuple(self.data), (self.wires, hyperparameter)
def __init__(self, phi: TensorLike, dim: int, wires: WiresLike, id: Optional[str] = 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))
@property
def resource_params(self) -> dict:
return {"num_wires": len(self.wires), "dim": self.hyperparameters["dimension"][0]}
[docs]
@staticmethod
def compute_matrix(phi: TensorLike, dimension: tuple[int, int]) -> TensorLike:
"""Get the matrix representation of Pi-controlled phase unitary."""
d, t = dimension
if math.get_interface(phi) == "tensorflow":
p = math.exp(1j * math.cast_like(phi, 1j))
minus_p = math.exp(-1j * math.cast_like(phi, 1j))
zeros = 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 = math.stack(columns, like="tensorflow", axis=-2)
return r
arg = 1j * phi
prefactors = math.array([1] * d + [-1] * (t - d), like=phi)
if math.ndim(arg) == 0:
return math.diag(math.exp(arg * prefactors))
diags = math.exp(math.outer(arg, prefactors))
return math.stack([math.diag(d) for d in diags])
[docs]
@staticmethod
def compute_eigvals(*params: TensorLike, **hyperparams) -> TensorLike:
"""Get the eigvals for the Pi-controlled phase unitary."""
phi = params[0]
d, t = hyperparams["dimension"]
if math.get_interface(phi) == "tensorflow":
phase = math.exp(1j * math.cast_like(phi, 1j))
minus_phase = math.exp(-1j * math.cast_like(phi, 1j))
return stack_last([phase if index < d else minus_phase for index in range(t)])
arg = 1j * phi
prefactors = math.array([1] * d + [-1] * (t - d), like=phi)
if math.ndim(phi) == 0:
product = arg * prefactors
else:
product = math.outer(arg, prefactors)
return math.exp(product)
[docs]
@staticmethod
def compute_decomposition(
*params: TensorLike, wires: WiresLike, **hyperparams
) -> list[Operator]:
r"""Representation of the PCPhase operator as a product of other operators (static method).
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
In short, this decomposition relies on decomposing the generator (see :meth:`~.generator`)
of the ``PCPhase`` gate into generators of multicontrolled :class:`~.PhaseShift` gates,
potentially complemented with (non-controlled) Pauli-X gates and/or a global phase.
For example, for ``dim=13`` on four qubits:
>>> op_13 = qml.PCPhase(1.23, dim=13, wires=[1, 2, 3, 4])
>>> print(qml.draw(op_13.decomposition)())
1: ──GlobalPhase(-1.23)─╭●─────────╭●───────────┤
2: ──GlobalPhase(-1.23)─╰Rϕ(-2.46)─├●───────────┤
3: ──GlobalPhase(-1.23)────────────├○───────────┤
4: ──GlobalPhase(-1.23)──X─────────╰Rϕ(2.46)──X─┤
In the following we provide a detailed example for illustration purposes.
**Detailed example**
Consider the projector-controlled phase gate on :math:`n=4` qubits and with
:math:`d=\texttt{dim}=3`, i.e,
>>> op_3 = qml.PCPhase(1.23, dim=3, wires=[0, 1, 2, 3])
It acts on :math:`N=2^n=16`-dimensional vectors and is described by
.. math:: \Pi(\phi) = \exp(i\phi G) = \exp(i\phi(2\Pi-\mathbb{I}_N)),
where :math:`G` is a diagonal matrix with :math:`d=3` ones, followed by
:math:`2^n-d = 16 - 3=13` negative ones. Accordingly, :math:`\Pi` is diagonal with
:math:`3` ones and :math:`13` zeros.
First, we implement the global phase generated by :math:`\mathbb{I}_N` with
a :class:`~.GlobalPhase` gate with angle :math:`-\phi`.
Then we decompose :math:`d` into powers of two with positive or negative sign, via
:math:`d=3=4-1 = 2^2-2^0`. This decomposition tells us that we can write the
target gate with two (multi-)controlled phase shift gates. For this, we rewrite
the projector :math:`\Pi` according to the decomposition as
.. math::
\Pi &= \text{diag}(1, 1, 1, 0, 0, \dots, 0)\\
&=\text{diag}(1, 1, 1, 1, 0, \dots, 0)
-\text{diag}(0, 0, 0, 1, 0, \dots, 0)
where :math:`0,\dots, 0` indicates :math:`12` zeros each time.
How do we realize this projector decomposition on the gate level?
A singly-controlled phase shift gate applies a phase to a quarter of all computational
basis states (the control filters by the state of one qubit, and the phase shift gate
itself filters by the :math:`|1\rangle` state of the target qubit, cutting the number
of states we are acting on in half each time).
For :math:`n=4`, this amounts to :math:`2^4/4=4` states, which is exactly
what we need for the first term above. To apply the phase to the *first* four states,
:math:`|0000\rangle`, :math:`|0001\rangle`, :math:`|0010\rangle`, and :math:`|0011\rangle`,
we want to "filter by" the first two qubits being in the :math:`|0\rangle` state.
For qubit :math:`0`, we do this by controlling on the :math:`|0\rangle` state.
For qubit :math:`1`, we pick it as the target of the controlled phase shift operation.
Generically, this would make it act on the :math:`|1\rangle` state, so we simply flip
qubit :math:`1` before and after the operation to apply the phase to the :math:`|0\rangle`
state instead.
Thus, we conclude this first step by applying the gates
``qml.X(1)``, ``qml.ctrl(qml.PhaseShift(2 * phi, 1), control=[0], control_values=[0])``,
and ``qml.X(1)``.
Next, we implement the second term in the projector decomposition, applying a phase
to a single computational basis state. This requires us to fully control a phase shift
gate, i.e., we use the last qubit as target and the other three as controls (there is
some freedom of choice here, but this is a convenient choice).
We want to apply the phase to the state :math:`|3\rangle=|0011\rangle`. So the controls
:math:`0` and :math:`1` are set to zero and the control :math:`2` is set to one.
As we want to effect the phase onto the :math:`|1\rangle` state of qubit :math:`3`,
we don't need to flip the target bit as we did before. However, given the negative sign
in the projector decomposition, we need to multiply the phase with :math:`-1`.
Overall, we apply the gate
``qml.ctrl(qml.PhaseShift(-2 * phi, 3), control=[0, 1, 2], control_values=[0, 0, 1])``,
which concludes the decomposition, now reading:
>>> print(qml.draw(op_3.decomposition)())
0: ──GlobalPhase(1.23)────╭○───────────╭○─────────┤
1: ──GlobalPhase(1.23)──X─╰Rϕ(2.46)──X─├○─────────┤
2: ──GlobalPhase(1.23)─────────────────├●─────────┤
3: ──GlobalPhase(1.23)─────────────────╰Rϕ(-2.46)─┤
"""
with queuing.AnnotatedQueue() as q:
_decompose_pcphase(*params, wires=wires, **hyperparams)
if queuing.QueuingManager.recording():
for op in q.queue:
queuing.apply(op)
return q.queue
[docs]
def adjoint(self) -> "PCPhase":
"""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: Union[int, float]) -> list[Operator]:
"""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) -> "PCPhase":
"""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: Optional[int] = None,
base_label: Optional[str] = None,
cache: Optional[dict] = None,
) -> str:
"""The label of the operator when displayed in a circuit."""
return super().label(decimals=decimals, base_label=base_label or "∏_ϕ", cache=cache)
def _ctrl_phase_shift_resource(subspace, n_control_wires, n_zero_control_values, n_work_wires):
if n_control_wires == 0:
return {qml.PhaseShift: 1}
return {
controlled_resource_rep(
qml.PhaseShift,
{},
num_control_wires=n_control_wires,
num_zero_control_values=n_zero_control_values,
num_work_wires=n_work_wires,
): 1,
qml.X: 2 * (1 - subspace),
}
def _ctrl_phase_shift(
phi, target_wire, subspace, control_wires, control_values, work_wires
): # pylint: disable=too-many-arguments
r"""Implement a ((multi-)controlled) phase shift on the specified subspace of a
target qubit/wire.
Args:
phi (float): Phase shift angle
target_wire (~.Wires): the target wire to apply phase shift to.
subspace (int): which subspace of the target wire the phase shift is applied to. 0 or 1.
control_wires (WiresLike): the control wires
control_values (Iterable[bool | int]): the control values.
work_wires (WiresLike): the work wires
Returns:
float: any global phase produced in the process.
The decomposition for subspace=1 always is a simple ``PhaseShift`` gate, or its controlled
counterpart. The decomposition of a non-controlled phase shift for subspace=0 can be achieved
in two ways: The first is to flip the angle of the phase shift and complementing it with a
global phase, so that the (diagonal of the) gate is decomposed as
.. math::
(\exp(i\phi), 1) = (1, \exp(-i\phi)) (\exp(i\phi), \exp(i\phi)).
The second is to conjugate a phase shift by Pauli-X operators on the same qubit, decomposing
the gate matrix as
/ e^(i\phi) 0 \ -- / 0 1 \/ 1 0 \/ 0 1 \
\ 0 1 / -- \ 1 0 /\ 0 e^(i\phi) /\ 1 0 /.
Without controls, the first approach is nicer, because global phases usually are free
operations. With controls, however, this approach would lead to a controlled global
phase, which is equivalent to a phase shift on top of the controlled phase shift. For the
second approach, we may use
`the ctrl(compute-uncompute) pattern <https://iopscience.iop.org/article/10.1088/2058-9565/aaa5cc>`__
to avoid controlling the Pauli-X operations, yielding two non-controlled Pauli-X gates and the
"main" controlled phase shift operation. We deem this decomposition to be better.
"""
if subspace == 1:
# If there are no control wires, we are dealing with the very first phase shift of
# the decomposition, which should be adding projectors. So subspace should have been 0.
assert len(control_wires) > 0
qml.ctrl(
qml.PhaseShift(phi, wires=target_wire),
control=control_wires,
control_values=control_values,
work_wires=work_wires,
)
return 0.0
if len(control_wires) == 0:
# Flip angle for phase_shift(subspace=0) = phase_shift(subspace=1)*global_phase
qml.PhaseShift(-phi, wires=target_wire)
return -phi
qml.X(target_wire)
qml.ctrl(
qml.PhaseShift(phi, wires=target_wire),
control=control_wires,
control_values=control_values,
work_wires=work_wires,
)
qml.X(target_wire)
return 0.0
def _decompose_pcphase_resource(num_wires, dim):
"""Decompose the PCPhase operation into controlled phase shifts and Pauli-X gates."""
gate_count = Counter()
flipped, *powers_of_two = decomp_int_to_powers_of_two(dim, num_wires + 1)
sigma = (-1) ** flipped
powers_of_two = [sigma * val for val in powers_of_two]
n_zero_control_values = 0
for i, c_i in enumerate(powers_of_two):
if c_i != 0:
subspace = int(c_i < 0)
if flipped:
subspace = 1 - subspace
gate_count.update(
_ctrl_phase_shift_resource(
subspace,
n_control_wires=i,
n_zero_control_values=n_zero_control_values,
n_work_wires=num_wires - i - 1,
)
)
d_i = next(iter(val for val in powers_of_two[i + 1 :] if val != 0), None)
next_cval = d_i == 1
if c_i == 0:
next_cval = not next_cval
if flipped:
next_cval = not next_cval
if not next_cval:
n_zero_control_values += 1
gate_count[qml.GlobalPhase] += 1
return dict(gate_count)
@register_resources(_decompose_pcphase_resource)
def _decompose_pcphase(phi, wires, dimension):
"""Decompose the PCPhase operation into controlled phase shifts and Pauli-X gates."""
dim, _ = dimension
# Use one more bit than there are wires, according to flipping all relevant bits for the
# projector decomposition, or a global phase on the gate level. Afterwards, we have
# dim <= 2**len(wires)=2**(n-1), which is a requirement of decomp_int_to_powers_of_two.
flipped, *powers_of_two = decomp_int_to_powers_of_two(dim, len(wires) + 1)
sigma = (-1) ** flipped
# Overall global phase to implement I_N part of the generator
global_phase = sigma * phi
phi = 2 * sigma * phi
# If in flipped (sigma=-1) mode, reverse the sign of all coefficients
powers_of_two = [sigma * val for val in powers_of_two]
assert len(powers_of_two) == len(wires)
control_values = []
for i, c_i in enumerate(powers_of_two):
if c_i != 0:
# Projector with rank 2**(n-1-i) needs to be added/subtracted
subspace = int(c_i < 0) # If c_i < 0, target |1> subspace, else target |0> subspace
if flipped: # Flip subspace if in flipped (sigma=-1) mode
subspace = 1 - subspace
global_phase += _ctrl_phase_shift(
c_i * phi,
target_wire=wires[i],
subspace=subspace,
control_wires=wires[:i],
control_values=control_values,
work_wires=wires[i + 1 :], # Unused wires of PCPhase can be used as work wires
)
# The control value to be used on the current wire (it will be the same for all
# subsequent operations) depends on whether we add or subtract next)
d_i = next(iter(val for val in powers_of_two[i + 1 :] if val != 0), None)
# If we add next, control into the |1> subspace, otherwise into |0> subspace
# The control value is modified both if we are in the flipped global phase mode and if
# the current loop iteration did not add a bit string/projector/gate (c_i==0)
next_cval = d_i == 1
if c_i == 0:
next_cval = not next_cval
if flipped:
next_cval = not next_cval
control_values.append(next_cval)
qml.GlobalPhase(global_phase)
add_decomps(PCPhase, _decompose_pcphase)
[docs]
class IsingXX(Operation):
r"""
Ising XX coupling gate
.. math:: XX(\phi) = \exp\left(-i \frac{\phi}{2} (X \otimes X)\right) =
\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."""
resource_keys = set()
grad_method = "A"
parameter_frequencies = [(1,)]
[docs]
def generator(self) -> "qml.Hamiltonian":
return qml.Hamiltonian([-0.5], [PauliX(wires=self.wires[0]) @ PauliX(wires=self.wires[1])])
def __init__(self, phi: TensorLike, wires: WiresLike, id: Optional[str] = None):
super().__init__(phi, wires=wires, id=id)
@property
def resource_params(self) -> dict:
return {}
[docs]
@staticmethod
def compute_matrix(phi: TensorLike) -> TensorLike: # 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 (TensorLike): phase angle
Returns:
TensorLike: 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 = math.cos(phi / 2)
s = math.sin(phi / 2)
eye = math.eye(4, like=phi)
rev_eye = math.convert_like(np.eye(4)[::-1].copy(), phi)
if math.get_interface(phi) == "tensorflow":
c = math.cast_like(c, 1j)
s = math.cast_like(s, 1j)
eye = math.cast_like(eye, 1j)
rev_eye = math.cast_like(rev_eye, 1j)
# The following avoids casting an imaginary quantity to reals when backpropagating
js = -1j * s
if math.ndim(phi) == 0:
return c * eye + js * rev_eye
return math.tensordot(c, eye, axes=0) + math.tensordot(js, rev_eye, axes=0)
[docs]
@staticmethod
def compute_decomposition(phi: TensorLike, wires: WiresLike) -> list[Operator]:
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 (TensorLike): 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 adjoint(self) -> "IsingXX":
(phi,) = self.parameters
return IsingXX(-phi, wires=self.wires)
[docs]
def pow(self, z: Union[int, float]) -> list[Operator]:
return [IsingXX(self.data[0] * z, wires=self.wires)]
[docs]
def simplify(self) -> "IsingXX":
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)
def _isingxx_to_cnot_rx_cnot_resources():
return {qml.CNOT: 2, qml.RX: 1}
@register_resources(_isingxx_to_cnot_rx_cnot_resources)
def _isingxx_to_cnot_rx_cnot(phi: TensorLike, wires: WiresLike, **__):
qml.CNOT(wires=wires)
qml.RX(phi, wires=[wires[0]])
qml.CNOT(wires=wires)
add_decomps(IsingXX, _isingxx_to_cnot_rx_cnot)
add_decomps("Adjoint(IsingXX)", adjoint_rotation)
add_decomps("Pow(IsingXX)", pow_rotation)
[docs]
class IsingYY(Operation):
r"""
Ising YY coupling gate
.. math:: \mathtt{YY}(\phi) = \exp\left(-i \frac{\phi}{2} (Y \otimes Y)\right) =
\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."""
resource_keys = set()
grad_method = "A"
parameter_frequencies = [(1,)]
[docs]
def generator(self) -> "qml.Hamiltonian":
return qml.Hamiltonian([-0.5], [PauliY(wires=self.wires[0]) @ PauliY(wires=self.wires[1])])
def __init__(self, phi: TensorLike, wires: WiresLike, id: Optional[str] = None):
super().__init__(phi, wires=wires, id=id)
@property
def resource_params(self) -> dict:
return {}
[docs]
@staticmethod
def compute_decomposition(phi: TensorLike, wires: WiresLike) -> list[Operator]:
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: TensorLike) -> TensorLike: # 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 (TensorLike): phase angle
Returns:
TensorLike: 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 = math.cos(phi / 2)
s = math.sin(phi / 2)
if math.get_interface(phi) == "tensorflow":
c = math.cast_like(c, 1j)
s = math.cast_like(s, 1j)
js = 1j * s
r_term = math.cast_like(
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 math.ndim(phi) == 0:
return c * math.cast_like(math.eye(4, like=c), c) + js * r_term
return math.tensordot(c, np.eye(4), axes=0) + math.tensordot(js, r_term, axes=0)
[docs]
def adjoint(self) -> "IsingYY":
(phi,) = self.parameters
return IsingYY(-phi, wires=self.wires)
[docs]
def pow(self, z: Union[int, float]) -> list[Operator]:
return [IsingYY(self.data[0] * z, wires=self.wires)]
[docs]
def simplify(self) -> "IsingYY":
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)
def _isingyy_to_cy_ry_cy_resources():
return {qml.CY: 2, RY: 1}
@register_resources(_isingyy_to_cy_ry_cy_resources)
def _isingyy_to_cy_ry_cy(phi: TensorLike, wires: WiresLike, **__):
qml.CY(wires=wires)
RY(phi, wires=[wires[0]])
qml.CY(wires=wires)
add_decomps(IsingYY, _isingyy_to_cy_ry_cy)
add_decomps("Adjoint(IsingYY)", adjoint_rotation)
add_decomps("Pow(IsingYY)", pow_rotation)
[docs]
class IsingZZ(Operation):
r"""
Ising ZZ coupling gate
.. math:: ZZ(\phi) = \exp\left(-i \frac{\phi}{2} (Z \otimes Z)\right) =
\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."""
resource_keys = set()
grad_method = "A"
parameter_frequencies = [(1,)]
[docs]
def generator(self) -> "qml.Hamiltonian":
return qml.Hamiltonian([-0.5], [PauliZ(wires=self.wires[0]) @ PauliZ(wires=self.wires[1])])
def __init__(self, phi: TensorLike, wires: WiresLike, id: Optional[str] = None):
super().__init__(phi, wires=wires, id=id)
@property
def resource_params(self) -> dict:
return {}
[docs]
@staticmethod
def compute_decomposition(phi: TensorLike, wires: WiresLike):
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: TensorLike) -> TensorLike: # 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 (TensorLike): phase angle
Returns:
TensorLike: 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 math.get_interface(phi) == "tensorflow":
p = math.exp(-0.5j * math.cast_like(phi, 1j))
if math.ndim(p) == 0:
return math.diag([p, math.conj(p), math.conj(p), p])
diags = stack_last([p, math.conj(p), math.conj(p), p])
return diags[:, :, np.newaxis] * math.cast_like(math.eye(4, like=diags), diags)
signs = math.array([1, -1, -1, 1], like=phi)
arg = -0.5j * phi
if math.ndim(arg) == 0:
return math.diag(math.exp(arg * signs))
diags = math.exp(math.outer(arg, signs))
return diags[:, :, np.newaxis] * math.cast_like(math.eye(4, like=diags), diags)
[docs]
@staticmethod
def compute_eigvals(phi: TensorLike) -> TensorLike: # 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 (TensorLike) phase angle
Returns:
TensorLike: 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 math.get_interface(phi) == "tensorflow":
phase = math.exp(-0.5j * math.cast_like(phi, 1j))
return stack_last([phase, math.conj(phase), math.conj(phase), phase])
prefactors = math.array([-0.5j, 0.5j, 0.5j, -0.5j], like=phi)
if math.ndim(phi) == 0:
product = phi * prefactors
else:
product = math.outer(phi, prefactors)
return math.exp(product)
[docs]
def adjoint(self) -> "IsingZZ":
(phi,) = self.parameters
return IsingZZ(-phi, wires=self.wires)
[docs]
def pow(self, z: Union[int, float]) -> list[Operator]:
return [IsingZZ(self.data[0] * z, wires=self.wires)]
[docs]
def simplify(self) -> "IsingZZ":
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)
def _isingzz_to_cnot_rz_cnot_resources():
return {qml.CNOT: 2, RZ: 1}
@register_resources(_isingzz_to_cnot_rz_cnot_resources)
def _isingzz_to_cnot_rz_cnot(phi: TensorLike, wires: WiresLike, **__):
qml.CNOT(wires=wires)
RZ(phi, wires=[wires[1]])
qml.CNOT(wires=wires)
add_decomps(IsingZZ, _isingzz_to_cnot_rz_cnot)
add_decomps("Adjoint(IsingZZ)", adjoint_rotation)
add_decomps("Pow(IsingZZ)", pow_rotation)
[docs]
class IsingXY(Operation):
r"""
Ising (XX + YY) coupling gate
.. math:: \mathtt{XY}(\phi) = \exp\left(i \frac{\phi}{4} (X \otimes X + Y \otimes Y)\right) =
\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."""
resource_keys = set()
grad_method = "A"
parameter_frequencies = [(0.5, 1.0)]
[docs]
def generator(self) -> "qml.Hamiltonian":
return qml.Hamiltonian(
[0.25, 0.25],
[
qml.X(wires=self.wires[0]) @ qml.X(wires=self.wires[1]),
qml.Y(wires=self.wires[0]) @ qml.Y(wires=self.wires[1]),
],
)
def __init__(self, phi: TensorLike, wires: WiresLike, id: Optional[str] = None):
super().__init__(phi, wires=wires, id=id)
@property
def resource_params(self) -> dict:
return {}
[docs]
@staticmethod
def compute_decomposition(phi: TensorLike, wires: WiresLike) -> list[Operator]:
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))
[H(0), CY(wires=[0, 1]), RY(0.615, wires=[0]), RX(-0.615, wires=[1]), CY(wires=[0, 1]), H(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: TensorLike) -> TensorLike: # 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 (TensorLike): phase angle
Returns:
TensorLike: 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 = math.cos(phi / 2)
s = math.sin(phi / 2)
if math.get_interface(phi) == "tensorflow":
c = math.cast_like(c, 1j)
s = math.cast_like(s, 1j)
js = 1j * s
off_diag = math.cast_like(
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 math.ndim(phi) == 0:
return math.diag([1, c, c, 1]) + js * off_diag
ones = math.ones_like(c)
diags = stack_last([ones, c, c, ones])[:, :, np.newaxis]
return diags * np.eye(4) + math.tensordot(js, off_diag, axes=0)
[docs]
@staticmethod
def compute_eigvals(phi: TensorLike) -> TensorLike: # 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 (TensorLike): phase angle
Returns:
TensorLike: eigenvalues
**Example**
>>> qml.IsingXY.compute_eigvals(0.5)
array([0.96891242+0.24740396j, 0.96891242-0.24740396j, 1. +0.j , 1. +0.j ])
"""
if math.get_interface(phi) == "tensorflow":
phi = math.cast_like(phi, 1j)
signs = np.array([1, -1, 0, 0])
if math.ndim(phi) == 0:
return math.exp(0.5j * phi * signs)
return math.exp(math.tensordot(0.5j * phi, signs, axes=0))
[docs]
def adjoint(self) -> "IsingXY":
(phi,) = self.parameters
return IsingXY(-phi, wires=self.wires)
[docs]
def pow(self, z: Union[int, float]) -> list[Operator]:
return [IsingXY(self.data[0] * z, wires=self.wires)]
[docs]
def simplify(self) -> "IsingXY":
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)
def _isingxy_to_h_cy_resources():
return {Hadamard: 2, qml.CY: 2, RY: 1, RX: 1}
@register_resources(_isingxy_to_h_cy_resources)
def _isingxy_to_h_cy(phi: TensorLike, wires: WiresLike, **__):
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]])
add_decomps(IsingXY, _isingxy_to_h_cy)
add_decomps("Adjoint(IsingXY)", adjoint_rotation)
add_decomps("Pow(IsingXY)", pow_rotation)
[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."""
ndim_params = (0,)
"""tuple[int]: Number of dimensions per trainable parameter that the operator depends on."""
resource_keys = set()
grad_method = "A"
grad_recipe = ([[0.5, 1, np.pi / 2], [-0.5, 1, -np.pi / 2]],)
def __init__(self, phi: TensorLike, wires: WiresLike, id: Optional[str] = None):
super().__init__(phi, wires=wires, id=id)
@property
def resource_params(self) -> dict:
return {}
[docs]
@staticmethod
def compute_decomposition(phi: TensorLike, wires: WiresLike) -> list[Operator]:
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: TensorLike) -> TensorLike: # 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 (TensorLike): phase angle
Returns:
TensorLike: 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 math.get_interface(phi) == "tensorflow":
phi = math.cast_like(phi, 1j)
e = math.exp(1j * phi)
zero = math.zeros_like(phi)
one = math.ones_like(phi)
return math.stack(
[
stack_last([one, zero, zero, zero]),
stack_last([zero, zero, e, zero]),
stack_last([zero, e, zero, zero]),
stack_last([zero, zero, zero, one]),
],
axis=-2,
)
[docs]
@staticmethod
def compute_eigvals(phi: TensorLike) -> TensorLike: # 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 (TensorLike): phase angle
Returns:
TensorLike: eigenvalues
**Example**
>>> qml.PSWAP.compute_eigvals(0.5)
array([ 1. +0.j , 1. +0.j, -0.87758256-0.47942554j, 0.87758256+0.47942554j])
"""
if math.get_interface(phi) == "tensorflow":
phi = math.cast_like(phi, 1j)
e = math.exp(1j * phi)
one = math.ones_like(phi)
return math.transpose(math.stack([one, one, -e, e]))
[docs]
def simplify(self) -> "PSWAP":
phi = self.data[0] % (2 * np.pi)
if _can_replace(phi, 0):
return qml.SWAP(wires=self.wires)
return PSWAP(phi, wires=self.wires)
def _pswap_to_swap_cnot_phaseshift_cnot_resources():
return {qml.SWAP: 1, qml.CNOT: 2, PhaseShift: 1}
@register_resources(_pswap_to_swap_cnot_phaseshift_cnot_resources)
def _pswap_to_swap_cnot_phaseshift_cnot(phi: TensorLike, wires: WiresLike, **__):
qml.SWAP(wires=wires)
qml.CNOT(wires=wires)
PhaseShift(phi, wires=[wires[1]])
qml.CNOT(wires=wires)
add_decomps(PSWAP, _pswap_to_swap_cnot_phaseshift_cnot)
add_decomps("Adjoint(PSWAP)", adjoint_rotation)
[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,)]
[docs]
def generator(self) -> "qml.Projector":
return qml.Projector(np.array([0, 0]), wires=self.wires)
resource_keys = set()
def __init__(self, phi: TensorLike, wires: WiresLike, id: Optional[str] = None):
super().__init__(phi, wires=wires, id=id)
@property
def resource_params(self) -> dict:
return {}
[docs]
def label(
self,
decimals: Optional[int] = None,
base_label: Optional[str] = None,
cache: Optional[dict] = None,
) -> str:
return super().label(decimals=decimals, base_label="Rϕ(00)", cache=cache)
[docs]
@staticmethod
def compute_matrix(phi: TensorLike) -> TensorLike: # 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 (TensorLike): phase shift
Returns:
TensorLike: 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 math.get_interface(phi) == "tensorflow":
phi = math.cast_like(phi, 1j)
exp_part = math.exp(1j * phi)
if math.ndim(phi) > 0:
ones = math.ones_like(exp_part)
zeros = math.zeros_like(exp_part)
matrix = [
[exp_part, zeros, zeros, zeros],
[zeros, ones, zeros, zeros],
[zeros, zeros, ones, zeros],
[zeros, zeros, zeros, ones],
]
return math.stack([stack_last(row) for row in matrix], axis=-2)
return math.diag([exp_part, 1, 1, 1])
[docs]
@staticmethod
def compute_eigvals(phi: TensorLike) -> TensorLike: # 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 (TensorLike): phase shift
Returns:
TensorLike: 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 math.get_interface(phi) == "tensorflow":
phi = math.cast_like(phi, 1j)
exp_part = math.exp(1j * phi)
ones = math.ones_like(exp_part)
return stack_last([exp_part, ones, ones, ones])
[docs]
@staticmethod
def compute_decomposition(phi: TensorLike, wires: WiresLike) -> list[Operator]:
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
[docs]
def pow(self, z: Union[int, float]) -> "CPhaseShift00":
return [CPhaseShift00(self.data[0] * z, wires=self.wires)]
@property
def control_values(self) -> str:
"""str: The control values of the operation"""
return "0"
@property
def control_wires(self) -> Wires:
return self.wires[0:1]
def _cphaseshift00_resources():
return {PauliX: 4, PhaseShift: 3, qml.CNOT: 2}
@register_resources(_cphaseshift00_resources)
def _cphaseshift00(phi: TensorLike, wires: WiresLike, **__):
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])
add_decomps(CPhaseShift00, _cphaseshift00)
add_decomps("Adjoint(CPhaseShift00)", adjoint_rotation)
add_decomps("Pow(CPhaseShift00)", pow_rotation)
[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,)]
[docs]
def generator(self) -> "qml.Projector":
return qml.Projector(np.array([0, 1]), wires=self.wires)
resource_keys = set()
def __init__(self, phi: TensorLike, wires: WiresLike, id: Optional[str] = None):
super().__init__(phi, wires=wires, id=id)
@property
def resource_params(self) -> dict:
return {}
[docs]
def label(
self,
decimals: Optional[int] = None,
base_label: Optional[str] = None,
cache: Optional[dict] = None,
) -> str:
return super().label(decimals=decimals, base_label="Rϕ(01)", cache=cache)
[docs]
@staticmethod
def compute_matrix(phi: TensorLike) -> TensorLike: # 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 (TensorLike): phase shift
Returns:
TensorLike: 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 math.get_interface(phi) == "tensorflow":
phi = math.cast_like(phi, 1j)
exp_part = math.exp(1j * phi)
if math.ndim(phi) > 0:
ones = math.ones_like(exp_part)
zeros = math.zeros_like(exp_part)
matrix = [
[ones, zeros, zeros, zeros],
[zeros, exp_part, zeros, zeros],
[zeros, zeros, ones, zeros],
[zeros, zeros, zeros, ones],
]
return math.stack([stack_last(row) for row in matrix], axis=-2)
return math.diag([1, exp_part, 1, 1])
[docs]
@staticmethod
def compute_eigvals(phi: TensorLike) -> TensorLike: # 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 (TensorLike): phase shift
Returns:
TensorLike: 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 math.get_interface(phi) == "tensorflow":
phi = math.cast_like(phi, 1j)
exp_part = math.exp(1j * phi)
ones = math.ones_like(exp_part)
return stack_last([ones, exp_part, ones, ones])
[docs]
@staticmethod
def compute_decomposition(phi: TensorLike, wires: WiresLike) -> list[Operator]:
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 (Tensorlike): 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
[docs]
def pow(self, z: Union[int, float]) -> "CPhaseShift01":
return [CPhaseShift01(self.data[0] * z, wires=self.wires)]
@property
def control_values(self) -> str:
"""str: The control values of the operation"""
return "0"
@property
def control_wires(self) -> Wires:
return self.wires[0:1]
def _cphaseshift01_resources():
return {PauliX: 2, PhaseShift: 3, qml.CNOT: 2}
@register_resources(_cphaseshift01_resources)
def _cphaseshift01(phi: TensorLike, wires: WiresLike, **__):
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])
add_decomps(CPhaseShift01, _cphaseshift01)
add_decomps("Adjoint(CPhaseShift01)", adjoint_rotation)
add_decomps("Pow(CPhaseShift01)", pow_rotation)
[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,)]
[docs]
def generator(self) -> "qml.Projector":
return qml.Projector(np.array([1, 0]), wires=self.wires)
resource_keys = set()
def __init__(self, phi: TensorLike, wires: WiresLike, id: Optional[str] = None):
super().__init__(phi, wires=wires, id=id)
@property
def resource_params(self) -> dict:
return {}
[docs]
def label(
self,
decimals: Optional[int] = None,
base_label: Optional[str] = None,
cache: Optional[dict] = None,
) -> str:
return super().label(decimals=decimals, base_label="Rϕ(10)", cache=cache)
[docs]
@staticmethod
def compute_matrix(phi: TensorLike) -> TensorLike: # 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 (TensorLike): phase shift
Returns:
TensorLike: 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 math.get_interface(phi) == "tensorflow":
phi = math.cast_like(phi, 1j)
exp_part = math.exp(1j * phi)
if math.ndim(phi) > 0:
ones = math.ones_like(exp_part)
zeros = math.zeros_like(exp_part)
matrix = [
[ones, zeros, zeros, zeros],
[zeros, ones, zeros, zeros],
[zeros, zeros, exp_part, zeros],
[zeros, zeros, zeros, ones],
]
return math.stack([stack_last(row) for row in matrix], axis=-2)
return math.diag([1, 1, exp_part, 1])
[docs]
@staticmethod
def compute_eigvals(phi: TensorLike) -> TensorLike: # 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 (TensorLike): phase shift
Returns:
TensorLike: 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 math.get_interface(phi) == "tensorflow":
phi = math.cast_like(phi, 1j)
exp_part = math.exp(1j * phi)
ones = math.ones_like(exp_part)
return stack_last([ones, ones, exp_part, ones])
[docs]
@staticmethod
def compute_decomposition(phi: TensorLike, wires: WiresLike) -> list[Operator]:
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 (TensorLike): 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
[docs]
def pow(self, z: Union[int, float]):
return [CPhaseShift10(self.data[0] * z, wires=self.wires)]
@property
def control_wires(self) -> Wires:
return self.wires[0:1]
def _cphaseshift10_resources():
return {PauliX: 2, PhaseShift: 3, qml.CNOT: 2}
@register_resources(_cphaseshift10_resources)
def _cphaseshift10(phi: TensorLike, wires: WiresLike, **__):
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])
add_decomps(CPhaseShift10, _cphaseshift10)
add_decomps("Adjoint(CPhaseShift10)", adjoint_rotation)
add_decomps("Pow(CPhaseShift10)", pow_rotation)
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