Source code for pennylane.ops.qubit.parametric_ops

# Copyright 2018-2021 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
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# limitations under the License.
"""
This submodule contains the discrete-variable quantum operations that are the
core parameterized gates.
"""
# pylint:disable=abstract-method,arguments-differ,protected-access
import cmath
import functools
import math
import numpy as np

import pennylane as qml
from pennylane.operation import AnyWires, DiagonalOperation, Operation
from pennylane.ops.qubit.non_parametric_ops import PauliX, PauliY, PauliZ, Hadamard
from pennylane.templates.decorator import template
from pennylane.utils import expand, pauli_eigs
from pennylane.wires import Wires


INV_SQRT2 = 1 / math.sqrt(2)


[docs]class RX(Operation): r"""RX(phi, wires) The single qubit X rotation .. math:: R_x(\phi) = e^{-i\phi\sigma_x/2} = \begin{bmatrix} \cos(\phi/2) & -i\sin(\phi/2) \\ -i\sin(\phi/2) & \cos(\phi/2) \end{bmatrix}. **Details:** * Number of wires: 1 * Number of parameters: 1 * Gradient recipe: :math:`\frac{d}{d\phi}f(R_x(\phi)) = \frac{1}{2}\left[f(R_x(\phi+\pi/2)) - f(R_x(\phi-\pi/2))\right]` where :math:`f` is an expectation value depending on :math:`R_x(\phi)`. Args: phi (float): rotation angle :math:`\phi` wires (Sequence[int] or int): the wire the operation acts on """ num_params = 1 num_wires = 1 par_domain = "R" is_composable_rotation = True basis = "X" grad_method = "A" generator = [PauliX, -1 / 2] has_unitary_generator = True @classmethod def _matrix(cls, *params): theta = params[0] c = math.cos(theta / 2) js = 1j * math.sin(-theta / 2) return np.array([[c, js], [js, c]])
[docs] def adjoint(self): return RX(-self.data[0], wires=self.wires)
def _controlled(self, wire): CRX(*self.parameters, wires=wire + self.wires)
[docs] def single_qubit_rot_angles(self): # RX(\theta) = RZ(-\pi/2) RY(\theta) RZ(\pi/2) return [np.pi / 2, self.data[0], -np.pi / 2]
[docs]class RY(Operation): r"""RY(phi, wires) The single qubit Y rotation .. math:: R_y(\phi) = e^{-i\phi\sigma_y/2} = \begin{bmatrix} \cos(\phi/2) & -\sin(\phi/2) \\ \sin(\phi/2) & \cos(\phi/2) \end{bmatrix}. **Details:** * Number of wires: 1 * Number of parameters: 1 * Gradient recipe: :math:`\frac{d}{d\phi}f(R_y(\phi)) = \frac{1}{2}\left[f(R_y(\phi+\pi/2)) - f(R_y(\phi-\pi/2))\right]` where :math:`f` is an expectation value depending on :math:`R_y(\phi)`. Args: phi (float): rotation angle :math:`\phi` wires (Sequence[int] or int): the wire the operation acts on """ num_params = 1 num_wires = 1 par_domain = "R" is_composable_rotation = True basis = "Y" grad_method = "A" generator = [PauliY, -1 / 2] has_unitary_generator = True @classmethod def _matrix(cls, *params): theta = params[0] c = math.cos(theta / 2) s = math.sin(theta / 2) return np.array([[c, -s], [s, c]])
[docs] def adjoint(self): return RY(-self.data[0], wires=self.wires)
def _controlled(self, wire): CRY(*self.parameters, wires=wire + self.wires)
[docs] def single_qubit_rot_angles(self): # RY(\theta) = RZ(0) RY(\theta) RZ(0) return [0.0, self.data[0], 0.0]
[docs]class RZ(DiagonalOperation): r"""RZ(phi, wires) The single qubit Z rotation .. math:: R_z(\phi) = e^{-i\phi\sigma_z/2} = \begin{bmatrix} e^{-i\phi/2} & 0 \\ 0 & e^{i\phi/2} \end{bmatrix}. **Details:** * Number of wires: 1 * Number of parameters: 1 * Gradient recipe: :math:`\frac{d}{d\phi}f(R_z(\phi)) = \frac{1}{2}\left[f(R_z(\phi+\pi/2)) - f(R_z(\phi-\pi/2))\right]` where :math:`f` is an expectation value depending on :math:`R_z(\phi)`. Args: phi (float): rotation angle :math:`\phi` wires (Sequence[int] or int): the wire the operation acts on """ num_params = 1 num_wires = 1 par_domain = "R" is_composable_rotation = True basis = "Z" grad_method = "A" generator = [PauliZ, -1 / 2] has_unitary_generator = True @classmethod def _matrix(cls, *params): theta = params[0] p = cmath.exp(-0.5j * theta) return np.array([[p, 0], [0, p.conjugate()]]) @classmethod def _eigvals(cls, *params): theta = params[0] p = cmath.exp(-0.5j * theta) return np.array([p, p.conjugate()])
[docs] def adjoint(self): return RZ(-self.data[0], wires=self.wires)
def _controlled(self, wire): CRZ(*self.parameters, wires=wire + self.wires)
[docs] def single_qubit_rot_angles(self): # RZ(\theta) = RZ(\theta) RY(0) RZ(0) return [self.data[0], 0.0, 0.0]
[docs]class PhaseShift(DiagonalOperation): r"""PhaseShift(phi, wires) Arbitrary single qubit local phase shift .. math:: R_\phi(\phi) = e^{i\phi/2}R_z(\phi) = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\phi} \end{bmatrix}. **Details:** * Number of wires: 1 * Number of parameters: 1 * Gradient recipe: :math:`\frac{d}{d\phi}f(R_\phi(\phi)) = \frac{1}{2}\left[f(R_\phi(\phi+\pi/2)) - f(R_\phi(\phi-\pi/2))\right]` where :math:`f` is an expectation value depending on :math:`R_{\phi}(\phi)`. Args: phi (float): rotation angle :math:`\phi` wires (Sequence[int] or int): the wire the operation acts on """ num_params = 1 num_wires = 1 par_domain = "R" is_composable_rotation = True basis = "Z" grad_method = "A" generator = [np.array([[0, 0], [0, 1]]), 1] has_unitary_generator = False @classmethod def _matrix(cls, *params): phi = params[0] return np.array([[1, 0], [0, cmath.exp(1j * phi)]]) @classmethod def _eigvals(cls, *params): phi = params[0] return np.array([1, cmath.exp(1j * phi)])
[docs] @staticmethod def decomposition(phi, wires): decomp_ops = [RZ(phi, wires=wires)] return decomp_ops
[docs] def adjoint(self): return PhaseShift(-self.data[0], wires=self.wires)
def _controlled(self, wire): ControlledPhaseShift(*self.parameters, wires=wire + self.wires)
[docs] def single_qubit_rot_angles(self): # PhaseShift(\theta) = RZ(\theta) RY(0) RZ(0) return [self.data[0], 0.0, 0.0]
[docs]class ControlledPhaseShift(DiagonalOperation): r"""ControlledPhaseShift(phi, wires) A qubit controlled phase shift. .. math:: CR_\phi(\phi) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & e^{i\phi} \end{bmatrix}. .. note:: The first wire provided corresponds to the **control qubit**. **Details:** * Number of wires: 2 * Number of parameters: 1 * Gradient recipe: :math:`\frac{d}{d\phi}f(CR_\phi(\phi)) = \frac{1}{2}\left[f(CR_\phi(\phi+\pi/2)) - f(CR_\phi(\phi-\pi/2))\right]` where :math:`f` is an expectation value depending on :math:`CR_{\phi}(\phi)`. Args: phi (float): rotation angle :math:`\phi` wires (Sequence[int]): the wire the operation acts on """ num_params = 1 num_wires = 2 par_domain = "R" is_composable_rotation = True basis = "Z" grad_method = "A" generator = [np.array([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 1]]), 1] has_unitary_generator = False @classmethod def _matrix(cls, *params): phi = params[0] return np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, cmath.exp(1j * phi)]]) @classmethod def _eigvals(cls, *params): phi = params[0] return np.array([1, 1, 1, cmath.exp(1j * phi)])
[docs] @staticmethod def decomposition(phi, wires): decomp_ops = [ qml.PhaseShift(phi / 2, wires=wires[0]), qml.CNOT(wires=wires), qml.PhaseShift(-phi / 2, wires=wires[1]), qml.CNOT(wires=wires), qml.PhaseShift(phi / 2, wires=wires[1]), ] return decomp_ops
[docs] def adjoint(self): return ControlledPhaseShift(-self.data[0], wires=self.wires)
@property def control_wires(self): return Wires(self.wires[0])
CPhase = ControlledPhaseShift
[docs]class Rot(Operation): r"""Rot(phi, theta, omega, wires) Arbitrary single qubit rotation .. math:: R(\phi,\theta,\omega) = RZ(\omega)RY(\theta)RZ(\phi)= \begin{bmatrix} e^{-i(\phi+\omega)/2}\cos(\theta/2) & -e^{i(\phi-\omega)/2}\sin(\theta/2) \\ e^{-i(\phi-\omega)/2}\sin(\theta/2) & e^{i(\phi+\omega)/2}\cos(\theta/2) \end{bmatrix}. **Details:** * Number of wires: 1 * Number of parameters: 3 * Gradient recipe: :math:`\frac{d}{d\phi}f(R(\phi, \theta, \omega)) = \frac{1}{2}\left[f(R(\phi+\pi/2, \theta, \omega)) - f(R(\phi-\pi/2, \theta, \omega))\right]` where :math:`f` is an expectation value depending on :math:`R(\phi, \theta, \omega)`. This gradient recipe applies for each angle argument :math:`\{\phi, \theta, \omega\}`. .. note:: If the ``Rot`` gate is not supported on the targeted device, PennyLane will attempt to decompose the gate into :class:`~.RZ` and :class:`~.RY` gates. Args: phi (float): rotation angle :math:`\phi` theta (float): rotation angle :math:`\theta` omega (float): rotation angle :math:`\omega` wires (Sequence[int] or int): the wire the operation acts on """ num_params = 3 num_wires = 1 par_domain = "R" is_composable_rotation = True grad_method = "A" @classmethod def _matrix(cls, *params): phi, theta, omega = params c = math.cos(theta / 2) s = math.sin(theta / 2) return np.array( [ [ cmath.exp(-0.5j * (phi + omega)) * c, -cmath.exp(0.5j * (phi - omega)) * s, ], [ cmath.exp(-0.5j * (phi - omega)) * s, cmath.exp(0.5j * (phi + omega)) * c, ], ] )
[docs] @staticmethod def decomposition(phi, theta, omega, wires): decomp_ops = [ RZ(phi, wires=wires), RY(theta, wires=wires), RZ(omega, wires=wires), ] return decomp_ops
[docs] def adjoint(self): phi, theta, omega = self.parameters return Rot(-omega, -theta, -phi, wires=self.wires)
def _controlled(self, wire): CRot(*self.parameters, wires=wire + self.wires)
[docs] def single_qubit_rot_angles(self): return self.data
[docs]class MultiRZ(DiagonalOperation): r"""MultiRZ(theta, wires) Arbitrary multi Z rotation. .. math:: MultiRZ(\theta) = \exp(-i \frac{\theta}{2} Z^{\otimes n}) **Details:** * Number of wires: Any * Number of parameters: 1 * 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 (float): rotation angle :math:`\theta` wires (Sequence[int] or int): the wires the operation acts on """ num_params = 1 num_wires = AnyWires par_domain = "R" grad_method = "A" @classmethod def _matrix(cls, theta, n): """Matrix representation of a MultiRZ gate. Args: theta (float): Rotation angle. n (int): Number of wires the rotation acts on. This has to be given explicitly in the static method as the wires object is not available. Returns: array[complex]: The matrix representation """ multi_Z_rot_eigs = MultiRZ._eigvals(theta, n) multi_Z_rot_matrix = np.diag(multi_Z_rot_eigs) return multi_Z_rot_matrix _generator = None @property def generator(self): if self._generator is None: self._generator = [np.diag(pauli_eigs(len(self.wires))), -1 / 2] return self._generator has_unitary_generator = True @property def matrix(self): # Redefine the property here to pass additionally the number of wires to the ``_matrix`` method if self.inverse: # The matrix is diagonal, so there is no need to transpose return self._matrix(*self.parameters, len(self.wires)).conj() return self._matrix(*self.parameters, len(self.wires)) @classmethod def _eigvals(cls, theta, n): return np.exp(-1j * theta / 2 * pauli_eigs(n)) @property def eigvals(self): # Redefine the property here to pass additionally the number of wires to the ``_eigvals`` method if self.inverse: return self._eigvals(*self.parameters, len(self.wires)).conj() return self._eigvals(*self.parameters, len(self.wires))
[docs] @staticmethod @template def decomposition(theta, wires): for i in range(len(wires) - 1, 0, -1): qml.CNOT(wires=[wires[i], wires[i - 1]]) RZ(theta, wires=wires[0]) for i in range(len(wires) - 1): qml.CNOT(wires=[wires[i + 1], wires[i]])
[docs] def adjoint(self): return MultiRZ(-self.parameters[0], wires=self.wires)
[docs]class PauliRot(Operation): r"""PauliRot(theta, pauli_word, wires) Arbitrary Pauli word rotation. .. math:: RP(\theta, P) = \exp(-i \frac{\theta}{2} P) **Details:** * Number of wires: Any * Number of parameters: 2 (1 differentiable parameter) * 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 """ num_params = 2 num_wires = AnyWires do_check_domain = False par_domain = "R" grad_method = "A" _ALLOWED_CHARACTERS = "IXYZ" _PAULI_CONJUGATION_MATRICES = { "X": Hadamard._matrix(), "Y": RX._matrix(np.pi / 2), "Z": np.array([[1, 0], [0, 1]]), } def __init__(self, *params, wires=None, do_queue=True): super().__init__(*params, wires=wires, do_queue=do_queue) pauli_word = params[1] if not PauliRot._check_pauli_word(pauli_word): raise ValueError( 'The given Pauli word "{}" contains characters that are not allowed.' " Allowed characters are I, X, Y and Z".format(pauli_word) ) num_wires = 1 if isinstance(wires, int) else len(wires) if not len(pauli_word) == num_wires: raise ValueError( "The given Pauli word has length {}, length {} was expected for wires {}".format( len(pauli_word), num_wires, wires ) ) @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 pauli_word) @classmethod def _matrix(cls, *params): theta = params[0] pauli_word = params[1] if not PauliRot._check_pauli_word(pauli_word): raise ValueError( 'The given Pauli word "{}" contains characters that are not allowed.' " Allowed characters are I, X, Y and Z".format(pauli_word) ) # Simplest case is if the Pauli is the identity matrix if pauli_word == "I" * len(pauli_word): return np.exp(-1j * theta / 2) * np.eye(2 ** 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._matrix(theta, len(non_identity_gates)) # now we conjugate with Hadamard and RX to create the Pauli string conjugation_matrix = functools.reduce( np.kron, [PauliRot._PAULI_CONJUGATION_MATRICES[gate] for gate in non_identity_gates], ) return expand( conjugation_matrix.T.conj() @ multi_Z_rot_matrix @ conjugation_matrix, non_identity_wires, list(range(len(pauli_word))), ) _generator = None @property def generator(self): if self._generator is None: pauli_word = self.parameters[1] # Simplest case is if the Pauli is the identity matrix if pauli_word == "I" * len(pauli_word): self._generator = [np.eye(2 ** len(pauli_word)), -1 / 2] return self._generator # 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"] ) # get MultiRZ's generator multi_Z_rot_generator = np.diag(pauli_eigs(len(non_identity_gates))) # now we conjugate with Hadamard and RX to create the Pauli string conjugation_matrix = functools.reduce( np.kron, [PauliRot._PAULI_CONJUGATION_MATRICES[gate] for gate in non_identity_gates], ) self._generator = [ expand( conjugation_matrix.T.conj() @ multi_Z_rot_generator @ conjugation_matrix, non_identity_wires, list(range(len(pauli_word))), ), -1 / 2, ] return self._generator has_unitary_generator = True @classmethod def _eigvals(cls, theta, pauli_word): # Identity must be treated specially because its eigenvalues are all the same if pauli_word == "I" * len(pauli_word): return np.exp(-1j * theta / 2) * np.ones(2 ** len(pauli_word)) return MultiRZ._eigvals(theta, len(pauli_word))
[docs] @staticmethod @template def decomposition(theta, pauli_word, wires): # Catch cases when the wire is passed as a single int. if isinstance(wires, int): wires = [wires] # Check for identity and do nothing if pauli_word == "I" * len(wires): 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": Hadamard(wires=[wire]) elif gate == "Y": RX(np.pi / 2, wires=[wire]) MultiRZ(theta, wires=list(active_wires)) for wire, gate in zip(active_wires, active_gates): if gate == "X": Hadamard(wires=[wire]) elif gate == "Y": RX(-np.pi / 2, wires=[wire])
[docs] def adjoint(self): return PauliRot(-self.parameters[0], self.parameters[1], wires=self.wires)
# Four term gradient recipe for controlled rotations c1 = INV_SQRT2 * (np.sqrt(2) + 1) / 4 c2 = INV_SQRT2 * (np.sqrt(2) - 1) / 4 a = np.pi / 2 b = 3 * np.pi / 2 four_term_grad_recipe = ([[c1, 1, a], [-c1, 1, -a], [-c2, 1, b], [c2, 1, -b]],)
[docs]class CRX(Operation): r"""CRX(phi, wires) The controlled-RX operator .. math:: \begin{align} CR_x(\phi) &= \begin{bmatrix} & 1 & 0 & 0 & 0 \\ & 0 & 1 & 0 & 0\\ & 0 & 0 & \cos(\phi/2) & -i\sin(\phi/2)\\ & 0 & 0 & -i\sin(\phi/2) & \cos(\phi/2) \end{bmatrix}. \end{align} **Details:** * Number of wires: 2 * Number of parameters: 1 * Gradient recipe: The controlled-RX operator satisfies a four-term parameter-shift rule (see Appendix F, https://arxiv.org/abs/2104.05695): .. math:: \frac{d}{d\phi}f(CR_x(\phi)) = c_+ \left[f(CR_x(\phi+a)) - f(CR_x(\phi-a))\right] - c_- \left[f(CR_x(\phi+b)) - f(CR_x(\phi-b))\right] where :math:`f` is an expectation value depending on :math:`CR_x(\phi)`, and - :math:`a = \pi/2` - :math:`b = 3\pi/2` - :math:`c_{\pm} = (\sqrt{2} \pm 1)/{4\sqrt{2}}` Args: phi (float): rotation angle :math:`\phi` wires (Sequence[int]): the wire the operation acts on """ num_params = 1 num_wires = 2 par_domain = "R" is_composable_rotation = True basis = "X" grad_method = "A" grad_recipe = four_term_grad_recipe generator = [ np.array([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]]), -1 / 2, ] has_unitary_generator = False @classmethod def _matrix(cls, *params): theta = params[0] c = math.cos(theta / 2) js = 1j * math.sin(-theta / 2) return np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, c, js], [0, 0, js, c]])
[docs] @staticmethod def decomposition(theta, wires): decomp_ops = [ RZ(np.pi / 2, wires=wires[1]), RY(theta / 2, wires=wires[1]), qml.CNOT(wires=wires), RY(-theta / 2, wires=wires[1]), qml.CNOT(wires=wires), RZ(-np.pi / 2, wires=wires[1]), ] return decomp_ops
[docs] def adjoint(self): return CRX(-self.data[0], wires=self.wires)
@property def control_wires(self): return Wires(self.wires[0])
[docs]class CRY(Operation): r"""CRY(phi, wires) The controlled-RY operator .. math:: \begin{align} CR_y(\phi) &= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & \cos(\phi/2) & -\sin(\phi/2)\\ 0 & 0 & \sin(\phi/2) & \cos(\phi/2) \end{bmatrix}. \end{align} **Details:** * Number of wires: 2 * Number of parameters: 1 * Gradient recipe: The controlled-RY operator satisfies a four-term parameter-shift rule (see Appendix F, https://arxiv.org/abs/2104.05695): .. math:: \frac{d}{d\phi}f(CR_y(\phi)) = c_+ \left[f(CR_y(\phi+a)) - f(CR_y(\phi-a))\right] - c_- \left[f(CR_y(\phi+b)) - f(CR_y(\phi-b))\right] where :math:`f` is an expectation value depending on :math:`CR_y(\phi)`, and - :math:`a = \pi/2` - :math:`b = 3\pi/2` - :math:`c_{\pm} = (\sqrt{2} \pm 1)/{4\sqrt{2}}` Args: phi (float): rotation angle :math:`\phi` wires (Sequence[int]): the wire the operation acts on """ num_params = 1 num_wires = 2 par_domain = "R" is_composable_rotation = True basis = "Y" grad_method = "A" grad_recipe = four_term_grad_recipe generator = [ np.array([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -1j], [0, 0, 1j, 0]]), -1 / 2, ] has_unitary_generator = False @classmethod def _matrix(cls, *params): theta = params[0] c = math.cos(theta / 2) s = math.sin(theta / 2) return np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, c, -s], [0, 0, s, c]])
[docs] @staticmethod def decomposition(theta, wires): decomp_ops = [ RY(theta / 2, wires=wires[1]), qml.CNOT(wires=wires), RY(-theta / 2, wires=wires[1]), qml.CNOT(wires=wires), ] return decomp_ops
[docs] def adjoint(self): return CRY(-self.data[0], wires=self.wires)
@property def control_wires(self): return Wires(self.wires[0])
[docs]class CRZ(DiagonalOperation): r"""CRZ(phi, wires) The controlled-RZ operator .. math:: \begin{align} CR_z(\phi) &= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & e^{-i\phi/2} & 0\\ 0 & 0 & 0 & e^{i\phi/2} \end{bmatrix}. \end{align} .. note:: The subscripts of the operations in the formula refer to the wires they act on, e.g. 1 corresponds to the first element in ``wires`` that is the **control qubit**. **Details:** * Number of wires: 2 * Number of parameters: 1 * Gradient recipe: The controlled-RZ operator satisfies a four-term parameter-shift rule (see Appendix F, https://arxiv.org/abs/2104.05695): .. math:: \frac{d}{d\phi}f(CR_z(\phi)) = c_+ \left[f(CR_z(\phi+a)) - f(CR_z(\phi-a))\right] - c_- \left[f(CR_z(\phi+b)) - f(CR_z(\phi-b))\right] where :math:`f` is an expectation value depending on :math:`CR_z(\phi)`, and - :math:`a = \pi/2` - :math:`b = 3\pi/2` - :math:`c_{\pm} = (\sqrt{2} \pm 1)/{4\sqrt{2}}` Args: phi (float): rotation angle :math:`\phi` wires (Sequence[int]): the wire the operation acts on """ num_params = 1 num_wires = 2 par_domain = "R" is_composable_rotation = True basis = "Z" grad_method = "A" grad_recipe = four_term_grad_recipe generator = [ np.array([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, -1]]), -1 / 2, ] has_unitary_generator = False @classmethod def _matrix(cls, *params): theta = params[0] return np.array( [ [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, cmath.exp(-0.5j * theta), 0], [0, 0, 0, cmath.exp(0.5j * theta)], ] ) @classmethod def _eigvals(cls, *params): theta = params[0] return np.array( [ 1, 1, cmath.exp(-0.5j * theta), cmath.exp(0.5j * theta), ] )
[docs] @staticmethod def decomposition(lam, wires): decomp_ops = [ PhaseShift(lam / 2, wires=wires[1]), qml.CNOT(wires=wires), PhaseShift(-lam / 2, wires=wires[1]), qml.CNOT(wires=wires), ] return decomp_ops
[docs] def adjoint(self): return CRZ(-self.data[0], wires=self.wires)
@property def control_wires(self): return Wires(self.wires[0])
[docs]class CRot(Operation): r"""CRot(phi, theta, omega, wires) The controlled-Rot operator .. math:: CR(\phi, \theta, \omega) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & e^{-i(\phi+\omega)/2}\cos(\theta/2) & -e^{i(\phi-\omega)/2}\sin(\theta/2)\\ 0 & 0 & e^{-i(\phi-\omega)/2}\sin(\theta/2) & e^{i(\phi+\omega)/2}\cos(\theta/2) \end{bmatrix}. .. note:: The first wire provided corresponds to the **control qubit**. **Details:** * Number of wires: 2 * Number of parameters: 3 * Gradient recipe: The controlled-Rot operator satisfies a four-term parameter-shift rule (see Appendix F, https://arxiv.org/abs/2104.05695): .. math:: \frac{d}{d\mathbf{x}_i}f(CR(\mathbf{x}_i)) = c_+ \left[f(CR(\mathbf{x}_i+a)) - f(CR(\mathbf{x}_i-a))\right] - c_- \left[f(CR(\mathbf{x}_i+b)) - f(CR(\mathbf{x}_i-b))\right] where :math:`f` is an expectation value depending on :math:`CR(\mathbf{x}_i)`, and - :math:`\mathbf{x} = (\phi, \theta, \omega)` and `i` is an index to :math:`\mathbf{x}` - :math:`a = \pi/2` - :math:`b = 3\pi/2` - :math:`c_{\pm} = (\sqrt{2} \pm 1)/{4\sqrt{2}}` Args: phi (float): rotation angle :math:`\phi` theta (float): rotation angle :math:`\theta` omega (float): rotation angle :math:`\omega` wires (Sequence[int]): the wire the operation acts on """ num_params = 3 num_wires = 2 par_domain = "R" grad_method = "A" grad_recipe = four_term_grad_recipe * 3 @classmethod def _matrix(cls, *params): phi, theta, omega = params c = math.cos(theta / 2) s = math.sin(theta / 2) return np.array( [ [1, 0, 0, 0], [0, 1, 0, 0], [ 0, 0, cmath.exp(-0.5j * (phi + omega)) * c, -cmath.exp(0.5j * (phi - omega)) * s, ], [ 0, 0, cmath.exp(-0.5j * (phi - omega)) * s, cmath.exp(0.5j * (phi + omega)) * c, ], ] )
[docs] @staticmethod def decomposition(phi, theta, omega, wires): decomp_ops = [ RZ((phi - omega) / 2, wires=wires[1]), qml.CNOT(wires=wires), RZ(-(phi + omega) / 2, wires=wires[1]), RY(-theta / 2, wires=wires[1]), qml.CNOT(wires=wires), RY(theta / 2, wires=wires[1]), RZ(omega, wires=wires[1]), ] return decomp_ops
[docs] def adjoint(self): phi, theta, omega = self.parameters return CRot(-omega, -theta, -phi, wires=self.wires)
[docs]class U1(Operation): r"""U1(phi) U1 gate. .. math:: U_1(\phi) = e^{i\phi/2}R_z(\phi) = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\phi} \end{bmatrix}. .. note:: The ``U1`` gate is an alias for the phase shift operation :class:`~.PhaseShift`. **Details:** * Number of wires: 1 * Number of parameters: 1 * Gradient recipe: :math:`\frac{d}{d\phi}f(U_1(\phi)) = \frac{1}{2}\left[f(U_1(\phi+\pi/2)) - f(U_1(\phi-\pi/2))\right]` where :math:`f` is an expectation value depending on :math:`U_1(\phi)`. Args: phi (float): rotation angle :math:`\phi` wires (Sequence[int] or int): the wire the operation acts on """ num_params = 1 num_wires = 1 par_domain = "R" grad_method = "A" generator = [np.array([[0, 0], [0, 1]]), 1] has_unitary_generator = False @classmethod def _matrix(cls, *params): phi = params[0] return np.array([[1, 0], [0, cmath.exp(1j * phi)]])
[docs] @staticmethod def decomposition(phi, wires): return [PhaseShift(phi, wires=wires)]
[docs] def adjoint(self): return U1(-self.data[0], wires=self.wires)
[docs]class U2(Operation): r"""U2(phi, lambda, wires) U2 gate. .. math:: U_2(\phi, \lambda) = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & -\exp(i \lambda) \\ \exp(i \phi) & \exp(i (\phi + \lambda)) \end{bmatrix} The :math:`U_2` gate is related to the single-qubit rotation :math:`R` (:class:`Rot`) and the :math:`R_\phi` (:class:`PhaseShift`) gates via the following relation: .. math:: U_2(\phi, \lambda) = R_\phi(\phi+\lambda) R(\lambda,\pi/2,-\lambda) .. note:: If the ``U2`` gate is not supported on the targeted device, PennyLane will attempt to decompose the gate into :class:`~.Rot` and :class:`~.PhaseShift` gates. **Details:** * Number of wires: 1 * Number of parameters: 2 * Gradient recipe: :math:`\frac{d}{d\phi}f(U_2(\phi, \lambda)) = \frac{1}{2}\left[f(U_2(\phi+\pi/2, \lambda)) - f(U_2(\phi-\pi/2, \lambda))\right]` where :math:`f` is an expectation value depending on :math:`U_2(\phi, \lambda)`. This gradient recipe applies for each angle argument :math:`\{\phi, \lambda\}`. Args: phi (float): azimuthal angle :math:`\phi` lambda (float): quantum phase :math:`\lambda` wires (Sequence[int] or int): the subsystem the gate acts on """ num_params = 2 num_wires = 1 par_domain = "R" grad_method = "A" @classmethod def _matrix(cls, *params): phi, lam = params return INV_SQRT2 * np.array( [ [1, -cmath.exp(1j * lam)], [cmath.exp(1j * phi), cmath.exp(1j * (phi + lam))], ] )
[docs] @staticmethod def decomposition(phi, lam, wires): decomp_ops = [ Rot(lam, np.pi / 2, -lam, wires=wires), PhaseShift(lam, wires=wires), PhaseShift(phi, wires=wires), ] return decomp_ops
[docs] def adjoint(self): phi, lam = self.parameters new_lam = (np.pi - phi) % (2 * np.pi) new_phi = (np.pi - lam) % (2 * np.pi) return U2(new_phi, new_lam, wires=self.wires)
[docs]class U3(Operation): r"""U3(theta, phi, lambda, wires) Arbitrary single qubit unitary. .. math:: U_3(\theta, \phi, \lambda) = \begin{bmatrix} \cos(\theta/2) & -\exp(i \lambda)\sin(\theta/2) \\ \exp(i \phi)\sin(\theta/2) & \exp(i (\phi + \lambda))\cos(\theta/2) \end{bmatrix} The :math:`U_3` gate is related to the single-qubit rotation :math:`R` (:class:`Rot`) and the :math:`R_\phi` (:class:`PhaseShift`) gates via the following relation: .. math:: U_3(\theta, \phi, \lambda) = R_\phi(\phi+\lambda) R(\lambda,\theta,-\lambda) .. note:: If the ``U3`` gate is not supported on the targeted device, PennyLane will attempt to decompose the gate into :class:`~.PhaseShift` and :class:`~.Rot` gates. **Details:** * Number of wires: 1 * Number of parameters: 3 * Gradient recipe: :math:`\frac{d}{d\phi}f(U_3(\theta, \phi, \lambda)) = \frac{1}{2}\left[f(U_3(\theta+\pi/2, \phi, \lambda)) - f(U_3(\theta-\pi/2, \phi, \lambda))\right]` where :math:`f` is an expectation value depending on :math:`U_3(\theta, \phi, \lambda)`. This gradient recipe applies for each angle argument :math:`\{\theta, \phi, \lambda\}`. Args: theta (float): polar angle :math:`\theta` phi (float): azimuthal angle :math:`\phi` lambda (float): quantum phase :math:`\lambda` wires (Sequence[int] or int): the subsystem the gate acts on """ num_params = 3 num_wires = 1 par_domain = "R" grad_method = "A" @classmethod def _matrix(cls, *params): theta, phi, lam = params c = math.cos(theta / 2) s = math.sin(theta / 2) return np.array( [ [c, -s * cmath.exp(1j * lam)], [s * cmath.exp(1j * phi), c * cmath.exp(1j * (phi + lam))], ] )
[docs] @staticmethod def decomposition(theta, phi, lam, wires): decomp_ops = [ Rot(lam, theta, -lam, wires=wires), PhaseShift(lam, wires=wires), PhaseShift(phi, wires=wires), ] return decomp_ops
[docs] def adjoint(self): theta, phi, lam = self.parameters new_lam = (np.pi - phi) % (2 * np.pi) new_phi = (np.pi - lam) % (2 * np.pi) return U3(theta, new_phi, new_lam, wires=self.wires)
[docs]class IsingXX(Operation): r"""IsingXX(phi, wires) Ising XX coupling gate .. math:: XX(\phi) = \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}. **Details:** * Number of wires: 2 * Number of parameters: 1 * 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 """ num_params = 1 num_wires = 2 par_domain = "R" grad_method = "A" generator = [ np.array([[0, 0, 0, 1], [0, 0, 1, 0], [0, 1, 0, 0], [1, 0, 0, 0]]), -1 / 2, ] has_unitary_generator = True @classmethod def _matrix(cls, *params): phi = params[0] c = math.cos(phi / 2) s = math.sin(phi / 2) return np.array( [ [c, 0, 0, -1j * s], [0, c, -1j * s, 0], [0, -1j * s, c, 0], [-1j * s, 0, 0, c], ] )
[docs] @staticmethod def decomposition(phi, wires): decomp_ops = [ qml.CNOT(wires=wires), RX(phi, wires=[wires[0]]), qml.CNOT(wires=wires), ] return decomp_ops
[docs] def adjoint(self): (phi,) = self.parameters return IsingXX(-phi, wires=self.wires)
[docs]class IsingYY(Operation): r"""IsingYY(phi, wires) Ising YY coupling gate .. math:: \mathtt{YY}(\phi) = \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}. **Details:** * Number of wires: 2 * Number of parameters: 1 * 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 """ num_params = 1 num_wires = 2 par_domain = "R" grad_method = "A" generator = [ np.array([[0, 0, 0, -1], [0, 0, 1, 0], [0, 1, 0, 0], [-1, 0, 0, 0]]), -1 / 2, ] has_unitary_generator = True
[docs] @staticmethod def decomposition(phi, wires): return [ qml.CY(wires=wires), qml.RY(phi, wires=[wires[0]]), qml.CY(wires=wires), ]
@classmethod def _matrix(cls, *params): phi = params[0] cos = np.cos(phi / 2) isin = 1.0j * np.sin(phi / 2) return np.array( [ [cos, 0.0, 0.0, isin], [0.0, cos, -isin, 0.0], [0.0, -isin, cos, 0.0], [isin, 0.0, 0.0, cos], ], dtype=complex, )
[docs] def adjoint(self): (phi,) = self.parameters return IsingYY(-phi, wires=self.wires)
[docs]class IsingZZ(Operation): r""" IsingZZ(phi, wires) Ising ZZ coupling gate .. math:: ZZ(\phi) = \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}. **Details:** * Number of wires: 2 * Number of parameters: 1 * 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 """ num_params = 1 num_wires = 2 par_domain = "R" grad_method = "A" generator = [ np.array([[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 1]]), -1 / 2, ] has_unitary_generator = True
[docs] @staticmethod def decomposition(phi, wires): return [ qml.CNOT(wires=wires), qml.RZ(phi, wires=[wires[1]]), qml.CNOT(wires=wires), ]
@classmethod def _matrix(cls, *params): phi = params[0] pos_phase = np.exp(1.0j * phi / 2) neg_phase = np.exp(-1.0j * phi / 2) return np.diag([neg_phase, pos_phase, pos_phase, neg_phase])
[docs] def adjoint(self): (phi,) = self.parameters return IsingZZ(-phi, wires=self.wires)