# This code is part of Qiskit. # # (C) Copyright IBM 2021. # # This code is licensed under the Apache License, Version 2.0. You may # obtain a copy of this license in the LICENSE.txt file in the root directory # of this source tree or at http://www.apache.org/licenses/LICENSE-2.0. # # Any modifications or derivative works of this code must retain this # copyright notice, and modified files need to carry a notice indicating # that they have been altered from the originals. """Two-qubit XX+YY gate.""" from __future__ import annotations import math from cmath import exp from typing import Optional import numpy from qiskit.circuit.gate import Gate from qiskit.circuit.parameterexpression import ParameterValueType, ParameterExpression from qiskit._accelerate.circuit import StandardGate class XXPlusYYGate(Gate): r"""XX+YY interaction gate. A 2-qubit parameterized XX+YY interaction, also known as an XY gate. Its action is to induce a coherent rotation by some angle between :math:`|01\rangle` and :math:`|10\rangle`. Circuit symbol: .. code-block:: text ┌───────────────┐ q_0: ┤0 ├ │ (XX+YY)(θ,β) │ q_1: ┤1 ├ └───────────────┘ Matrix representation: .. math:: \newcommand{\rotationangle}{\frac{\theta}{2}} R_{XX+YY}(\theta, \beta)\ q_0, q_1 = RZ_0(-\beta) \cdot \exp\left(-i \frac{\theta}{2} \frac{XX+YY}{2}\right) \cdot RZ_0(\beta) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\left(\rotationangle\right) & -i\sin\left(\rotationangle\right)e^{-i\beta} & 0 \\ 0 & -i\sin\left(\rotationangle\right)e^{i\beta} & \cos\left(\rotationangle\right) & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} .. note:: In Qiskit's convention, higher qubit indices are more significant (little endian convention). In the above example we apply the gate on (q_0, q_1) which results in adding the (optional) phase defined by :math:`\beta` on q_0. Instead, if we apply it on (q_1, q_0), the phase is added on q_1. If :math:`\beta` is set to its default value of :math:`0`, the gate is equivalent in big and little endian. .. code-block:: text ┌───────────────┐ q_0: ┤1 ├ │ (XX+YY)(θ,β) │ q_1: ┤0 ├ └───────────────┘ .. math:: \newcommand{\rotationangle}{\frac{\theta}{2}} R_{XX+YY}(\theta, \beta)\ q_0, q_1 = RZ_1(-\beta) \cdot \exp\left(-i \frac{\theta}{2} \frac{XX+YY}{2}\right) \cdot RZ_1(\beta) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\left(\rotationangle\right) & -i\sin\left(\rotationangle\right)e^{i\beta} & 0 \\ 0 & -i\sin\left(\rotationangle\right)e^{-i\beta} & \cos\left(\rotationangle\right) & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} """ _standard_gate = StandardGate.XXPlusYY def __init__( self, theta: ParameterValueType, beta: ParameterValueType = 0, label: Optional[str] = "(XX+YY)", ): """ Args: theta: The rotation angle. beta: The phase angle. label: The label of the gate. """ super().__init__("xx_plus_yy", 2, [theta, beta], label=label) def _define(self): """Default definition""" # pylint: disable=cyclic-import from qiskit.circuit import QuantumCircuit # ┌───────┐┌───┐ ┌───┐┌──────────┐┌───┐┌─────┐┌────────┐ # q_0: ┤ Rz(β) ├┤ S ├──────┤ X ├┤ Ry(-θ/2) ├┤ X ├┤ Sdg ├┤ Rz(-β) ├───── # └┬─────┬┘├───┴┐┌───┐└─┬─┘├──────────┤└─┬─┘├─────┤└┬──────┬┘┌───┐ # q_1: ─┤ Sdg ├─┤ √X ├┤ S ├──■──┤ Ry(-θ/2) ├──■──┤ Sdg ├─┤ √Xdg ├─┤ S ├ # └─────┘ └────┘└───┘ └──────────┘ └─────┘ └──────┘ └───┘ self.definition = QuantumCircuit._from_circuit_data( StandardGate.XXPlusYY._get_definition(self.params), legacy_qubits=True, name=self.name ) def control( self, num_ctrl_qubits: int = 1, label: str | None = None, ctrl_state: str | int | None = None, annotated: bool | None = None, ): """Return a (multi-)controlled-(XX+YY) gate. Args: num_ctrl_qubits: number of control qubits. label: An optional label for the gate [Default: ``None``] ctrl_state: control state expressed as integer, string (e.g.``'110'``), or ``None``. If ``None``, use all 1s. annotated: indicates whether the controlled gate should be implemented as an annotated gate. If ``None``, this is set to ``True`` if the gate contains free parameters, in which case it cannot yet be synthesized. Returns: ControlledGate: controlled version of this gate. """ if annotated is None: annotated = any(isinstance(p, ParameterExpression) for p in self.params) gate = super().control( num_ctrl_qubits=num_ctrl_qubits, label=label, ctrl_state=ctrl_state, annotated=annotated, ) return gate def inverse(self, annotated: bool = False): """Return inverse XX+YY gate (i.e. with the negative rotation angle and same phase angle). Args: annotated: when set to ``True``, this is typically used to return an :class:`.AnnotatedOperation` with an inverse modifier set instead of a concrete :class:`.Gate`. However, for this class this argument is ignored as the inverse of this gate is always a :class:`.XXPlusYYGate` with inverse parameter values. Returns: XXPlusYYGate: inverse gate. """ return XXPlusYYGate(-self.params[0], self.params[1]) def __array__(self, dtype=None, copy=None): """Return a numpy.array for the XX+YY gate.""" if copy is False: raise ValueError("unable to avoid copy while creating an array as requested") half_theta = float(self.params[0]) / 2 beta = float(self.params[1]) cos = math.cos(half_theta) sin = math.sin(half_theta) return numpy.array( [ [1, 0, 0, 0], [0, cos, -1j * sin * exp(-1j * beta), 0], [0, -1j * sin * exp(1j * beta), cos, 0], [0, 0, 0, 1], ], dtype=dtype, ) def power(self, exponent: float, annotated: bool = False): theta, beta = self.params return XXPlusYYGate(exponent * theta, beta) def __eq__(self, other): if isinstance(other, XXPlusYYGate): return self._compare_parameters(other) return False