# 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 math import pi from typing import Optional import numpy from qiskit.circuit.gate import Gate from qiskit.circuit.quantumregister import QuantumRegister 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.XXPlusYYGate def __init__( self, theta: ParameterValueType, beta: ParameterValueType = 0, label: Optional[str] = "(XX+YY)", *, duration=None, unit="dt", ): """Create new XX+YY gate. 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, duration=duration, unit=unit) def _define(self): """ gate xx_plus_yy(theta, beta) a, b { rz(beta) b; rz(-pi/2) a; sx a; rz(pi/2) a; s b; cx a, b; ry(theta/2) a; ry(theta/2) b; cx a, b; sdg b; rz(-pi/2) a; sxdg a; rz(pi/2) a; rz(-beta) b; } """ # pylint: disable=cyclic-import from qiskit.circuit.quantumcircuit import QuantumCircuit from .x import CXGate from .s import SGate, SdgGate from .sx import SXGate, SXdgGate from .rz import RZGate from .ry import RYGate theta = self.params[0] beta = self.params[1] q = QuantumRegister(2, "q") qc = QuantumCircuit(q, name=self.name) rules = [ (RZGate(beta), [q[0]], []), (RZGate(-pi / 2), [q[1]], []), (SXGate(), [q[1]], []), (RZGate(pi / 2), [q[1]], []), (SGate(), [q[0]], []), (CXGate(), [q[1], q[0]], []), (RYGate(-theta / 2), [q[1]], []), (RYGate(-theta / 2), [q[0]], []), (CXGate(), [q[1], q[0]], []), (SdgGate(), [q[0]], []), (RZGate(-pi / 2), [q[1]], []), (SXdgGate(), [q[1]], []), (RZGate(pi / 2), [q[1]], []), (RZGate(-beta), [q[0]], []), ] for instr, qargs, cargs in rules: qc._append(instr, qargs, cargs) self.definition = qc 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