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This code is part of Qiskit.","#","# (C) Copyright IBM 2017.","#","# 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.","","\"\"\"Sqrt(X) and C-Sqrt(X) gates.\"\"\"","","from math import pi","from typing import Optional, Union","from qiskit.circuit.singleton import SingletonGate, SingletonControlledGate, stdlib_singleton_key","from qiskit.circuit.quantumregister import QuantumRegister","from qiskit.circuit._utils import with_gate_array, with_controlled_gate_array","","","_SX_ARRAY = [[0.5 + 0.5j, 0.5 - 0.5j], [0.5 - 0.5j, 0.5 + 0.5j]]","_SXDG_ARRAY = [[0.5 - 0.5j, 0.5 + 0.5j], [0.5 + 0.5j, 0.5 - 0.5j]]","","","@with_gate_array(_SX_ARRAY)","class SXGate(SingletonGate):"," r\"\"\"The single-qubit Sqrt(X) gate (:math:`\\sqrt{X}`).",""," Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`"," with the :meth:`~qiskit.circuit.QuantumCircuit.sx` method.",""," **Matrix Representation:**",""," .. math::",""," \\sqrt{X} = \\frac{1}{2} \\begin{pmatrix}"," 1 + i & 1 - i \\\\"," 1 - i & 1 + i"," \\end{pmatrix}",""," **Circuit symbol:**",""," .. parsed-literal::",""," ┌────┐"," q_0: ┤ √X ├"," └────┘",""," .. note::",""," A global phase difference exists between the definitions of"," :math:`RX(\\pi/2)` and :math:`\\sqrt{X}`.",""," .. math::",""," RX(\\pi/2) = \\frac{1}{\\sqrt{2}} \\begin{pmatrix}"," 1 & -i \\\\"," -i & 1"," \\end{pmatrix}"," = e^{-i \\pi/4} \\sqrt{X}",""," \"\"\"",""," def __init__(self, label: Optional[str] = None, *, duration=None, unit=\"dt\"):"," \"\"\"Create new SX gate.\"\"\""," super().__init__(\"sx\", 1, [], label=label, duration=duration, unit=unit)",""," _singleton_lookup_key = stdlib_singleton_key()",""," def _define(self):"," \"\"\""," gate sx a { rz(-pi/2) a; h a; rz(-pi/2); }"," \"\"\""," # pylint: disable=cyclic-import"," from qiskit.circuit.quantumcircuit import QuantumCircuit"," from .s import SdgGate"," from .h import HGate",""," q = QuantumRegister(1, \"q\")"," qc = QuantumCircuit(q, name=self.name, global_phase=pi / 4)"," rules = [(SdgGate(), [q[0]], []), (HGate(), [q[0]], []), (SdgGate(), [q[0]], [])]"," for operation, qubits, clbits in rules:"," qc._append(operation, qubits, clbits)"," self.definition = qc",""," def inverse(self, annotated: bool = False):"," \"\"\"Return inverse SX gate (i.e. SXdg).",""," 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:`.SXdgGate`.",""," Returns:"," SXdgGate: inverse of :class:`.SXGate`."," \"\"\""," return SXdgGate()",""," def control("," self,"," num_ctrl_qubits: int = 1,"," label: Optional[str] = None,"," ctrl_state: Optional[Union[str, int]] = None,"," annotated: bool = False,"," ):"," \"\"\"Return a (multi-)controlled-SX gate.",""," One control returns a CSX 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 can be implemented"," as an annotated gate.",""," Returns:"," SingletonControlledGate: controlled version of this gate."," \"\"\""," if not annotated and num_ctrl_qubits == 1:"," gate = CSXGate(label=label, ctrl_state=ctrl_state, _base_label=self.label)"," else:"," gate = super().control("," num_ctrl_qubits=num_ctrl_qubits,"," label=label,"," ctrl_state=ctrl_state,"," annotated=annotated,"," )"," return gate",""," def __eq__(self, other):"," return isinstance(other, SXGate)","","","@with_gate_array(_SXDG_ARRAY)","class SXdgGate(SingletonGate):"," r\"\"\"The inverse single-qubit Sqrt(X) gate.",""," Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`"," with the :meth:`~qiskit.circuit.QuantumCircuit.sxdg` method.",""," .. math::",""," \\sqrt{X}^{\\dagger} = \\frac{1}{2} \\begin{pmatrix}"," 1 - i & 1 + i \\\\"," 1 + i & 1 - i"," \\end{pmatrix}","",""," .. note::",""," A global phase difference exists between the definitions of"," :math:`RX(-\\pi/2)` and :math:`\\sqrt{X}^{\\dagger}`.",""," .. math::",""," RX(-\\pi/2) = \\frac{1}{\\sqrt{2}} \\begin{pmatrix}"," 1 & i \\\\"," i & 1"," \\end{pmatrix}"," = e^{-i \\pi/4} \\sqrt{X}^{\\dagger}"," \"\"\"",""," def __init__(self, label: Optional[str] = None, *, duration=None, unit=\"dt\"):"," \"\"\"Create new SXdg gate.\"\"\""," super().__init__(\"sxdg\", 1, [], label=label, duration=duration, unit=unit)",""," _singleton_lookup_key = stdlib_singleton_key()",""," def _define(self):"," \"\"\""," gate sxdg a { rz(pi/2) a; h a; rz(pi/2); }"," \"\"\""," # pylint: disable=cyclic-import"," from qiskit.circuit.quantumcircuit import QuantumCircuit"," from .s import SGate"," from .h import HGate",""," q = QuantumRegister(1, \"q\")"," qc = QuantumCircuit(q, name=self.name, global_phase=-pi / 4)"," rules = [(SGate(), [q[0]], []), (HGate(), [q[0]], []), (SGate(), [q[0]], [])]"," for operation, qubits, clbits in rules:"," qc._append(operation, qubits, clbits)"," self.definition = qc",""," def inverse(self, annotated: bool = False):"," \"\"\"Return inverse SXdg gate (i.e. SX).",""," 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:`.SXGate`.",""," Returns:"," SXGate: inverse of :class:`.SXdgGate`"," \"\"\""," return SXGate()",""," def __eq__(self, other):"," return isinstance(other, SXdgGate)","","","@with_controlled_gate_array(_SX_ARRAY, num_ctrl_qubits=1)","class CSXGate(SingletonControlledGate):"," r\"\"\"Controlled-√X gate.",""," Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`"," with the :meth:`~qiskit.circuit.QuantumCircuit.csx` method.",""," **Circuit symbol:**",""," .. parsed-literal::",""," q_0: ──■──"," ┌─┴──┐"," q_1: ┤ √X ├"," └────┘",""," **Matrix representation:**",""," .. math::",""," C\\sqrt{X} \\ q_0, q_1 ="," I \\otimes |0 \\rangle\\langle 0| + \\sqrt{X} \\otimes |1 \\rangle\\langle 1| ="," \\begin{pmatrix}"," 1 & 0 & 0 & 0 \\\\"," 0 & (1 + i) / 2 & 0 & (1 - i) / 2 \\\\"," 0 & 0 & 1 & 0 \\\\"," 0 & (1 - i) / 2 & 0 & (1 + i) / 2"," \\end{pmatrix}","",""," .. note::",""," In Qiskit's convention, higher qubit indices are more significant"," (little endian convention). In many textbooks, controlled gates are"," presented with the assumption of more significant qubits as control,"," which in our case would be `q_1`. Thus a textbook matrix for this"," gate will be:",""," .. parsed-literal::"," ┌────┐"," q_0: ┤ √X ├"," └─┬──┘"," q_1: ──■──",""," .. math::",""," C\\sqrt{X}\\ q_1, q_0 ="," |0 \\rangle\\langle 0| \\otimes I + |1 \\rangle\\langle 1| \\otimes \\sqrt{X} ="," \\begin{pmatrix}"," 1 & 0 & 0 & 0 \\\\"," 0 & 1 & 0 & 0 \\\\"," 0 & 0 & (1 + i) / 2 & (1 - i) / 2 \\\\"," 0 & 0 & (1 - i) / 2 & (1 + i) / 2"," \\end{pmatrix}",""," \"\"\"",""," def __init__("," self,"," label: Optional[str] = None,"," ctrl_state: Optional[Union[str, int]] = None,"," *,"," duration=None,"," unit=\"dt\","," _base_label=None,"," ):"," \"\"\"Create new CSX gate.\"\"\""," super().__init__("," \"csx\","," 2,"," [],"," num_ctrl_qubits=1,"," label=label,"," ctrl_state=ctrl_state,"," base_gate=SXGate(label=_base_label),"," duration=duration,"," unit=unit,"," )",""," _singleton_lookup_key = stdlib_singleton_key(num_ctrl_qubits=1)",""," def _define(self):"," \"\"\""," gate csx a,b { h b; cu1(pi/2) a,b; h b; }"," \"\"\""," # 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