# 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
import numpy
from qiskit.circuit.controlledgate import ControlledGate
from qiskit.circuit.gate import Gate
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]]
[문서]@with_gate_array(_SX_ARRAY)
class SXGate(Gate):
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):
"""Create new SX gate."""
super().__init__("sx", 1, [], label=label)
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
[문서] def inverse(self):
"""Return inverse SX gate (i.e. SXdg)."""
return SXdgGate()
[문서] def control(
self,
num_ctrl_qubits: int = 1,
label: Optional[str] = None,
ctrl_state: Optional[Union[str, int]] = None,
):
"""Return a (multi-)controlled-SX gate.
One control returns a CSX gate.
Args:
num_ctrl_qubits (int): number of control qubits.
label (str or None): An optional label for the gate [Default: None]
ctrl_state (int or str or None): control state expressed as integer,
string (e.g. '110'), or None. If None, use all 1s.
Returns:
ControlledGate: controlled version of this gate.
"""
if num_ctrl_qubits == 1:
gate = CSXGate(label=label, ctrl_state=ctrl_state)
gate.base_gate.label = self.label
return gate
return super().control(num_ctrl_qubits=num_ctrl_qubits, label=label, ctrl_state=ctrl_state)
[문서]@with_gate_array(_SXDG_ARRAY)
class SXdgGate(Gate):
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}
"""
_ARRAY = numpy.array(
[[0.5 - 0.5j, 0.5 + 0.5j], [0.5 + 0.5j, 0.5 - 0.5j]], dtype=numpy.complex128
)
_ARRAY.setflags(write=False)
def __init__(self, label: Optional[str] = None):
"""Create new SXdg gate."""
super().__init__("sxdg", 1, [], label=label)
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
[문서] def inverse(self):
"""Return inverse SXdg gate (i.e. SX)."""
return SXGate()
[문서]@with_controlled_gate_array(_SX_ARRAY, num_ctrl_qubits=1)
class CSXGate(ControlledGate):
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):
"""Create new CSX gate."""
super().__init__(
"csx", 2, [], num_ctrl_qubits=1, label=label, ctrl_state=ctrl_state, base_gate=SXGate()
)
def _define(self):
"""
gate csx a,b { h b; cu1(pi/2) a,b; h b; }
"""
# pylint: disable=cyclic-import
from qiskit.circuit.quantumcircuit import QuantumCircuit
from .h import HGate
from .u1 import CU1Gate
q = QuantumRegister(2, "q")
qc = QuantumCircuit(q, name=self.name)
rules = [(HGate(), [q[1]], []), (CU1Gate(pi / 2), [q[0], q[1]], []), (HGate(), [q[1]], [])]
for operation, qubits, clbits in rules:
qc._append(operation, qubits, clbits)
self.definition = qc