# 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."""
import numpy
from qiskit.qasm import pi
from qiskit.circuit.controlledgate import ControlledGate
from qiskit.circuit.gate import Gate
from qiskit.circuit.quantumregister import QuantumRegister
[docs]class SXGate(Gate):
r"""The single-qubit Sqrt(X) gate (:math:`\sqrt{X}`).
**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}
"""
[docs] def __init__(self, label=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]], [])
]
qc.data = rules
self.definition = qc
[docs] def inverse(self):
"""Return inverse SX gate (i.e. SXdg)."""
return SXdgGate()
[docs] def control(self, num_ctrl_qubits=1, label=None, ctrl_state=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)
[docs] def to_matrix(self):
"""Return a numpy.array for the SX gate."""
return numpy.array([[1 + 1j, 1 - 1j],
[1 - 1j, 1 + 1j]], dtype=complex) / 2
[docs]class SXdgGate(Gate):
r"""The inverse single-qubit Sqrt(X) gate.
.. 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}
"""
[docs] def __init__(self, label=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]], [])
]
qc.data = rules
self.definition = qc
[docs] def inverse(self):
"""Return inverse SXdg gate (i.e. SX)."""
return SXGate()
[docs] def to_matrix(self):
"""Return a numpy.array for the SXdg gate."""
return numpy.array([[1 - 1j, 1 + 1j],
[1 + 1j, 1 - 1j]], dtype=complex) / 2
[docs]class CSXGate(ControlledGate):
r"""Controlled-√X gate.
**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}
"""
# Define class constants. This saves future allocation time.
_matrix1 = numpy.array([[1, 0, 0, 0],
[0, (1 + 1j) / 2, 0, (1 - 1j) / 2],
[0, 0, 1, 0],
[0, (1 - 1j) / 2, 0, (1 + 1j) / 2]], dtype=complex)
_matrix0 = numpy.array([[(1 + 1j) / 2, 0, (1 - 1j) / 2, 0],
[0, 1, 0, 0],
[(1 - 1j) / 2, 0, (1 + 1j) / 2, 0],
[0, 0, 0, 1]], dtype=complex)
[docs] def __init__(self, label=None, ctrl_state=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]], [])
]
qc.data = rules
self.definition = qc
[docs] def to_matrix(self):
"""Return a numpy.array for the CSX gate."""
if self.ctrl_state:
return self._matrix1
else:
return self._matrix0