# 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.
"""Hadamard gate."""
from math import sqrt, 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 .t import TGate, TdgGate
from .s import SGate, SdgGate
[docs]class HGate(Gate):
r"""Single-qubit Hadamard gate.
This gate is a \pi rotation about the X+Z axis, and has the effect of
changing computation basis from :math:`|0\rangle,|1\rangle` to
:math:`|+\rangle,|-\rangle` and vice-versa.
Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`
with the :meth:`~qiskit.circuit.QuantumCircuit.h` method.
**Circuit symbol:**
.. parsed-literal::
┌───┐
q_0: ┤ H ├
└───┘
**Matrix Representation:**
.. math::
H = \frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & 1 \\
1 & -1
\end{pmatrix}
"""
def __init__(self, label: Optional[str] = None):
"""Create new H gate."""
super().__init__("h", 1, [], label=label)
def _define(self):
"""
gate h a { u2(0,pi) a; }
"""
# pylint: disable=cyclic-import
from qiskit.circuit.quantumcircuit import QuantumCircuit
from .u2 import U2Gate
q = QuantumRegister(1, "q")
qc = QuantumCircuit(q, name=self.name)
rules = [(U2Gate(0, pi), [q[0]], [])]
for instr, qargs, cargs in rules:
qc._append(instr, qargs, cargs)
self.definition = qc
[docs] def control(
self,
num_ctrl_qubits: int = 1,
label: Optional[str] = None,
ctrl_state: Optional[Union[int, str]] = None,
):
"""Return a (multi-)controlled-H gate.
One control qubit returns a CH 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 = CHGate(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 inverse(self):
r"""Return inverted H gate (itself)."""
return HGate() # self-inverse
def __array__(self, dtype=None):
"""Return a Numpy.array for the H gate."""
return numpy.array([[1, 1], [1, -1]], dtype=dtype) / numpy.sqrt(2)
[docs]class CHGate(ControlledGate):
r"""Controlled-Hadamard gate.
Applies a Hadamard on the target qubit if the control is
in the :math:`|1\rangle` state.
Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`
with the :meth:`~qiskit.circuit.QuantumCircuit.ch` method.
**Circuit symbol:**
.. parsed-literal::
q_0: ──■──
┌─┴─┐
q_1: ┤ H ├
└───┘
**Matrix Representation:**
.. math::
CH\ q_0, q_1 =
I \otimes |0\rangle\langle 0| + H \otimes |1\rangle\langle 1| =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\
0 & 0 & 1 & 0 \\
0 & \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{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: ┤ H ├
└─┬─┘
q_1: ──■──
.. math::
CH\ q_1, q_0 =
|0\rangle\langle 0| \otimes I + |1\rangle\langle 1| \otimes H =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}
\end{pmatrix}
"""
# Define class constants. This saves future allocation time.
_sqrt2o2 = 1 / sqrt(2)
_matrix1 = numpy.array(
[[1, 0, 0, 0], [0, _sqrt2o2, 0, _sqrt2o2], [0, 0, 1, 0], [0, _sqrt2o2, 0, -_sqrt2o2]],
dtype=complex,
)
_matrix0 = numpy.array(
[[_sqrt2o2, 0, _sqrt2o2, 0], [0, 1, 0, 0], [_sqrt2o2, 0, -_sqrt2o2, 0], [0, 0, 0, 1]],
dtype=complex,
)
def __init__(self, label: Optional[str] = None, ctrl_state: Optional[Union[int, str]] = None):
"""Create new CH gate."""
super().__init__(
"ch", 2, [], num_ctrl_qubits=1, label=label, ctrl_state=ctrl_state, base_gate=HGate()
)
def _define(self):
"""
gate ch a,b {
s b;
h b;
t b;
cx a, b;
tdg b;
h b;
sdg b;
}
"""
# pylint: disable=cyclic-import
from qiskit.circuit.quantumcircuit import QuantumCircuit
from .x import CXGate # pylint: disable=cyclic-import
q = QuantumRegister(2, "q")
qc = QuantumCircuit(q, name=self.name)
rules = [
(SGate(), [q[1]], []),
(HGate(), [q[1]], []),
(TGate(), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(TdgGate(), [q[1]], []),
(HGate(), [q[1]], []),
(SdgGate(), [q[1]], []),
]
for instr, qargs, cargs in rules:
qc._append(instr, qargs, cargs)
self.definition = qc
[docs] def inverse(self):
"""Return inverted CH gate (itself)."""
return CHGate(ctrl_state=self.ctrl_state) # self-inverse
def __array__(self, dtype=None):
"""Return a numpy.array for the CH gate."""
mat = self._matrix1 if self.ctrl_state else self._matrix0
if dtype:
return numpy.asarray(mat, dtype=dtype)
return mat