"""This module implements the HGate."""
from __future__ import annotations
import numpy as np
from bqskit.ir.gates.constantgate import ConstantGate
from bqskit.ir.gates.quditgate import QuditGate
from bqskit.qis.unitary.unitarymatrix import UnitaryMatrix
from bqskit.utils.typing import is_integer
[docs]
class HGate(ConstantGate, QuditGate):
"""
The one-qudit Hadamard gate. This is a Clifford gate.
By default, the HGate is a qubit Hadamard gate. However, it can be
generalized to qudits by setting the radix parameter during construction.
It is represented by the following matrix for qubits:
.. math::
\\begin{pmatrix}
\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2} \\\\
\\frac{\\sqrt{2}}{2} & -\\frac{\\sqrt{2}}{2} \\\\
\\end{pmatrix}
However, generally it is given by the following formula:
.. math::
H = \\frac{1}{\\sqrt(d)} \\sum_{ij} \\omega_d^{ij} |i\\rangle\\langle j|
where
.. math::
\\omega = \\exp(\\frac{2\\pi i}{d})
and `d` is the number of levels (2 levels is a qubit,
3 levels is a qutrit, etc.)
References:
- https://www.frontiersin.org/articles/10.3389/fphy.2020.589504/full
- https://pubs.aip.org/aip/jmp/article-abstract/56/3/032202/763827
"""
_num_qudits = 1
_qasm_name = 'h'
[docs]
def __init__(self, radix: int = 2) -> None:
"""
Construct a HGate.
Args:
radix (int): The number of qudit levels (>=2). By default, this
is 2, which is gives a qubit Hadamard gate.
Raises:
ValueError: if radix < 2
"""
if not is_integer(radix):
raise TypeError(f'Expected integer for radix, got {type(radix)}.')
if radix < 2:
raise ValueError(f'Radix must be greater than 1, got {radix}.')
self._radix = radix
# Calculate unitary
if radix == 2:
matrix = np.array(
[
[np.sqrt(2) / 2, np.sqrt(2) / 2],
[np.sqrt(2) / 2, -np.sqrt(2) / 2],
],
dtype=np.complex128,
)
self._utry = UnitaryMatrix(matrix)
else:
matrix = np.zeros([radix] * 2, dtype=np.complex128)
omega = np.exp(2j * np.pi / radix)
for i in range(radix):
for j in range(i, radix):
val = omega ** (i * j)
matrix[i, j] = val
matrix[j, i] = val
matrix *= 1 / np.sqrt(radix)
self._utry = UnitaryMatrix(matrix, self.radixes)