# 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 __future__ import annotations from math import sqrt from typing import Optional, Union import numpy from qiskit.circuit.singleton import SingletonGate, SingletonControlledGate, stdlib_singleton_key from qiskit.circuit._utils import with_gate_array, with_controlled_gate_array from qiskit._accelerate.circuit import StandardGate _H_ARRAY = 1 / sqrt(2) * numpy.array([[1, 1], [1, -1]], dtype=numpy.complex128) @with_gate_array(_H_ARRAY) class HGate(SingletonGate): 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: .. code-block:: text ┌───┐ q_0: ┤ H ├ └───┘ Matrix representation: .. math:: H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} """ _standard_gate = StandardGate.H def __init__(self, label: Optional[str] = None): """ Args: label: An optional label for the gate. """ super().__init__("h", 1, [], label=label) _singleton_lookup_key = stdlib_singleton_key() def _define(self): """Default definition""" # pylint: disable=cyclic-import from qiskit.circuit import QuantumCircuit # ┌────────────┐ # q: ┤ U(π/2,0,π) ├ # └────────────┘ self.definition = QuantumCircuit._from_circuit_data( StandardGate.H._get_definition(self.params), legacy_qubits=True, name=self.name ) def control( self, num_ctrl_qubits: int = 1, label: str | None = None, ctrl_state: int | str | None = None, annotated: bool | None = None, ): """Return a (multi-)controlled-H gate. One control qubit returns a CH 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 should be implemented as an annotated gate. If ``None``, this is handled as ``False``. Returns: ControlledGate: controlled version of this gate. """ if not annotated and num_ctrl_qubits == 1: gate = CHGate(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 inverse(self, annotated: bool = False): r"""Return inverted H gate (itself). 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 this gate is self-inverse. Returns: HGate: inverse gate (self-inverse). """ return HGate() # self-inverse def __eq__(self, other): return isinstance(other, HGate) @with_controlled_gate_array(_H_ARRAY, num_ctrl_qubits=1) class CHGate(SingletonControlledGate): 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: .. code-block:: text 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: .. code-block:: text ┌───┐ 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} """ _standard_gate = StandardGate.CH def __init__( self, label: Optional[str] = None, ctrl_state: Optional[Union[int, str]] = None, *, _base_label=None, ): """Create new CH gate.""" super().__init__( "ch", 2, [], num_ctrl_qubits=1, label=label, ctrl_state=ctrl_state, base_gate=HGate(label=_base_label), _base_label=_base_label, ) _singleton_lookup_key = stdlib_singleton_key(num_ctrl_qubits=1) def _define(self): """Default definition""" # pylint: disable=cyclic-import from qiskit.circuit import QuantumCircuit # q_0: ─────────────────■───────────────────── # ┌───┐┌───┐┌───┐┌─┴─┐┌─────┐┌───┐┌─────┐ # q_1: ┤ S ├┤ H ├┤ T ├┤ X ├┤ Tdg ├┤ H ├┤ Sdg ├ # └───┘└───┘└───┘└───┘└─────┘└───┘└─────┘ self.definition = QuantumCircuit._from_circuit_data( StandardGate.CH._get_definition(self.params), legacy_qubits=True, name=self.name ) def inverse(self, annotated: bool = False): """Return inverted CH gate (itself).""" return CHGate(ctrl_state=self.ctrl_state) # self-inverse def __eq__(self, other): return isinstance(other, CHGate) and self.ctrl_state == other.ctrl_state