Source code for qiskit.circuit.library.standard_gates.z

# 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.

"""Z and CZ 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


class ZGate(Gate):
    r"""The single-qubit Pauli-Z gate (:math:`\sigma_z`).

    **Matrix Representation:**

    .. math::

        Z = \begin{pmatrix}
                1 & 0 \\
                0 & -1
            \end{pmatrix}

    **Circuit symbol:**

    .. parsed-literal::

             ┌───┐
        q_0: ┤ Z ├
             └───┘

    Equivalent to a :math:`\pi` radian rotation about the Z axis.

    .. note::

        A global phase difference exists between the definitions of
        :math:`RZ(\pi)` and :math:`Z`.

        .. math::

            RZ(\pi) = \begin{pmatrix}
                        -1 & 0 \\
                        0 & 1
                      \end{pmatrix}
                    = -Z

    The gate is equivalent to a phase flip.

    .. math::

        |0\rangle \rightarrow |0\rangle \\
        |1\rangle \rightarrow -|1\rangle
    """

    def __init__(self, label=None):
        """Create new Z gate."""
        super().__init__('z', 1, [], label=label)

    def _define(self):
        # pylint: disable=cyclic-import
        from qiskit.circuit.quantumcircuit import QuantumCircuit
        from .u1 import U1Gate
        q = QuantumRegister(1, 'q')
        qc = QuantumCircuit(q, name=self.name)
        rules = [
            (U1Gate(pi), [q[0]], [])
        ]
        for instr, qargs, cargs in rules:
            qc._append(instr, qargs, cargs)

        self.definition = qc

    def control(self, num_ctrl_qubits=1, label=None, ctrl_state=None):
        """Return a (mutli-)controlled-Z gate.

        One control returns a CZ 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 = CZGate(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)

    def inverse(self):
        """Return inverted Z gate (itself)."""
        return ZGate()  # self-inverse

    def to_matrix(self):
        """Return a numpy.array for the Z gate."""
        return numpy.array([[1, 0],
                            [0, -1]], dtype=complex)


class CZGate(ControlledGate):
    r"""Controlled-Z gate.

    This is a Clifford and symmetric gate.

    **Circuit symbol:**

    .. parsed-literal::

        q_0: ─■─
              │
        q_1: ─■─

    **Matrix representation:**

    .. math::

        CZ\ q_1, q_0 =
            |0\rangle\langle 0| \otimes I + |1\rangle\langle 1| \otimes Z =
            \begin{pmatrix}
                1 & 0 & 0 & 0 \\
                0 & 1 & 0 & 0 \\
                0 & 0 & 1 & 0 \\
                0 & 0 & 0 & -1
            \end{pmatrix}

    In the computational basis, this gate flips the phase of
    the target qubit if the control qubit is in the :math:`|1\rangle` state.
    """

    def __init__(self, label=None, ctrl_state=None):
        """Create new CZ gate."""
        super().__init__('cz', 2, [], label=label, num_ctrl_qubits=1,
                         ctrl_state=ctrl_state, base_gate=ZGate())

    def _define(self):
        """
        gate cz a,b { h b; cx a,b; h b; }
        """
        # pylint: disable=cyclic-import
        from qiskit.circuit.quantumcircuit import QuantumCircuit
        from .h import HGate
        from .x import CXGate
        q = QuantumRegister(2, 'q')
        qc = QuantumCircuit(q, name=self.name)
        rules = [
            (HGate(), [q[1]], []),
            (CXGate(), [q[0], q[1]], []),
            (HGate(), [q[1]], [])
        ]
        for instr, qargs, cargs in rules:
            qc._append(instr, qargs, cargs)

        self.definition = qc

    def inverse(self):
        """Return inverted CZ gate (itself)."""
        return CZGate(ctrl_state=self.ctrl_state)  # self-inverse

    def to_matrix(self):
        """Return a numpy.array for the CZ gate."""
        if self.ctrl_state:
            return numpy.array([[1, 0, 0, 0],
                                [0, 1, 0, 0],
                                [0, 0, 1, 0],
                                [0, 0, 0, -1]], dtype=complex)
        else:
            return numpy.array([[1, 0, 0, 0],
                                [0, 1, 0, 0],
                                [0, 0, -1, 0],
                                [0, 0, 0, 1]], dtype=complex)