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

# This code is part of Qiskit.
#
#
# obtain a copy of this license in the LICENSE.txt file in the root directory
#
# 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, CZ and CCZ gates."""

from math import pi
from typing import Optional, Union

import numpy

from qiskit.circuit._utils import with_gate_array, with_controlled_gate_array
from qiskit.circuit.controlledgate import ControlledGate
from qiskit.circuit.gate import Gate
from qiskit.circuit.quantumregister import QuantumRegister

from .p import PhaseGate

_Z_ARRAY = [[1, 0], [0, -1]]

[docs]@with_gate_array(_Z_ARRAY)
class ZGate(Gate):
r"""The single-qubit Pauli-Z gate (:math:\sigma_z).

Can be applied to a :class:~qiskit.circuit.QuantumCircuit
with the :meth:~qiskit.circuit.QuantumCircuit.z method.

**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}
-i & 0 \\
0 & i
\end{pmatrix}
= -i Z

The gate is equivalent to a phase flip.

.. math::

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

def __init__(self, label: Optional[str] = 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

[docs]    def control(
self,
num_ctrl_qubits: int = 1,
label: Optional[str] = None,
ctrl_state: Optional[Union[str, int]] = None,
):
"""Return a (multi-)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)

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

[docs]    def power(self, exponent: float):
"""Raise gate to a power."""
return PhaseGate(numpy.pi * exponent)

[docs]@with_controlled_gate_array(_Z_ARRAY, num_ctrl_qubits=1)
class CZGate(ControlledGate):
r"""Controlled-Z gate.

This is a Clifford and symmetric gate.

Can be applied to a :class:~qiskit.circuit.QuantumCircuit
with the :meth:~qiskit.circuit.QuantumCircuit.cz method.

**Circuit symbol:**

.. parsed-literal::

q_0: ─■─
│
q_1: ─■─

**Matrix representation:**

.. math::

CZ\ q_0, q_1 =
I \otimes |0\rangle\langle 0| + Z \otimes |1\rangle\langle 1| =
\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: Optional[str] = None, ctrl_state: Optional[Union[str, int]] = 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

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

[docs]@with_controlled_gate_array(_Z_ARRAY, num_ctrl_qubits=2, cached_states=(3,))
class CCZGate(ControlledGate):
r"""CCZ gate.

This is a symmetric gate.

Can be applied to a :class:~qiskit.circuit.QuantumCircuit
with the :meth:~qiskit.circuit.QuantumCircuit.ccz method.

**Circuit symbol:**

.. parsed-literal::

q_0: ─■─
│
q_1: ─■─
│
q_2: ─■─

**Matrix representation:**

.. math::

CCZ\ q_0, q_1, q_2 =
I \otimes I \otimes |0\rangle\langle 0| + CZ \otimes |1\rangle\langle 1| =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1
\end{pmatrix}

In the computational basis, this gate flips the phase of
the target qubit if the control qubits are in the :math:|11\rangle state.
"""

def __init__(self, label: Optional[str] = None, ctrl_state: Optional[Union[str, int]] = None):
"""Create new CCZ gate."""
super().__init__(
"ccz", 3, [], label=label, num_ctrl_qubits=2, ctrl_state=ctrl_state, base_gate=ZGate()
)

def _define(self):
"""
gate ccz a,b,c { h c; ccx a,b,c; h c; }
"""
# pylint: disable=cyclic-import
from qiskit.circuit.quantumcircuit import QuantumCircuit

from .h import HGate
from .x import CCXGate

q = QuantumRegister(3, "q")
qc = QuantumCircuit(q, name=self.name)
rules = [(HGate(), [q[2]], []), (CCXGate(), [q[0], q[1], q[2]], []), (HGate(), [q[2]], [])]
for instr, qargs, cargs in rules:
qc._append(instr, qargs, cargs)

self.definition = qc

[docs]    def inverse(self):
"""Return inverted CCZ gate (itself)."""
return CCZGate(ctrl_state=self.ctrl_state)  # self-inverse