# Source code for qiskit.circuit.library.standard_gates.rz

# 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.

"""Rotation around the Z axis."""
from cmath import exp
from typing import Optional, Union
from qiskit.circuit.gate import Gate
from qiskit.circuit.controlledgate import ControlledGate
from qiskit.circuit.quantumregister import QuantumRegister
from qiskit.circuit.parameterexpression import ParameterValueType

[docs]class RZGate(Gate):
r"""Single-qubit rotation about the Z axis.

This is a diagonal gate. It can be implemented virtually in hardware
via framechanges (i.e. at zero error and duration).

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

**Circuit symbol:**

.. parsed-literal::

┌───────┐
q_0: ┤ Rz(λ) ├
└───────┘

**Matrix Representation:**

.. math::

RZ(\lambda) = \exp\left(-i\frac{\lambda}{2}Z\right) =
\begin{pmatrix}
e^{-i\frac{\lambda}{2}} & 0 \\
0 & e^{i\frac{\lambda}{2}}
\end{pmatrix}

.. seealso::

:class:~qiskit.circuit.library.standard_gates.U1Gate
This gate is equivalent to U1 up to a phase factor.

.. math::

U1(\lambda) = e^{i{\lambda}/2}RZ(\lambda)

Reference for virtual Z gate implementation:
1612.00858 <https://arxiv.org/abs/1612.00858>_
"""

def __init__(self, phi: ParameterValueType, label: Optional[str] = None):
"""Create new RZ gate."""
super().__init__("rz", 1, [phi], label=label)

def _define(self):
"""
gate rz(phi) a { u1(phi) a; }
"""
# pylint: disable=cyclic-import
from qiskit.circuit.quantumcircuit import QuantumCircuit
from .u1 import U1Gate

q = QuantumRegister(1, "q")
theta = self.params[0]
qc = QuantumCircuit(q, name=self.name, global_phase=-theta / 2)
rules = [(U1Gate(theta), [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-RZ 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 = CRZGate(self.params[0], 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 RZ gate

:math:RZ(\lambda)^{\dagger} = RZ(-\lambda)
"""
return RZGate(-self.params[0])

def __array__(self, dtype=None):
"""Return a numpy.array for the RZ gate."""
import numpy as np

ilam2 = 0.5j * float(self.params[0])
return np.array([[exp(-ilam2), 0], [0, exp(ilam2)]], dtype=dtype)

[docs]    def power(self, exponent: float):
"""Raise gate to a power."""
(theta,) = self.params
return RZGate(exponent * theta)

[docs]class CRZGate(ControlledGate):
r"""Controlled-RZ gate.

This is a diagonal but non-symmetric gate that induces a
phase on the state of the target qubit, depending on the control state.

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

**Circuit symbol:**

.. parsed-literal::

q_0: ────■────
┌───┴───┐
q_1: ┤ Rz(λ) ├
└───────┘

**Matrix representation:**

.. math::

CRZ(\lambda)\ q_0, q_1 =
I \otimes |0\rangle\langle 0| + RZ(\lambda) \otimes |1\rangle\langle 1| =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & e^{-i\frac{\lambda}{2}} & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & e^{i\frac{\lambda}{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: ┤ Rz(λ) ├
└───┬───┘
q_1: ────■────

.. math::

CRZ(\lambda)\ q_1, q_0 =
|0\rangle\langle 0| \otimes I + |1\rangle\langle 1| \otimes RZ(\lambda) =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & e^{-i\frac{\lambda}{2}} & 0 \\
0 & 0 & 0 & e^{i\frac{\lambda}{2}}
\end{pmatrix}

.. seealso::

:class:~qiskit.circuit.library.standard_gates.CU1Gate:
Due to the global phase difference in the matrix definitions
of U1 and RZ, CU1 and CRZ are different gates with a relative
phase difference.
"""

def __init__(
self,
theta: ParameterValueType,
label: Optional[str] = None,
ctrl_state: Optional[Union[str, int]] = None,
):
"""Create new CRZ gate."""
super().__init__(
"crz",
2,
[theta],
num_ctrl_qubits=1,
label=label,
ctrl_state=ctrl_state,
base_gate=RZGate(theta),
)

def _define(self):
"""
gate crz(lambda) a,b
{ rz(lambda/2) b; cx a,b;
rz(-lambda/2) b; cx a,b;
}
"""
# pylint: disable=cyclic-import
from qiskit.circuit.quantumcircuit import QuantumCircuit
from .x import CXGate

# q_0: ─────────────■────────────────■──
#      ┌─────────┐┌─┴─┐┌──────────┐┌─┴─┐
# q_1: ┤ Rz(λ/2) ├┤ X ├┤ Rz(-λ/2) ├┤ X ├
#      └─────────┘└───┘└──────────┘└───┘
q = QuantumRegister(2, "q")
qc = QuantumCircuit(q, name=self.name)
rules = [
(RZGate(self.params[0] / 2), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(RZGate(-self.params[0] / 2), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
]
for instr, qargs, cargs in rules:
qc._append(instr, qargs, cargs)

self.definition = qc

[docs]    def inverse(self):
"""Return inverse CRZ gate (i.e. with the negative rotation angle)."""
return CRZGate(-self.params[0], ctrl_state=self.ctrl_state)

def __array__(self, dtype=None):
"""Return a numpy.array for the CRZ gate."""
import numpy

arg = 1j * float(self.params[0]) / 2
if self.ctrl_state:
return numpy.array(
[[1, 0, 0, 0], [0, exp(-arg), 0, 0], [0, 0, 1, 0], [0, 0, 0, exp(arg)]],
dtype=dtype,
)
else:
return numpy.array(
[[exp(-arg), 0, 0, 0], [0, 1, 0, 0], [0, 0, exp(arg), 0], [0, 0, 0, 1]],
dtype=dtype,
)