r"""Generic single-qubit rotation gate with 3 Euler angles.

Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`

with the :meth:`~qiskit.circuit.QuantumCircuit.u` method.

**Circuit symbol:**

.. parsed-literal::

┌──────────┐

q_0: ┤ U(ϴ,φ,λ) ├

└──────────┘

**Matrix Representation:**

.. math::

\newcommand{\rotationangle}{\frac{\theta}{2}}

U(\theta, \phi, \lambda) =

\begin{pmatrix}

\cos\left(\rotationangle\right) & -e^{i\lambda}\sin\left(\rotationangle\right) \\

e^{i\phi}\sin\left(\rotationangle\right) & e^{i(\phi+\lambda)}\cos\left(\rotationangle\right)

\end{pmatrix}

.. note::

The matrix representation shown here is the same as in the `OpenQASM 3.0 specification

<https://openqasm.com/language/gates.html#built-in-gates>`_,

which differs from the `OpenQASM 2.0 specification

<https://doi.org/10.48550/arXiv.1707.03429>`_ by a global phase of

:math:`e^{i(\phi+\lambda)/2}`.

**Examples:**

.. math::

U\left(\theta, -\frac{\pi}{2}, \frac{\pi}{2}\right) = RX(\theta)

.. math::

U(\theta, 0, 0) = RY(\theta)

"""

def __init__(

self,

theta: ParameterValueType,

phi: ParameterValueType,

lam: ParameterValueType,

label: Optional[str] = None,

*,

duration=None,

unit="dt",

):

"""Create new U gate."""

super().__init__("u", 1, [theta, phi, lam], label=label, duration=duration, unit=unit)

def inverse(self, annotated: bool = False):

r"""Return inverted U gate.

:math:`U(\theta,\phi,\lambda)^{\dagger} =U(-\theta,-\lambda,-\phi))`

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 the

inverse of this gate is always a :class:`.UGate` with inverse parameter values.

Returns:

UGate: inverse gate.

"""

return UGate(-self.params[0], -self.params[2], -self.params[1])

def control(

self,

num_ctrl_qubits: int = 1,

label: Optional[str] = None,

ctrl_state: Optional[Union[str, int]] = None,

annotated: bool = False,

):

"""Return a (multi-)controlled-U 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 can be implemented

as an annotated gate.

Returns:

ControlledGate: controlled version of this gate.

"""

if not annotated and num_ctrl_qubits == 1:

gate = CUGate(

self.params[0],

self.params[1],

self.params[2],

0,

label=label,

ctrl_state=ctrl_state,

)

gate.base_gate.label = self.label

else:

gate = super().control(

num_ctrl_qubits=num_ctrl_qubits,

label=label,

ctrl_state=ctrl_state,

annotated=annotated,

)

return gate

def __array__(self, dtype=None, copy=None):

"""Return a numpy.array for the U gate."""

if copy is False:

raise ValueError("unable to avoid copy while creating an array as requested")

theta, phi, lam = (float(param) for param in self.params)

cos = math.cos(theta / 2)

sin = math.sin(theta / 2)

return numpy.array(

[

[cos, -exp(1j * lam) * sin],

[exp(1j * phi) * sin, exp(1j * (phi + lam)) * cos],

],

dtype=dtype or complex,

)

def __eq__(self, other):

if isinstance(other, UGate):

return self._compare_parameters(other)

# This awful class is to let `CUGate.params` have its keys settable (as

# `QuantumCircuit.assign_parameters` requires), while accounting for the problem that `CUGate`

# was defined to have a different number of parameters to its `base_gate`, which breaks

# `ControlledGate`'s assumptions, and would make most parametric `CUGate`s invalid.

#

# It's constructed only as part of the `CUGate.params` getter, and given that the general

# circuit model assumes that that's a directly mutable list that _must_ be kept in sync with the

# gate's requirements, we don't need this to support arbitrary mutation, just enough for

# `QuantumCircuit.assign_parameters` to work.

__slots__ = ("_gate",)

def __init__(self, gate):

super().__init__(gate._params)

self._gate = gate

def __setitem__(self, key, value):

super().__setitem__(key, value)

self._gate._params[key] = value

# Magic numbers: CUGate has 4 parameters, UGate has 3, with the last of CUGate's missing.

if isinstance(key, slice):

# We don't need to worry about the case of the slice being used to insert extra / remove

# elements because that would be "undefined behaviour" in a gate already, so we're

# within our rights to do anything at all.

for i, base_key in enumerate(range(*key.indices(4))):

if base_key < 0:

base_key = 4 + base_key

if base_key < 3:

self._gate.base_gate.params[base_key] = value[i]

else:

if key < 0:

key = 4 + key

if key < 3:

r"""Controlled-U gate (4-parameter two-qubit gate).

This is a controlled version of the U gate (generic single qubit rotation),

including a possible global phase :math:`e^{i\gamma}` of the U gate.

Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`

with the :meth:`~qiskit.circuit.QuantumCircuit.cu` method.

**Circuit symbol:**

.. parsed-literal::

q_0: ──────■──────

┌─────┴──────┐

q_1: ┤ U(ϴ,φ,λ,γ) ├

└────────────┘

**Matrix representation:**

.. math::

\newcommand{\rotationangle}{\frac{\theta}{2}}

CU(\theta, \phi, \lambda, \gamma)\ q_0, q_1 =

I \otimes |0\rangle\langle 0| +

e^{i\gamma} U(\theta,\phi,\lambda) \otimes |1\rangle\langle 1| =

\begin{pmatrix}

1 & 0 & 0 & 0 \\

0 & e^{i\gamma}\cos(\rotationangle) &

0 & -e^{i(\gamma + \lambda)}\sin(\rotationangle) \\

0 & 0 & 1 & 0 \\

0 & e^{i(\gamma+\phi)}\sin(\rotationangle) &

0 & e^{i(\gamma+\phi+\lambda)}\cos(\rotationangle)

\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: ┤ U(ϴ,φ,λ,γ) ├

└─────┬──────┘

q_1: ──────■───────

.. math::

\newcommand{\rotationangle}{\frac{\theta}{2}}

CU(\theta, \phi, \lambda, \gamma)\ q_1, q_0 =

|0\rangle\langle 0| \otimes I +

e^{i\gamma}|1\rangle\langle 1| \otimes U(\theta,\phi,\lambda) =

\begin{pmatrix}

1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & e^{i\gamma} \cos(\rotationangle) & -e^{i(\gamma + \lambda)}\sin(\rotationangle) \\

0 & 0 &

e^{i(\gamma + \phi)}\sin(\rotationangle) & e^{i(\gamma + \phi+\lambda)}\cos(\rotationangle)

\end{pmatrix}

"""

def __init__(

self,

theta: ParameterValueType,

phi: ParameterValueType,

lam: ParameterValueType,

gamma: ParameterValueType,

label: Optional[str] = None,

ctrl_state: Optional[Union[str, int]] = None,

*,

duration=None,

unit="dt",

_base_label=None,

):

"""Create new CU gate."""

super().__init__(

"cu",

2,

[theta, phi, lam, gamma],

num_ctrl_qubits=1,

label=label,

ctrl_state=ctrl_state,

base_gate=UGate(theta, phi, lam, label=_base_label),

duration=duration,

unit=unit,

)

def _define(self):

"""

gate cu(theta,phi,lambda,gamma) c, t

{ phase(gamma) c;

phase((lambda+phi)/2) c;

phase((lambda-phi)/2) t;

cx c,t;

u(-theta/2,0,-(phi+lambda)/2) t;

cx c,t;

u(theta/2,phi,0) t;

}

"""

# pylint: disable=cyclic-import

from qiskit.circuit.quantumcircuit import QuantumCircuit

# ┌──────┐ ┌──────────────┐

# q_0: ────┤ P(γ) ├────┤ P(λ/2 + φ/2) ├──■────────────────────────────■────────────────

# ┌───┴──────┴───┐└──────────────┘┌─┴─┐┌──────────────────────┐┌─┴─┐┌────────────┐

# q_1: ┤ P(λ/2 - φ/2) ├────────────────┤ X ├┤ U(-0/2,0,-λ/2 - φ/2) ├┤ X ├┤ U(0/2,φ,0) ├

# └──────────────┘ └───┘└──────────────────────┘└───┘└────────────┘

q = QuantumRegister(2, "q")

qc = QuantumCircuit(q, name=self.name)

qc.p(self.params[3], 0)

qc.p((self.params[2] + self.params[1]) / 2, 0)

qc.p((self.params[2] - self.params[1]) / 2, 1)

qc.cx(0, 1)

qc.u(-self.params[0] / 2, 0, -(self.params[1] + self.params[2]) / 2, 1)

qc.cx(0, 1)

qc.u(self.params[0] / 2, self.params[1], 0, 1)

self.definition = qc

def inverse(self, annotated: bool = False):

r"""Return inverted CU gate.

:math:`CU(\theta,\phi,\lambda,\gamma)^{\dagger} = CU(-\theta,-\phi,-\lambda,-\gamma))`

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 the inverse

of this gate is always a :class:`.CUGate` with inverse parameter

values.

Returns:

CUGate: inverse gate.

"""

return CUGate(

-self.params[0],

-self.params[2],

-self.params[1],

-self.params[3],

ctrl_state=self.ctrl_state,

)

def __array__(self, dtype=None, copy=None):

"""Return a numpy.array for the CU gate."""

if copy is False:

raise ValueError("unable to avoid copy while creating an array as requested")

theta, phi, lam, gamma = (float(param) for param in self.params)

cos = math.cos(theta / 2)

sin = math.sin(theta / 2)

a = cmath.exp(1j * gamma) * cos

b = -cmath.exp(1j * (gamma + lam)) * sin

c = cmath.exp(1j * (gamma + phi)) * sin

d = cmath.exp(1j * (gamma + phi + lam)) * cos

if self.ctrl_state:

return numpy.array(

[[1, 0, 0, 0], [0, a, 0, b], [0, 0, 1, 0], [0, c, 0, d]], dtype=dtype

)

else:

return numpy.array(

[[a, 0, b, 0], [0, 1, 0, 0], [c, 0, d, 0], [0, 0, 0, 1]], dtype=dtype

)

@property

def params(self):

return _CUGateParams(self)

@params.setter

def params(self, parameters):

# We need to skip `ControlledGate` in the inheritance tree, since it defines

# that all controlled gates are `(1-|c><c|).1 + |c><c|.base` for control-state `c`, which

# this class does _not_ satisfy (so it shouldn't really be a `ControlledGate`).

super(ControlledGate, type(self)).params.fset(self, parameters)

self.base_gate.params = parameters[:-1]

def __deepcopy__(self, memo=None):

# We have to override this because `ControlledGate` doesn't copy the `_params` list,

# assuming that `params` will be a view onto the base gate's `_params`.

memo = memo if memo is not None else {}

out = super().__deepcopy__(memo)

out._params = _copy.deepcopy(out._params, memo)