prompt: You will be provided with a partial code base and an issue statement explaining a problem to resolve.
  <issue>
  Documentation of RVGate is incorrect
  ### Environment
  
  N/A
  
  ### What is happening?
  
  Received this in an email:
  >Hi, I think I found some errors in the Qiskit documentation at
  <https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.RVGate>
  and I'm contacting you because you look like the two people who most recently edited the source file at
  <https://github.com/Qiskit/qiskit/blob/stable/0.46/qiskit/circuit/library/generalized_gates/rv.py>
  The matrix representation given in the documentation seems to be wrong.  I compared it to the definition given in
  <https://arxiv.org/pdf/2104.14875.pdf>
  on page 4, equation 1, we see the definition of the rotation matrix.  It almost matches the definition given in the documentation at
  <https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.RVGate>
  except for two mistakes: the "sinc" function should be "sin", and the angle should be divided by two.  This can be compared to the source code at
  <https://github.com/Qiskit/qiskit/blob/stable/0.46/qiskit/circuit/library/generalized_gates/rv.py>
  at lines 86 and 87, where we see the angle divided by two, and we see the use of the sin and cos functions.
  
  ### How can we reproduce the issue?
  
  N/A
  
  ### What should happen?
  
  N/A
  
  ### Any suggestions?
  
  _No response_
  
  </issue>
  <code>
  [start of qiskit/circuit/library/generalized_gates/rv.py]
  1 # This code is part of Qiskit.
  2 #
  3 # (C) Copyright IBM 2017, 2020
  4 #
  5 # This code is licensed under the Apache License, Version 2.0. You may
  6 # obtain a copy of this license in the LICENSE.txt file in the root directory
  7 # of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
  8 #
  9 # Any modifications or derivative works of this code must retain this
  10 # copyright notice, and modified files need to carry a notice indicating
  11 # that they have been altered from the originals.
  12 
  13 """Rotation around an arbitrary axis on the Bloch sphere."""
  14 
  15 import numpy
  16 from qiskit.circuit.gate import Gate
  17 from qiskit.circuit.exceptions import CircuitError
  18 
  19 
  20 class RVGate(Gate):
  21     r"""Rotation around arbitrary rotation axis :math:`v` where :math:`|v|` is
  22     angle of rotation in radians.
  23 
  24     Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`
  25     with the :meth:`~qiskit.circuit.QuantumCircuit.rv` method.
  26 
  27     **Circuit symbol:**
  28 
  29     .. parsed-literal::
  30 
  31              ┌─────────────────┐
  32         q_0: ┤ RV(v_x,v_y,v_z) ├
  33              └─────────────────┘
  34 
  35     **Matrix Representation:**
  36 
  37     .. math::
  38 
  39         \newcommand{\rotationangle}{|\vec{v}|}
  40         \newcommand{\sinc}{\text{sinc}}
  41             R(\vec{v}) = e^{-i \vec{v}\cdot\vec{\sigma}} =
  42                 \begin{pmatrix}
  43                     \cos\left(\rotationangle\right) -i v_z \sinc\left(\rotationangle\right)
  44                     & -(i v_x + v_y) \sinc\left(\rotationangle\right) \\
  45                     -(i v_x - v_y) \sinc\left(\rotationangle\right)
  46                     & \cos\left(\rotationangle\right) + i v_z \sinc\left(\rotationangle\right)
  47                 \end{pmatrix}
  48     """
  49 
  50     def __init__(self, v_x, v_y, v_z, basis="U"):
  51         """Create new rv single-qubit gate.
  52 
  53         Args:
  54             v_x (float): x-component
  55             v_y (float): y-component
  56             v_z (float): z-component
  57             basis (str, optional): basis (see
  58                 :class:`~qiskit.synthesis.one_qubit.one_qubit_decompose.OneQubitEulerDecomposer`)
  59         """
  60         # pylint: disable=cyclic-import
  61         from qiskit.synthesis.one_qubit.one_qubit_decompose import OneQubitEulerDecomposer
  62 
  63         super().__init__("rv", 1, [v_x, v_y, v_z])
  64         self._decomposer = OneQubitEulerDecomposer(basis=basis)
  65 
  66     def _define(self):
  67         try:
  68             self.definition = self._decomposer(self.to_matrix())
  69         except TypeError as ex:
  70             raise CircuitError(
  71                 f"The {self.name} gate cannot be decomposed with unbound parameters"
  72             ) from ex
  73 
  74     def inverse(self):
  75         """Invert this gate."""
  76         vx, vy, vz = self.params
  77         return RVGate(-vx, -vy, -vz)
  78 
  79     def to_matrix(self):
  80         """Return a numpy.array for the R(v) gate."""
  81         v = numpy.asarray(self.params, dtype=float)
  82         angle = numpy.sqrt(v.dot(v))
  83         if angle == 0:
  84             return numpy.array([[1, 0], [0, 1]])
  85         nx, ny, nz = v / angle
  86         sin = numpy.sin(angle / 2)
  87         cos = numpy.cos(angle / 2)
  88         return numpy.array(
  89             [
  90                 [cos - 1j * nz * sin, (-ny - 1j * nx) * sin],
  91                 [(ny - 1j * nx) * sin, cos + 1j * nz * sin],
  92             ]
  93         )
  94
  [end of qiskit/circuit/library/generalized_gates/rv.py]
  </code>
  
  I need you to solve this issue by generating a single patch file that I can apply directly to this repository using git apply. Please respond with a single patch file in the following format.
  <patch>
  diff --git a/file.py b/file.py
  --- a/file.py
  +++ b/file.py
  @@ -1,27 +1,35 @@
   def euclidean(a, b):
  -    while b:
  -        a, b = b, a % b
  -    return a
  +    if b == 0:
  +        return a
  +    return euclidean(b, a % b)
   
   
   def bresenham(x0, y0, x1, y1):
       points = []
       dx = abs(x1 - x0)
       dy = abs(y1 - y0)
  -    sx = 1 if x0 < x1 else -1
  -    sy = 1 if y0 < y1 else -1
  -    err = dx - dy
  +    x, y = x0, y0
  +    sx = -1 if x0 > x1 else 1
  +    sy = -1 if y0 > y1 else 1
   
  -    while True:
  -        points.append((x0, y0))
  -        if x0 == x1 and y0 == y1:
  -            break
  -        e2 = 2 * err
  -        if e2 > -dy:
  -            err -= dy
  -            x0 += sx
  -        if e2 < dx:
  -            err += dx
  -            y0 += sy
  +    if dx > dy:
  +        err = dx / 2.0
  +        while x != x1:
  +            points.append((x, y))
  +            err -= dy
  +            if err < 0:
  +                y += sy
  +                err += dx
  +            x += sx
  +    else:
  +        err = dy / 2.0
  +        while y != y1:
  +            points.append((x, y))
  +            err -= dx
  +            if err < 0:
  +                x += sx
  +                err += dy
  +            y += sy
  +
  +    points.append((x, y))
       return points
  
  </patch>
golden_diff: diff --git a/qiskit/circuit/library/generalized_gates/rv.py b/qiskit/circuit/library/generalized_gates/rv.py
  --- a/qiskit/circuit/library/generalized_gates/rv.py
  +++ b/qiskit/circuit/library/generalized_gates/rv.py
  @@ -18,7 +18,7 @@
   
   
   class RVGate(Gate):
  -    r"""Rotation around arbitrary rotation axis :math:`v` where :math:`|v|` is
  +    r"""Rotation around arbitrary rotation axis :math:`\vec{v}` where :math:`\|\vec{v}\|_2` is
       angle of rotation in radians.
   
       Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`
  @@ -36,14 +36,17 @@
   
       .. math::
   
  -        \newcommand{\rotationangle}{|\vec{v}|}
  -        \newcommand{\sinc}{\text{sinc}}
  -            R(\vec{v}) = e^{-i \vec{v}\cdot\vec{\sigma}} =
  +        \newcommand{\rotationangle}{\frac{\|\vec{v}\|_2}{2}}
  +            R(\vec{v}) = e^{-i \vec{v}\cdot\vec{\sigma} / 2} =
                   \begin{pmatrix}
  -                    \cos\left(\rotationangle\right) -i v_z \sinc\left(\rotationangle\right)
  -                    & -(i v_x + v_y) \sinc\left(\rotationangle\right) \\
  -                    -(i v_x - v_y) \sinc\left(\rotationangle\right)
  -                    & \cos\left(\rotationangle\right) + i v_z \sinc\left(\rotationangle\right)
  +                    \cos\left(\rotationangle\right)
  +                    -i \frac{v_z}{\|\vec{v}\|_2} \sin\left(\rotationangle\right)
  +                    & -(i \frac{v_x}{\|\vec{v}\|_2}
  +                    + \frac{v_y}{\|\vec{v}\|_2}) \sin\left(\rotationangle\right) \\
  +                    -(i \frac{v_x}{\|\vec{v}\|_2}
  +                    - \frac{v_y}{\|\vec{v}\|_2}) \sin\left(\rotationangle\right)
  +                    & \cos\left(\rotationangle\right)
  +                    + i \frac{v_z}{\|\vec{v}\|_2} \sin\left(\rotationangle\right)
                   \end{pmatrix}
       """
  
verification_info: {"golden_diff": "diff --git a/qiskit/circuit/library/generalized_gates/rv.py b/qiskit/circuit/library/generalized_gates/rv.py\n--- a/qiskit/circuit/library/generalized_gates/rv.py\n+++ b/qiskit/circuit/library/generalized_gates/rv.py\n@@ -18,7 +18,7 @@\n \n \n class RVGate(Gate):\n-    r\"\"\"Rotation around arbitrary rotation axis :math:`v` where :math:`|v|` is\n+    r\"\"\"Rotation around arbitrary rotation axis :math:`\\vec{v}` where :math:`\\|\\vec{v}\\|_2` is\n     angle of rotation in radians.\n \n     Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`\n@@ -36,14 +36,17 @@\n \n     .. math::\n \n-        \\newcommand{\\rotationangle}{|\\vec{v}|}\n-        \\newcommand{\\sinc}{\\text{sinc}}\n-            R(\\vec{v}) = e^{-i \\vec{v}\\cdot\\vec{\\sigma}} =\n+        \\newcommand{\\rotationangle}{\\frac{\\|\\vec{v}\\|_2}{2}}\n+            R(\\vec{v}) = e^{-i \\vec{v}\\cdot\\vec{\\sigma} / 2} =\n                 \\begin{pmatrix}\n-                    \\cos\\left(\\rotationangle\\right) -i v_z \\sinc\\left(\\rotationangle\\right)\n-                    & -(i v_x + v_y) \\sinc\\left(\\rotationangle\\right) \\\\\n-                    -(i v_x - v_y) \\sinc\\left(\\rotationangle\\right)\n-                    & \\cos\\left(\\rotationangle\\right) + i v_z \\sinc\\left(\\rotationangle\\right)\n+                    \\cos\\left(\\rotationangle\\right)\n+                    -i \\frac{v_z}{\\|\\vec{v}\\|_2} \\sin\\left(\\rotationangle\\right)\n+                    & -(i \\frac{v_x}{\\|\\vec{v}\\|_2}\n+                    + \\frac{v_y}{\\|\\vec{v}\\|_2}) \\sin\\left(\\rotationangle\\right) \\\\\n+                    -(i \\frac{v_x}{\\|\\vec{v}\\|_2}\n+                    - \\frac{v_y}{\\|\\vec{v}\\|_2}) \\sin\\left(\\rotationangle\\right)\n+                    & \\cos\\left(\\rotationangle\\right)\n+                    + i \\frac{v_z}{\\|\\vec{v}\\|_2} \\sin\\left(\\rotationangle\\right)\n                 \\end{pmatrix}\n     \"\"\"\n", "issue": "Documentation of RVGate is incorrect\n### Environment\n\nN/A\n\n### What is happening?\n\nReceived this in an email:\r\n>Hi, I think I found some errors in the Qiskit documentation at\r\n<https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.RVGate>\r\nand I'm contacting you because you look like the two people who most recently edited the source file at\r\n<https://github.com/Qiskit/qiskit/blob/stable/0.46/qiskit/circuit/library/generalized_gates/rv.py>\r\nThe matrix representation given in the documentation seems to be wrong.  I compared it to the definition given in\r\n<https://arxiv.org/pdf/2104.14875.pdf>\r\non page 4, equation 1, we see the definition of the rotation matrix.  It almost matches the definition given in the documentation at\r\n<https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.RVGate>\r\nexcept for two mistakes: the \"sinc\" function should be \"sin\", and the angle should be divided by two.  This can be compared to the source code at\r\n<https://github.com/Qiskit/qiskit/blob/stable/0.46/qiskit/circuit/library/generalized_gates/rv.py>\r\nat lines 86 and 87, where we see the angle divided by two, and we see the use of the sin and cos functions.\n\n### How can we reproduce the issue?\n\nN/A\n\n### What should happen?\n\nN/A\n\n### Any suggestions?\n\n_No response_\n", "before_files": [{"content": "# This code is part of Qiskit.\n#\n# (C) Copyright IBM 2017, 2020\n#\n# This code is licensed under the Apache License, Version 2.0. You may\n# obtain a copy of this license in the LICENSE.txt file in the root directory\n# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.\n#\n# Any modifications or derivative works of this code must retain this\n# copyright notice, and modified files need to carry a notice indicating\n# that they have been altered from the originals.\n\n\"\"\"Rotation around an arbitrary axis on the Bloch sphere.\"\"\"\n\nimport numpy\nfrom qiskit.circuit.gate import Gate\nfrom qiskit.circuit.exceptions import CircuitError\n\n\nclass RVGate(Gate):\n    r\"\"\"Rotation around arbitrary rotation axis :math:`v` where :math:`|v|` is\n    angle of rotation in radians.\n\n    Can be applied to a :class:`~qiskit.circuit.QuantumCircuit`\n    with the :meth:`~qiskit.circuit.QuantumCircuit.rv` method.\n\n    **Circuit symbol:**\n\n    .. parsed-literal::\n\n             \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510\n        q_0: \u2524 RV(v_x,v_y,v_z) \u251c\n             \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518\n\n    **Matrix Representation:**\n\n    .. math::\n\n        \\newcommand{\\rotationangle}{|\\vec{v}|}\n        \\newcommand{\\sinc}{\\text{sinc}}\n            R(\\vec{v}) = e^{-i \\vec{v}\\cdot\\vec{\\sigma}} =\n                \\begin{pmatrix}\n                    \\cos\\left(\\rotationangle\\right) -i v_z \\sinc\\left(\\rotationangle\\right)\n                    & -(i v_x + v_y) \\sinc\\left(\\rotationangle\\right) \\\\\n                    -(i v_x - v_y) \\sinc\\left(\\rotationangle\\right)\n                    & \\cos\\left(\\rotationangle\\right) + i v_z \\sinc\\left(\\rotationangle\\right)\n                \\end{pmatrix}\n    \"\"\"\n\n    def __init__(self, v_x, v_y, v_z, basis=\"U\"):\n        \"\"\"Create new rv single-qubit gate.\n\n        Args:\n            v_x (float): x-component\n            v_y (float): y-component\n            v_z (float): z-component\n            basis (str, optional): basis (see\n                :class:`~qiskit.synthesis.one_qubit.one_qubit_decompose.OneQubitEulerDecomposer`)\n        \"\"\"\n        # pylint: disable=cyclic-import\n        from qiskit.synthesis.one_qubit.one_qubit_decompose import OneQubitEulerDecomposer\n\n        super().__init__(\"rv\", 1, [v_x, v_y, v_z])\n        self._decomposer = OneQubitEulerDecomposer(basis=basis)\n\n    def _define(self):\n        try:\n            self.definition = self._decomposer(self.to_matrix())\n        except TypeError as ex:\n            raise CircuitError(\n                f\"The {self.name} gate cannot be decomposed with unbound parameters\"\n            ) from ex\n\n    def inverse(self):\n        \"\"\"Invert this gate.\"\"\"\n        vx, vy, vz = self.params\n        return RVGate(-vx, -vy, -vz)\n\n    def to_matrix(self):\n        \"\"\"Return a numpy.array for the R(v) gate.\"\"\"\n        v = numpy.asarray(self.params, dtype=float)\n        angle = numpy.sqrt(v.dot(v))\n        if angle == 0:\n            return numpy.array([[1, 0], [0, 1]])\n        nx, ny, nz = v / angle\n        sin = numpy.sin(angle / 2)\n        cos = numpy.cos(angle / 2)\n        return numpy.array(\n            [\n                [cos - 1j * nz * sin, (-ny - 1j * nx) * sin],\n                [(ny - 1j * nx) * sin, cos + 1j * nz * sin],\n            ]\n        )\n", "path": "qiskit/circuit/library/generalized_gates/rv.py"}]}