Added options for docstring

pennylane/ops/qutrit/channel.py

@@ -27,21 +27,110 @@ class QutritDepolarizingChannel(Channel):
  r"""
  Single-qutrit symmetrically depolarizing error channel.
  This channel is modelled by the following Kraus matrices:
+
+ .. math::
+ K_0 = \sqrt{1-p} \begin{bmatrix}
+ 1 & 0 & 0\\
+ 0 & 1 & 0\\
+ 0 & 0 & 1
+ \end{bmatrix}
+
+ .. math::
+ K_1 = \sqrt{p/9}\begin{bmatrix}
+ 0 & 1 & 0 \\
+ 0 & 0 & 1 \\
+ 1 & 0 & 0
+ \end{bmatrix}
+
+ .. math::
+ K_2 = \sqrt{p/9}\begin{bmatrix}
+ 0 & 0 & 1 \\
+ 1 & 0 & 0 \\
+ 0 & 1 & 0
+ \end{bmatrix}
+
+ .. math::
+ K_3 = \sqrt{p/9}\begin{bmatrix}
+ 1 & 0 & 0\\
+ 0 & \omega & 0\\
+ 0 & 0 & \omega^2
+ \end{bmatrix}
+ .. math::
+ K_4 = \sqrt{p/9}\begin{bmatrix}
+ 1 & 0 & 0\\
+ 0 & \omega^2 & 0\\
+ 0 & 0 & \omega^4
+ \end{bmatrix}
+
+ .. math::
+ K_5 = \sqrt{p/9}\begin{bmatrix}
+ 0 & \omega & 0 \\
+ 0 & 0 & \omega^2 \\
+ 1 & 0 & 0
+ \end{bmatrix}
+
+ .. math::
+ K_6 = \sqrt{p/9}\begin{bmatrix}
+ 0 & 0 & omega^2 \\
+ 1 & 0 & 0 \\
+ 0 & omega & 0
+ \end{bmatrix}
+
+ .. math::
+ K_7 = \sqrt{p/9}\begin{bmatrix}
+ 0 & \omega^2 & 0 \\
+ 0 & 0 & \omega \\
+ 1 & 0 & 0
+ \end{bmatrix}
+
+ .. math::
+ K_8 = \sqrt{p/9}\begin{bmatrix}
+ 0 & 0 & \omega \\
+ 1 & 0 & 0 \\
+ 0 & \omega^2 & 0
+ \end{bmatrix}
+
+ OR:
+ .. math::
+ X = \sqrt{p/9}\begin{bmatrix}
+ 0 & 1 & 0 \\
+ 0 & 0 & 1 \\
+ 1 & 0 & 0
+ \end{bmatrix}
+
+ .. math::
+ Z = \sqrt{p/9}\begin{bmatrix}
+ 1 & 0 & 0\\
+ 0 & \omega & 0\\
+ 0 & 0 & \omega^2
+ \end{bmatrix}
+
+ .. math::
+ K_0 = \sqrt{1-p} \begin{bmatrix}
+ 1 & 0 & 0\\
+ 0 & 1 & 0\\
+ 0 & 0 & 1
+ \end{bmatrix}
+
+ .. math::
+ K_{i,j} = \sqrt{p/9}X^iZ^j, \{i,j\} \neq \{0,0\}
+
+ Where :math:`\omega=\exp(\frac{2\pi}{3})` is the phase defining the third root of identity.
  where :math:`p \in [0, 1]` is the depolarization probability and is equally
- divided in the application of all TODO operations.
+ divided in the application of all qutrit Pauli operators.
  .. note::
- Multiple equivalent definitions of the Kraus operators :math:`\{K_0 \ldots K_8\}` exist in #TODO fix liturature
- the literature [`1 <https://michaelnielsen.org/qcqi/>`_] (Eqs. 8.102-103). Here, we adopt the
- one from Eq. 8.103, which is also presented in [`2 <http://theory.caltech.edu/~preskill/ph219/chap3_15.pdf>`_] (Eq. 3.85). # TODO change sources
+ The Kraus operators :math:`\{K_0 \ldots K_8\} used are the representations of the single qutrit Pauli group.
+ These Kraus operators are defined in [`1 <https://arxiv.org/pdf/quant-ph/9802007>`_] (Eq. 5).
+ [`2 <https://ui.adsabs.harvard.edu/link_gateway/1988JMP....29..665P/doi:10.1063/1.52800611.>`_] (Eq. 3.2)
  For this definition, please make a note of the following:
  * For :math:`p = 0`, the channel will be an Identity channel, i.e., a noise-free channel.
- * For :math:`p = TODO \frac{3}{4}`, the channel will be a fully depolarizing channel.
- * For :math:`p = 1`, the channel will be a uniform TODO error channel.
+ * For :math:`p = \frac{8}{9}`, the channel will be a fully depolarizing channel.
+ * For :math:`p = 1`, the channel will be a uniform error channel.
  **Details:**
  * Number of wires: 1
  * Number of parameters: 1
  Args:
- p (float): Each TODO gate is applied with probability :math:`\frac{p}{3}`
+ p (float): Each gate is applied with probability :math:`\frac{p}{9}`
  wires (Sequence[int] or int): the wire the channel acts on
  id (str or None): String representing the operation (optional)
  """
@@ -56,48 +145,46 @@ def __init__(self, p, wires, id=None):
  def compute_kraus_matrices(p): # pylint:disable=arguments-differ
  r"""Kraus matrices representing the depolarizing channel.
  Args:
- p (float): each TODO gate is applied with probability :math:`\frac{p}{9}`
+ p (float): each gate is applied with probability :math:`\frac{p}{9}`
  Returns:
  list (array): list of Kraus matrices
  **Example**
- >>> qml.QutritDepolarizingChannel.compute_kraus_matrices(0.5)
- [array([[0.74535599+0.j, 0. +0.j, 0. +0.j],
- [0. +0.j, 0.74535599+0.j, 0. +0.j],
- [0. +0.j, 0. +0.j, 0.74535599+0.j]]), array([[0. , 0.23570226, 0. ],
- [0. , 0. , 0.23570226],
- [0.23570226, 0. , 0. ]]), array([[0. , 0. , 0.23570226],
- [0.23570226, 0. , 0. ],
- [0. , 0.23570226, 0. ]]), array([[ 0.23570226+0.j , 0. +0.j ,
- 0. +0.j ],
- [ 0. +0.j , -0.11785113+0.20412415j,
- 0. +0.j ],
- [ 0. +0.j , 0. +0.j ,
- -0.11785113-0.20412415j]]), array([[ 0. +0.j , -0.11785113+0.20412415j,
- 0. +0.j ],
- [ 0. +0.j , 0. +0.j ,
- -0.11785113-0.20412415j],
- [ 0.23570226+0.j , 0. +0.j ,
- 0. +0.j ]]), array([[ 0. +0.j , 0. +0.j ,
- -0.11785113-0.20412415j],
- [ 0.23570226+0.j , 0. +0.j ,
- 0. +0.j ],
- [ 0. +0.j , -0.11785113+0.20412415j,
- 0. +0.j ]]), array([[ 0.23570226+0.j , 0. +0.j ,
- 0. +0.j ],
- [ 0. +0.j , -0.11785113-0.20412415j,
- 0. +0.j ],
- [ 0. +0.j , 0. +0.j ,
- -0.11785113+0.20412415j]]), array([[ 0. +0.j , -0.11785113-0.20412415j,
- 0. +0.j ],
- [ 0. +0.j , 0. +0.j ,
- -0.11785113+0.20412415j],
- [ 0.23570226+0.j , 0. +0.j ,
- 0. +0.j ]]), array([[ 0. +0.j , 0. +0.j ,
- -0.11785113+0.20412415j],
- [ 0.23570226+0.j , 0. +0.j ,
- 0. +0.j ],
- [ 0. +0.j , -0.11785113-0.20412415j,
- 0. +0.j ]])]
+ >>> np.round(qml.QutritDepolarizingChannel.compute_kraus_matrices(0.5), 3)
+ array([[[ 0.745+0.j , 0. +0.j , 0. +0.j ],
+ [ 0. +0.j , 0.745+0.j , 0. +0.j ],
+ [ 0. +0.j , 0. +0.j , 0.745+0.j ]],
+
+ [[ 0. +0.j , 0.236+0.j , 0. +0.j ],
+ [ 0. +0.j , 0. +0.j , 0.236+0.j ],
+ [ 0.236+0.j , 0. +0.j , 0. +0.j ]],
+
+ [[ 0. +0.j , 0. +0.j , 0.236+0.j ],
+ [ 0.236+0.j , 0. +0.j , 0. +0.j ],
+ [ 0. +0.j , 0.236+0.j , 0. +0.j ]],
+
+ [[ 0.236+0.j , 0. +0.j , 0. +0.j ],
+ [ 0. +0.j , -0.118+0.204j, 0. +0.j ],
+ [ 0. +0.j , 0. +0.j , -0.118-0.204j]],
+
+ [[ 0. +0.j , -0.118+0.204j, 0. +0.j ],
+ [ 0. +0.j , 0. +0.j , -0.118-0.204j],
+ [ 0.236+0.j , 0. +0.j , 0. +0.j ]],
+
+ [[ 0. +0.j , 0. +0.j , -0.118-0.204j],
+ [ 0.236+0.j , 0. +0.j , 0. +0.j ],
+ [ 0. +0.j , -0.118+0.204j, 0. +0.j ]],
+
+ [[ 0.236+0.j , 0. +0.j , 0. +0.j ],
+ [ 0. +0.j , -0.118-0.204j, 0. +0.j ],
+ [ 0. +0.j , 0. +0.j , -0.118+0.204j]],
+
+ [[ 0. +0.j , -0.118-0.204j, 0. +0.j ],
+ [ 0. +0.j , 0. +0.j , -0.118+0.204j],
+ [ 0.236+0.j , 0. +0.j , 0. +0.j ]],
+
+ [[ 0. +0.j , 0. +0.j , -0.118+0.204j],
+ [ 0.236+0.j , 0. +0.j , 0. +0.j ],
+ [ 0. +0.j , -0.118-0.204j, 0. +0.j ]]])
  """
  if not math.is_abstract(p) and not 0.0 <= p <= 1.0:
  raise ValueError("p must be in the interval [0,1]")