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Merge pull request #1441 from qiboteam/doc_fix
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Fix math rendering in some docsstrings of the `quantum_info` module
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@renatomello
renatomello authored Sep 13, 2024
2 parents 218f298 + 5a7fe1e commit 33efb91
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2 changes: 1 addition & 1 deletion doc/source/api-reference/qibo.rst
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Expand Up @@ -2382,7 +2382,7 @@ Hellinger fidelity
Hellinger shot error
""""""""""""""""""""

.. autofunction:: qibo.quantum_info.hellinger_fidelity
.. autofunction:: qibo.quantum_info.hellinger_shot_error


Haar integral
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7 changes: 5 additions & 2 deletions src/qibo/gates/channels.py
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Expand Up @@ -48,8 +48,11 @@ def to_choi(self, nqubits: Optional[int] = None, order: str = "row", backend=Non
of the Kraus channel :math:`\\{K_{\\alpha}\\}_{\\alpha}`.
.. math::
\\mathcal{E} = \\sum_{\\alpha} \\, |K_{\\alpha}\\rangle\\rangle
\\langle\\langle K_{\\alpha}|
\\mathcal{E} = \\sum_{\\alpha} \\, |K_{\\alpha})(K_{\\alpha}| \\, ,
where :math:`|K_{\\alpha})` is the vectorization of the Kraus operator
:math:`K_{\\alpha}`.
For a definition of vectorization, see :func:`qibo.quantum_info.vectorization`.
Args:
nqubits (int, optional): total number of qubits to be considered
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40 changes: 21 additions & 19 deletions src/qibo/quantum_info/basis.py
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Expand Up @@ -28,15 +28,15 @@ def pauli_basis(
vectorize (bool, optional): If ``False``, returns a nested array with
all Pauli matrices. If ``True``, retuns an array where every
row is a vectorized Pauli matrix. Defaults to ``False``.
sparse (bool, optional) If ``True``, retuns Pauli basis in a sparse
representation. Default is ``False``.
sparse (bool, optional): If ``True``, retuns Pauli basis in a sparse
representation. Defaults to ``False``.
order (str, optional): If ``"row"``, vectorization of Pauli basis is
performed row-wise. If ``"column"``, vectorization is performed
column-wise. If ``"system"``, system-wise vectorization is
performed. If ``vectorization=False``, then ``order=None`` is
forced. Default is ``None``.
forced. Defaults to ``None``.
pauli_order (str, optional): corresponds to the order of 4 single-qubit
Pauli elements. Default is "IXYZ".
Pauli elements. Defaults to ``"IXYZ"``.
backend (:class:`qibo.backends.abstract.Backend`, optional): backend
to be used in the execution. If ``None``, it uses
:class:`qibo.backends.GlobalBackend`. Defaults to ``None``.
Expand Down Expand Up @@ -148,15 +148,12 @@ def comp_basis_to_pauli(
The unitary :math:`U` is given by
.. math::
U = \\sum_{k = 0}^{d^{2} - 1} \\, \\ketbra{k}{P_{k}} \\,\\, ,
U = \\sum_{k = 0}^{d^{2} - 1} \\, |k)(P_{k}| \\,\\, ,
where :math:`\\ket{P_{k}}` is the system-vectorization of the :math:`k`-th
Pauli operator :math:`P_{k}`, and :math:`\\ket{k}` is the computational
basis element.
When converting a state :math:`\\ket{\\rho}` to its Pauli-Liouville
representation :math:`\\ket{\\rho'}`, one should use ``order="system"``
in :func:`vectorization`.
where :math:`|P_{k})` is the vectorization of the :math:`k`-th
Pauli operator :math:`P_{k}`, and :math:`|k)` is the vectorization
of the :math:`k`-th computational basis element.
For a definition of vectorization, see :func:`qibo.quantum_info.vectorization`.
Example:
.. code-block:: python
Expand All @@ -174,13 +171,13 @@ def comp_basis_to_pauli(
normalize (bool, optional): If ``True``, converts to the
Pauli basis. Defaults to ``False``.
sparse (bool, optional): If ``True``, returns unitary matrix in
sparse representation. Default is ``False``.
sparse representation. Defaults to ``False``.
order (str, optional): If ``"row"``, vectorization of Pauli basis is
performed row-wise. If ``"column"``, vectorization is performed
column-wise. If ``"system"``, system-wise vectorization is
performed. Default is ``"row"``.
performed. Defaults to ``"row"``.
pauli_order (str, optional): corresponds to the order of 4 single-qubit
Pauli elements. Default is "IXYZ".
Pauli elements. Defaults to ``"IXYZ"``.
backend (:class:`qibo.backends.abstract.Backend`, optional): backend to be
used in the execution. If ``None``, it uses
:class:`qibo.backends.GlobalBackend`. Defaults to ``None``.
Expand Down Expand Up @@ -237,20 +234,25 @@ def pauli_to_comp_basis(
The unitary :math:`U` is given by
.. math::
U = \\sum_{k = 0}^{d^{2} - 1} \\, \\ketbra{P_{k}}{b_{k}} \\, .
U = \\sum_{k = 0}^{d^{2} - 1} \\, |P_{k})(b_{k}| \\, ,
where :math:`|P_{k})` is the vectorization of the :math:`k`-th
Pauli operator :math:`P_{k}`, and :math:`|k)` is the vectorization
of the :math:`k`-th computational basis element.
For a definition of vectorization, see :func:`qibo.quantum_info.vectorization`.
Args:
nqubits (int): number of qubits.
normalize (bool, optional): If ``True``, converts to the
Pauli basis. Defaults to ``False``.
sparse (bool, optional): If ``True``, returns unitary matrix in
sparse representation. Default is ``False``.
sparse representation. Defaults to ``False``.
order (str, optional): If ``"row"``, vectorization of Pauli basis is
performed row-wise. If ``"column"``, vectorization is performed
column-wise. If ``"system"``, system-wise vectorization is
performed. Default is ``"row"``.
performed. Defaults to ``"row"``.
pauli_order (str, optional): corresponds to the order of 4 single-qubit
Pauli elements. Default is "IXYZ".
Pauli elements. Defaults to ``"IXYZ"``.
backend (:class:`qibo.backends.abstract.Backend`, optional): backend to be
used in the execution. If ``None``, it uses
:class:`qibo.backends.GlobalBackend`. Defaults to ``None``.
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4 changes: 2 additions & 2 deletions src/qibo/quantum_info/entropies.py
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Expand Up @@ -78,7 +78,7 @@ def classical_relative_entropy(prob_dist_p, prob_dist_q, base: float = 2, backen
For probabilities :math:`\\mathbf{p}` and :math:`\\mathbf{q}`, it is defined as
..math::
.. math::
D(\\mathbf{p} \\, \\| \\, \\mathbf{q}) = \\sum_{x} \\, \\mathbf{p}(x) \\,
\\log\\left( \\frac{\\mathbf{p}(x)}{\\mathbf{q}(x)} \\right) \\, .
Expand Down Expand Up @@ -680,7 +680,7 @@ def relative_renyi_entropy(
This is known as the `min-relative entropy <https://arxiv.org/abs/1310.7178>`_.
.. note::
Function raises ``NotImplementedError`` when ``target`` :math:`sigma`
Function raises ``NotImplementedError`` when ``target`` :math:`\\sigma`
is a pure state and :math:`\\alpha > 1`. This is due to the fact that
it is not possible to calculate :math:`\\sigma^{1 - \\alpha}` when
:math:`\\alpha > 1` and :math:`\\sigma` is a projector, i.e. a singular matrix.
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34 changes: 20 additions & 14 deletions src/qibo/quantum_info/superoperator_transformations.py
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Expand Up @@ -14,18 +14,17 @@


def vectorization(state, order: str = "row", backend=None):
"""Returns state :math:`\\rho` in its Liouville
representation :math:`|\\rho\\rangle\\rangle`.
"""Returns state :math:`\\rho` in its Liouville representation :math:`|\\rho)`.
If ``order="row"``, then:
.. math::
|\\rho\\rangle\\rangle = \\sum_{k, l} \\, \\rho_{kl} \\, \\ket{k} \\otimes \\ket{l}
|\\rho) = \\sum_{k, l} \\, \\rho_{kl} \\, \\ket{k} \\otimes \\ket{l}
If ``order="column"``, then:
.. math::
|\\rho\\rangle\\rangle = \\sum_{k, l} \\, \\rho_{kl} \\, \\ket{l} \\otimes \\ket{k}
|\\rho) = \\sum_{k, l} \\, \\rho_{kl} \\, \\ket{l} \\otimes \\ket{k}
Args:
state: state vector or density matrix.
Expand Down Expand Up @@ -88,12 +87,12 @@ def vectorization(state, order: str = "row", backend=None):

def unvectorization(state, order: str = "row", backend=None):
"""Returns state :math:`\\rho` from its Liouville
representation :math:`|\\rho\\rangle\\rangle`. This operation is
representation :math:`|\\rho)`. This operation is
the inverse function of :func:`vectorization`, i.e.
.. math::
\\begin{align}
\\rho &= \\text{unvectorization}(|\\rho\\rangle\\rangle) \\nonumber \\\\
\\rho &= \\text{unvectorization}(|\\rho)) \\nonumber \\\\
&= \\text{unvectorization}(\\text{vectorization}(\\rho)) \\nonumber
\\end{align}
Expand Down Expand Up @@ -152,9 +151,9 @@ def to_choi(channel, order: str = "row", backend=None):
"""Converts quantum ``channel`` :math:`U` to its Choi representation :math:`\\Lambda`.
.. math::
\\Lambda = | U \\rangle\\rangle \\langle\\langle U | \\, ,
\\Lambda = | U ) ( U | \\, ,
where :math:`| \\cdot \\rangle\\rangle` is the :func:`qibo.quantum_info.vectorization`
where :math:`| \\cdot )` is the :func:`qibo.quantum_info.vectorization`
operation.
Args:
Expand Down Expand Up @@ -376,7 +375,9 @@ def choi_to_kraus(
.. math::
\\Lambda = \\sum_{\\alpha} \\, \\lambda_{\\alpha}^{2} \\,
|\\tilde{K}_{\\alpha}\\rangle\\rangle \\langle\\langle \\tilde{K}_{\\alpha}| \\, .
|\\tilde{K}_{\\alpha})(\\tilde{K}_{\\alpha}| \\, .
where :math:`|\\cdot)` is the :func:`qibo.quantum_info.vectorization` operation.
This is the spectral decomposition of :math:`\\Lambda`, Hence, the set
:math:`\\{\\lambda_{\\alpha}, \\, \\tilde{K}_{\\alpha}\\}_{\\alpha}`
Expand All @@ -385,7 +386,7 @@ def choi_to_kraus(
.. math::
K_{\\alpha} = \\lambda_{\\alpha} \\,
\\text{unvectorization}(|\\tilde{K}_{\\alpha}\\rangle\\rangle) \\, .
\\text{unvectorization}(|\\tilde{K}_{\\alpha})) \\, .
If :math:`\\mathcal{E}` is not CP, then spectral composition is replaced by
a singular value decomposition (SVD), i.e.
Expand Down Expand Up @@ -638,7 +639,11 @@ def kraus_to_choi(kraus_ops, order: str = "row", backend=None):
of quantum channel to its Choi representation :math:`\\Lambda`.
.. math::
\\Lambda = \\sum_{\\alpha} \\, |K_{\\alpha}\\rangle\\rangle \\langle\\langle K_{\\alpha}|
\\Lambda = \\sum_{\\alpha} \\, |K_{\\alpha})( K_{\\alpha}|
where :math:`|K_{\\alpha})` is the vectorization of the Kraus operator
:math:`K_{\\alpha}`.
For a definition of vectorization, see :func:`qibo.quantum_info.vectorization`.
Args:
kraus_ops (list): List of Kraus operators as pairs ``(qubits, Ak)``
Expand Down Expand Up @@ -757,10 +762,11 @@ def kraus_to_chi(
of quantum channel to its :math:`\\chi`-matrix representation.
.. math::
\\chi = \\sum_{\\alpha} \\, |c_{\\alpha}\\rangle\\rangle \\langle\\langle c_{\\alpha}|,
\\chi = \\sum_{\\alpha} \\, |c_{\\alpha})( c_{\\alpha}|,
where :math:`|c_{\\alpha}\\rangle\\rangle \\cong |K_{\\alpha}\\rangle\\rangle`
in Pauli-Liouville basis.
where :math:`|c_{\\alpha}) \\cong |K_{\\alpha})` in Pauli-Liouville basis,
and :math:`| \\cdot )` is the :func:`qibo.quantum_info.vectorization`
operation.
Args:
kraus_ops (list): List of Kraus operators as pairs ``(qubits, Ak)``
Expand Down

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