fix: conflicts with master

doc/source/api-reference/qibo.rst

@@ -473,7 +473,8 @@ factor results in the mitigated Pauli expectation value :math:`\langle O\rangle_
 .. math::
    \langle O\rangle_{ideal} = \frac{\langle O\rangle_{noisy}}{\lambda}
 
-.. autofunction:: qibo.models.error_mitigation.apply_randomized_readout_mitigation
+This process can be implemented with the aforementioned
+:func:`qibo.models.error_mitigation.apply_randomized_readout_mitigation`.
 
 
 Zero Noise Extrapolation (ZNE)
@@ -2382,7 +2383,7 @@ Hellinger fidelity
 Hellinger shot error
 """"""""""""""""""""
 
-.. autofunction:: qibo.quantum_info.hellinger_fidelity
+.. autofunction:: qibo.quantum_info.hellinger_shot_error
 
 
 Haar integral

examples/shor/functions.py

@@ -393,8 +393,9 @@ def quantum_order_finding_semiclassical(N, a):
         circuit.add(gates.U1(q_reg, -angle))
         circuit.add(gates.H(q_reg))
         results.append(circuit.add(gates.M(q_reg, collapse=True)))
+    circuit.add(gates.M(q_reg))
 
-    circuit()  # execute
+    circuit(nshots=1)  # execute
     s = sum(int(r.symbols[0].outcome()) * (2**i) for i, r in enumerate(results))
     print(f"The quantum circuit measures s = {s}.\n")
     return s

src/qibo/backends/abstract.py

@@ -193,7 +193,7 @@ def execute_circuit(
     def execute_circuits(
         self, circuits, initial_states=None, nshots=None
     ):  # pragma: no cover
-        """Execute multiple :class:`qibo.models.circuit.Circuit`s in parallel."""
+        """Execute multiple :class:`qibo.models.circuit.Circuit` in parallel."""
         raise_error(NotImplementedError)
 
     @abc.abstractmethod

src/qibo/backends/clifford.py

@@ -90,13 +90,13 @@ def calculate_frequencies(self, samples):
         return collections.Counter(dict(zip(res, counts)))
 
     def zero_state(self, nqubits: int):
-        """Construct the zero state |00...00>.
+        """Construct the zero state :math`\\ket{00...00}`.
 
         Args:
-            nqubits (int): Number of qubits.
+            nqubits (int): number of qubits.
 
         Returns:
-            (ndarray): Symplectic matrix for the zero state.
+            ndarray: Symplectic matrix for the zero state.
         """
         identity = self.np.eye(nqubits)
         symplectic_matrix = self.np.zeros(

src/qibo/gates/channels.py

@@ -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

src/qibo/gates/measurements.py

@@ -17,27 +17,26 @@ class M(Gate):
             If the qubits to measure are held in an iterable (eg. list) the ``*``
             operator can be used, for example ``gates.M(*[0, 1, 4])`` or
             ``gates.M(*range(5))``.
-        register_name (str): Optional name of the register to distinguish it
+        register_name (str, optional): Optional name of the register to distinguish it
             from other registers when used in circuits.
         collapse (bool): Collapse the state vector after the measurement is
             performed. Can be used only for single shot measurements.
             If ``True`` the collapsed state vector is returned. If ``False``
             the measurement result is returned.
-        basis (:class:`qibo.gates.Gate`, str, list): Basis to measure.
+        basis (:class:`qibo.gates.Gate` or str or list, optional): Basis to measure.
             Can be either:
             - a qibo gate
             - the string representing the gate
             - a callable that accepts a qubit, for example: ``lambda q: gates.RX(q, 0.2)``
-            - a list of the above, if a different basis will be used for each
-              measurement qubit.
-            Default is Z.
-        p0 (dict): Optional bitflip probability map. Can be:
+            - a list of the above, if a different basis will be used for each measurement qubit.
+            Defaults is to :class:`qibo.gates.Z`.
+        p0 (dict, optional): bitflip probability map. Can be:
             A dictionary that maps each measured qubit to the probability
             that it is flipped, a list or tuple that has the same length
             as the tuple of measured qubits or a single float number.
             If a single float is given the same probability will be used
             for all qubits.
-        p1 (dict): Optional bitflip probability map for asymmetric bitflips.
+        p1 (dict, optional): bitflip probability map for asymmetric bitflips.
             Same as ``p0`` but controls the 1->0 bitflip probability.
             If ``p1`` is ``None`` then ``p0`` will be used both for 0->1 and
             1->0 bitflips.

src/qibo/models/error_mitigation.py

@@ -658,24 +658,24 @@ def apply_randomized_readout_mitigation(
         nshots (int, optional): number of shots. Defaults to :math:`10000`.
         ncircuits (int, optional): number of randomized circuits. Each of them uses
             ``int(nshots / ncircuits)`` shots. Defaults to 10.
-        qubit_map (list, optional): the qubit map. If None, a list of range of circuit's qubits is used.
-            Defaults to ``None``.
+        qubit_map (list, optional): the qubit map. If ``None``, a list of range of circuit's
+            qubits is used. Defaults to ``None``.
         seed (int or :class:`numpy.random.Generator`, optional): Either a generator of random
             numbers or a fixed seed to initialize a generator. If ``None``, initializes
             a generator with a random seed. Default: ``None``.
         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``.
 
-    Return:
+    Returns:
         :class:`qibo.measurements.CircuitResult`: the state of the input circuit with
             mitigated frequencies.
 
 
-    Reference:
+    References:
         1. Ewout van den Berg, Zlatko K. Minev et al,
         *Model-free readout-error mitigation for quantum expectation values*.
-           `arXiv:2012.09738 [quant-ph] <https://arxiv.org/abs/2012.09738>`_.
+        `arXiv:2012.09738 [quant-ph] <https://arxiv.org/abs/2012.09738>`_.
     """
     from qibo import Circuit  # pylint: disable=import-outside-toplevel
     from qibo.quantum_info import (  # pylint: disable=import-outside-toplevel

src/qibo/quantum_info/basis.py

@@ -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``.
@@ -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
@@ -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``.
@@ -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``.

src/qibo/quantum_info/clifford.py

@@ -297,16 +297,16 @@ def frequencies(self, binary: bool = True, registers: bool = False):
             of times each measured value/bitstring appears.
 
             If ``binary`` is ``True``
-                the keys of the `Counter` are in binary form, as strings of
-                :math:`0`s and :math`1`s.
+                the keys of the :class:`collections.Counter` are in binary form,
+                as strings of :math:`0` and :math`1`.
             If ``binary`` is ``False``
-                the keys of the ``Counter`` are integers.
+                the keys of the :class:`collections.Counter` are integers.
             If ``registers`` is ``True``
-                a `dict` of `Counter` s is returned where keys are the name of
-                each register.
+                a `dict` of :class:`collections.Counter` is returned where keys are
+                the name of each register.
             If ``registers`` is ``False``
-                a single ``Counter`` is returned which contains samples from all
-                the measured qubits, independently of their registers.
+                a single :class:`collections.Counter` is returned which contains samples
+                from all the measured qubits, independently of their registers.
         """
         measured_qubits = self.measurement_gate.target_qubits
         freq = self._backend.calculate_frequencies(self.samples(False))

src/qibo/quantum_info/entropies.py

@@ -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) \\, .
 
@@ -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.

src/qibo/quantum_info/superoperator_transformations.py

@@ -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.
@@ -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}
 
@@ -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:
@@ -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}`
@@ -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.
@@ -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)``
@@ -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)``
@@ -2130,8 +2136,6 @@ def _reshuffling(super_op, order: str = "row", backend=None):
     Returns:
         ndarray: Choi (Liouville) representation of the quantum channel.
     """
-    super_op = backend.cast(super_op)
-
     if not isinstance(order, str):
         raise_error(TypeError, f"order must be type str, but it is type {type(order)}.")
 
@@ -2150,6 +2154,8 @@ def _reshuffling(super_op, order: str = "row", backend=None):
 
     backend = _check_backend(backend)
 
+    super_op = backend.cast(super_op, dtype=super_op.dtype)
+
     dim = np.sqrt(super_op.shape[0])
 
     if (