Deprecate classical_fisher and quantum_fisher (#5985)

- Duplicated functions in new module `qml.gradients.fisher`. Moved tests
to `test.gradient.core.test_fisher`.
- Added new test file `test.qinfo.test_fisher_deprecation` with
deprecation tests.
- Added entry for functions in `qml.gradients` doc page.
- Removed entries of functions in `qml.qinfo`.

[[sc-67213](https://app.shortcut.com/xanaduai/story/67213)]
[[sc-67214](https://app.shortcut.com/xanaduai/story/67214)]

---------

Co-authored-by: Mudit Pandey <mudit.pandey@xanadu.ai>

doc/code/qml_qinfo.rst

@@ -18,3 +18,5 @@ Transforms
     :skip: metric_tensor
     :skip: adjoint_metric_tensor
     :skip: transform
+    :skip: classical_fisher
+    :skip: quantum_fisher

doc/development/deprecations.rst

@@ -9,6 +9,12 @@ deprecations are listed below.
 Pending deprecations
 --------------------
 
+* The functions ``qml.qinfo.classical_fisher`` and ``qml.qinfo.quantum_fisher`` are deprecated since they are being migrated
+  to the ``qml.gradients`` module. Therefore, ``qml.gradients.classical_fisher`` and ``qml.gradients.quantum_fisher`` should be used instead.
+
+  - Deprecated and Duplicated in v0.38
+  - Will be removed in v0.39
+
 * The ``simplify`` argument in ``qml.Hamiltonian`` and ``qml.ops.LinearCombination`` is deprecated. 
   Instead, ``qml.simplify()`` can be called on the constructed operator.
 

doc/releases/changelog-dev.md

@@ -49,6 +49,10 @@
 
 <h3>Deprecations 👋</h3>
 
+* `pennylane.qinfo.classical_fisher` and `pennylane.qinfo.quantum_fisher` have been deprecated.
+  Instead, use `pennylane.gradients.classical_fisher` and `pennylane.gradients.quantum_fisher`.
+  [(#5985)](https://github.com/PennyLaneAI/pennylane/pull/5985)
+
 <h3>Documentation 📝</h3>
 
 * Improves the docstring for `QuantumScript.expand` and `qml.tape.tape.expand_tape`.

pennylane/gradients/__init__.py

@@ -85,6 +85,8 @@
     batch_jvp
     jvp
     classical_jacobian
+    classical_fisher
+    quantum_fisher
 
 
 Registering autodifferentiation gradients
@@ -341,6 +343,7 @@ def my_custom_gradient(tape: qml.tape.QuantumTape, **kwargs) -> (Sequence[qml.ta
 from .adjoint_metric_tensor import adjoint_metric_tensor
 from .classical_jacobian import classical_jacobian
 from .finite_difference import finite_diff, finite_diff_coeffs
+from .fisher import classical_fisher, quantum_fisher
 from .general_shift_rules import (
     eigvals_to_frequencies,
     generate_multi_shift_rule,