Add helper func apply_operation for default mixed (#6379)

**Context:**
Before getting into the implementation of new `default_mixed` we need to
separate most of the useful logics from the older version one. One of
the most important things is the `apply_operation`. Basically, what we
want to do is to condensate everything into a new sub-module under
`qml/devices/qubit_mixed` such that previous empirical heuristics
regarding how to select between `einsum` and `tensordot` can be carried
over and wrapped independently.

**Description of the Change:**
New script containing the helper funcs as described above.

**Benefits:**
Better modularization. Getting ready for the implementaion of a new
`default_mixed` in the future.

**Possible Drawbacks:**

**Related GitHub Issues:**

**Related Shortcut Stories**
[sc-73319]

---------

Co-authored-by: Mudit Pandey <[email protected]>
Co-authored-by: Astral Cai <[email protected]>
Co-authored-by: Christina Lee <[email protected]>

doc/releases/changelog-dev.md

@@ -6,6 +6,20 @@
 
 <h3>Improvements 🛠</h3>
 
+<h4>Other Improvements</h4>
+
+* Added `qml.devices.qubit_mixed` module for mixed-state qubit device support. This module introduces:
+  - A new API for mixed-state operations
+  - An `apply_operation` helper function featuring:
+    - Two density matrix contraction methods using `einsum` and `tensordot`
+    - Optimized handling of special cases including:
+      - Diagonal operators
+      - Identity operators 
+      - CX (controlled-X)
+      - Multi-controlled X gates
+      - Grover operators
+  [(#6379)](https://github.com/PennyLaneAI/pennylane/pull/6379)
+
 <h3>Breaking changes 💔</h3>
 
 <h3>Deprecations 👋</h3>

pennylane/devices/qubit_mixed/__init__.py

@@ -0,0 +1,26 @@
+# Copyright 2018-2024 Xanadu Quantum Technologies Inc.
+
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+
+#     http://www.apache.org/licenses/LICENSE-2.0
+
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+"""
+Submodule for performing qubit mixed-state simulations of quantum circuits.
+
+This submodule is internal and subject to change without a deprecation cycle. Use
+at your own discretion.
+
+.. currentmodule:: pennylane.devices.qubit_mixed
+.. autosummary::
+    :toctree: api
+
+    apply_operation
+"""
+from .apply_operation import apply_operation