Change casting rules for StateMP to avoid copy in Lightning (#5995)

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**Context:**
The changes made in #5547 led to a performance regression in Lightning
because a superfluous state vector copy is made when adding `0.j`, which
was introduced to cast the output to complex.

**Description of the Change:**
Use `qml.math.cast` instead.

**Benefits:**
Avoid unnecessary sum of `0.j`

**Possible Drawbacks:**

**Related GitHub Issues:**
[sc-65127]

doc/releases/changelog-dev.md

@@ -24,6 +24,9 @@
 
 <h3>Improvements 🛠</h3>
 
+* `StateMP.process_state` defines rules in `cast_to_complex` for complex casting, avoiding a superfluous state vector copy in Lightning simulations
+  [(#5995)](https://github.com/PennyLaneAI/pennylane/pull/5995)
+
 * Port the fast `apply_operation` implementation of `PauliZ` to `PhaseShift`, `S` and `T`.
   [(#5876)](https://github.com/PennyLaneAI/pennylane/pull/5876)
 

pennylane/measurements/state.py

@@ -167,10 +167,18 @@ def shape(self, device, shots):
 
     def process_state(self, state: Sequence[complex], wire_order: Wires):
         # pylint:disable=redefined-outer-name
-        is_tf_interface = qml.math.get_deep_interface(state) == "tensorflow"
+        def cast_to_complex(state):
+            dtype = str(state.dtype)
+            if "complex" in dtype:
+                return state
+            if qml.math.get_interface(state) == "tensorflow":
+                return qml.math.cast(state, "complex128")
+            floating_single = "float32" in dtype or "complex64" in dtype
+            return qml.math.cast(state, "complex64" if floating_single else "complex128")
+
         wires = self.wires
         if not wires or wire_order == wires:
-            return qml.math.cast(state, "complex128") if is_tf_interface else state + 0.0j
+            return cast_to_complex(state)
 
         if set(wires) != set(wire_order):
             raise WireError(
@@ -190,7 +198,7 @@ def process_state(self, state: Sequence[complex], wire_order: Wires):
         state = qml.math.reshape(state, shape)
         state = qml.math.transpose(state, desired_axes)
         state = qml.math.reshape(state, flat_shape)
-        return qml.math.cast(state, "complex128") if is_tf_interface else state + 0.0j
+        return cast_to_complex(state)
 
     def process_density_matrix(self, density_matrix: Sequence[complex], wire_order: Wires):
         # pylint:disable=redefined-outer-name