diff --git a/pennylane/devices/qubit/measure.py b/pennylane/devices/qubit/measure.py
index ae2ddb07c02..9547945d36a 100644
--- a/pennylane/devices/qubit/measure.py
+++ b/pennylane/devices/qubit/measure.py
@@ -18,6 +18,7 @@
 
 from scipy.sparse import csr_matrix
 
+import pennylane as qml
 from pennylane import math
 from pennylane.measurements import (
     ExpectationMP,
@@ -195,13 +196,30 @@ def get_measurement_function(
 
             backprop_mode = math.get_interface(state, *measurementprocess.obs.data) != "numpy"
             if isinstance(measurementprocess.obs, (Hamiltonian, LinearCombination)):
-                # need to work out thresholds for when its faster to use "backprop mode" measurements
-                return sum_of_terms_method if backprop_mode else csr_dot_products
+
+                # need to work out thresholds for when it's faster to use "backprop mode"
+                if backprop_mode:
+                    return sum_of_terms_method
+
+                if not all(obs.has_sparse_matrix for obs in measurementprocess.obs.terms()[1]):
+                    return sum_of_terms_method
+
+                if isinstance(measurementprocess.obs, Hamiltonian) and any(
+                    any(len(o.wires) > 1 for o in qml.operation.Tensor(op).obs)
+                    for op in measurementprocess.obs.ops
+                ):
+                    return sum_of_terms_method
+
+                return csr_dot_products
 
             if isinstance(measurementprocess.obs, Sum):
                 if backprop_mode:
                     # always use sum_of_terms_method for Sum observables in backprop mode
                     return sum_of_terms_method
+
+                if not all(obs.has_sparse_matrix() for _, obs in measurementprocess.obs.terms()):
+                    return sum_of_terms_method
+
                 if (
                     measurementprocess.obs.has_overlapping_wires
                     and len(measurementprocess.obs.wires) > 7
diff --git a/pennylane/operation.py b/pennylane/operation.py
index 77f989e6383..46a8d4e5bb7 100644
--- a/pennylane/operation.py
+++ b/pennylane/operation.py
@@ -870,6 +870,18 @@ def compute_sparse_matrix(
         """
         raise SparseMatrixUndefinedError
 
+    # pylint: disable=no-self-argument, comparison-with-callable
+    @classproperty
+    def has_sparse_matrix(cls) -> bool:
+        r"""Bool: Whether the Operator returns a defined sparse matrix.
+
+        Note: Child classes may have this as an instance property instead of as a class property.
+        """
+        return (
+            cls.compute_sparse_matrix != Operator.compute_sparse_matrix
+            or cls.sparse_matrix != Operator.sparse_matrix
+        )
+
     def sparse_matrix(self, wire_order: Optional[WiresLike] = None) -> csr_matrix:
         r"""Representation of the operator as a sparse matrix in the computational basis.