diff --git a/crates/accelerate/src/sparse_pauli_op.rs b/crates/accelerate/src/sparse_pauli_op.rs
index 1a85daf036d9..89f033338951 100644
--- a/crates/accelerate/src/sparse_pauli_op.rs
+++ b/crates/accelerate/src/sparse_pauli_op.rs
@@ -10,6 +10,7 @@
 // copyright notice, and modified files need to carry a notice indicating
 // that they have been altered from the originals.
 
+use ahash::RandomState;
 use pyo3::exceptions::{PyRuntimeError, PyValueError};
 use pyo3::prelude::*;
 use pyo3::types::PyTuple;
@@ -20,6 +21,7 @@ use numpy::prelude::*;
 use numpy::{PyArray1, PyArray2, PyReadonlyArray1, PyReadonlyArray2, PyUntypedArrayMethods};
 
 use hashbrown::HashMap;
+use indexmap::IndexMap;
 use ndarray::{s, ArrayView1, ArrayView2, Axis};
 use num_complex::Complex64;
 use num_traits::Zero;
@@ -298,7 +300,11 @@ impl MatrixCompressedPaulis {
     /// explicitly stored operations, if there are duplicates.  After the summation, any terms that
     /// have become zero are dropped.
     pub fn combine(&mut self) {
-        let mut hash_table = HashMap::<(u64, u64), Complex64>::with_capacity(self.coeffs.len());
+        let mut hash_table =
+            IndexMap::<(u64, u64), Complex64, RandomState>::with_capacity_and_hasher(
+                self.coeffs.len(),
+                RandomState::new(),
+            );
         for (key, coeff) in self
             .x_like
             .drain(..)
diff --git a/releasenotes/notes/spo-to-matrix-determinism-554389d6fc98627c.yaml b/releasenotes/notes/spo-to-matrix-determinism-554389d6fc98627c.yaml
new file mode 100644
index 000000000000..61961d492e72
--- /dev/null
+++ b/releasenotes/notes/spo-to-matrix-determinism-554389d6fc98627c.yaml
@@ -0,0 +1,6 @@
+---
+fixes:
+  - |
+    Fixed a per-process based non-determinism in `SparsePauliOp.to_matrix`.  The exact order of the
+    floating-point operations in the summation would previously vary per process, but will now be
+    identical between different invocations of the same script.  See `#13413 <https://github.com/Qiskit/qiskit/issues/13413>`__.