diff --git a/crates/accelerate/src/dense_layout.rs b/crates/accelerate/src/dense_layout.rs
index a7466748417a..cbe8b9ff8cc7 100644
--- a/crates/accelerate/src/dense_layout.rs
+++ b/crates/accelerate/src/dense_layout.rs
@@ -30,6 +30,7 @@ struct SubsetResult {
     pub error: f64,
     pub map: Vec<usize>,
     pub subgraph: Vec<[usize; 2]>,
+    pub index: usize,
 }
 
 fn bfs_sort(adj_matrix: ArrayView2<f64>, start: usize, num_qubits: usize) -> Vec<usize> {
@@ -190,6 +191,7 @@ pub fn best_subset_inner(
             error,
             map: bfs,
             subgraph,
+            index: k,
         }
     };
 
@@ -199,17 +201,20 @@ pub fn best_subset_inner(
             map: Vec::new(),
             error: f64::INFINITY,
             subgraph: Vec::new(),
+            index: usize::MAX,
         }
     };
 
     let reduce_fn = |best: SubsetResult, curr: SubsetResult| -> SubsetResult {
         if use_error {
-            if curr.count >= best.count && curr.error < best.error {
+            if (curr.count >= best.count && curr.error < best.error)
+                || (curr.count == best.count && curr.error == best.error && curr.index < best.index)
+            {
                 curr
             } else {
                 best
             }
-        } else if curr.count > best.count {
+        } else if curr.count > best.count || (curr.count == best.count && curr.index < best.index) {
             curr
         } else {
             best
diff --git a/releasenotes/notes/fix-potential-non-determinism-dense-layout-da66de0217121146.yaml b/releasenotes/notes/fix-potential-non-determinism-dense-layout-da66de0217121146.yaml
new file mode 100644
index 000000000000..ddf68fc9d49b
--- /dev/null
+++ b/releasenotes/notes/fix-potential-non-determinism-dense-layout-da66de0217121146.yaml
@@ -0,0 +1,9 @@
+---
+fixes:
+  - |
+    Fixed a potential source of non-determinism in :class:`~.DenseLayout` (and
+    by extension :class:`~.SabreLayout`) when targeting a
+    :class:`.CouplingMap` or :class:`.Target` that has more than one subgraph
+    with the same degree of connectivity. In these case the exact output
+    layout from the pass could previously fluctuate based on the number of
+    local CPUs and thread execution speed.