diff --git a/crates/accelerate/src/basis/basis_translator/basis_search.rs b/crates/accelerate/src/basis/basis_translator/basis_search.rs
index 82b8fc474b93..a1c9a6d192a3 100644
--- a/crates/accelerate/src/basis/basis_translator/basis_search.rs
+++ b/crates/accelerate/src/basis/basis_translator/basis_search.rs
@@ -14,14 +14,14 @@ use std::cell::RefCell;
 
 use hashbrown::{HashMap, HashSet};
 use pyo3::prelude::*;
+
+use crate::equivalence::{CircuitRep, EdgeData, Equivalence, EquivalenceLibrary, Key, NodeData};
 use qiskit_circuit::operations::{Operation, Param};
 use rustworkx_core::petgraph::stable_graph::{EdgeReference, NodeIndex, StableDiGraph};
 use rustworkx_core::petgraph::visit::Control;
 use rustworkx_core::traversal::{dijkstra_search, DijkstraEvent};
 use smallvec::SmallVec;
 
-use crate::equivalence::{CircuitRep, EdgeData, Equivalence, EquivalenceLibrary, Key, NodeData};
-
 #[pyfunction]
 #[pyo3(name = "basis_search")]
 /// Search for a set of transformations from source_basis to target_basis.
@@ -68,8 +68,9 @@ pub(crate) fn basis_search(
 ) -> Option<BasisTransforms> {
     // Build the visitor attributes:
     let mut num_gates_remaining_for_rule: HashMap<usize, usize> = HashMap::default();
-    let predecessors: RefCell<HashMap<(&str, u32), Equivalence>> = RefCell::new(HashMap::default());
-    let opt_cost_map: RefCell<HashMap<(&str, u32), u32>> = RefCell::new(HashMap::default());
+    let predecessors: RefCell<HashMap<(String, u32), Equivalence>> =
+        RefCell::new(HashMap::default());
+    let opt_cost_map: RefCell<HashMap<(String, u32), u32>> = RefCell::new(HashMap::default());
     let mut basis_transforms: Vec<(String, u32, SmallVec<[Param; 3]>, CircuitRep)> = vec![];
 
     // Initialize visitor attributes:
@@ -124,21 +125,26 @@ pub(crate) fn basis_search(
     });
 
     // Edge cost function for Visitor
-    let edge_weight =
-        |edge: EdgeReference<Option<EdgeData>>| -> Result<u32, ()> {
-            if edge.weight().is_none() {
-                return Ok(1);
-            }
-            let edge_data = edge.weight().as_ref().unwrap();
-            let mut cost_tot = 0;
-            let borrowed_cost = opt_cost_map.borrow();
-            for instruction in edge_data.rule.circuit.data.iter() {
-                let instruction_op = instruction.op.view();
-                cost_tot += borrowed_cost[&(instruction_op.name(), instruction_op.num_qubits())];
-            }
-            Ok(cost_tot
-                - borrowed_cost[&(edge_data.source.name.as_str(), edge_data.source.num_qubits)])
-        };
+    let edge_weight = |edge: EdgeReference<Option<EdgeData>>| -> Result<u32, ()> {
+        if edge.weight().is_none() {
+            return Ok(1);
+        }
+        let edge_data = edge.weight().as_ref().unwrap();
+        let mut cost_tot = 0;
+        let borrowed_cost = opt_cost_map.borrow();
+        for instruction in edge_data.rule.circuit.data.iter() {
+            let instruction_op = instruction.op.view();
+            cost_tot += borrowed_cost[&(
+                instruction_op.name().to_string(),
+                instruction_op.num_qubits(),
+            )];
+        }
+        Ok(cost_tot
+            - borrowed_cost[&(
+                edge_data.source.name.to_string(),
+                edge_data.source.num_qubits,
+            )])
+    };
 
     let basis_transforms = match dijkstra_search(
         &equiv_lib.graph,
@@ -148,14 +154,15 @@ pub(crate) fn basis_search(
             match event {
                 DijkstraEvent::Discover(n, score) => {
                     let gate_key = &equiv_lib.graph[n].key;
-                    let gate = &(gate_key.name.as_str(), gate_key.num_qubits);
+                    let gate = (gate_key.name.to_string(), gate_key.num_qubits);
                     source_basis_remain.remove(gate_key);
                     let mut borrowed_cost_map = opt_cost_map.borrow_mut();
-                    borrowed_cost_map
-                        .entry(*gate)
-                        .and_modify(|cost_ref| *cost_ref = score)
-                        .or_insert(score);
-                    if let Some(rule) = predecessors.borrow().get(gate) {
+                    if let Some(entry) = borrowed_cost_map.get_mut(&gate) {
+                        *entry = score;
+                    } else {
+                        borrowed_cost_map.insert(gate.clone(), score);
+                    }
+                    if let Some(rule) = predecessors.borrow().get(&gate) {
                         // TODO: Logger
                         basis_transforms.push((
                             gate_key.name.to_string(),
@@ -174,7 +181,7 @@ pub(crate) fn basis_search(
                     let gate = &equiv_lib.graph[target].key;
                     predecessors
                         .borrow_mut()
-                        .entry((gate.name.as_str(), gate.num_qubits))
+                        .entry((gate.name.to_string(), gate.num_qubits))
                         .and_modify(|value| *value = edata.rule.clone())
                         .or_insert(edata.rule.clone());
                 }