diff --git a/crates/accelerate/src/basis/basis_translator/basis_search.rs b/crates/accelerate/src/basis/basis_translator/basis_search.rs
index 5b87cd68f429..815d8d460c73 100644
--- a/crates/accelerate/src/basis/basis_translator/basis_search.rs
+++ b/crates/accelerate/src/basis/basis_translator/basis_search.rs
@@ -12,29 +12,33 @@
 
 use std::cell::RefCell;
 
-use crate::equivalence::{CircuitRep, EdgeData, Equivalence, EquivalenceLibrary, Key, NodeData};
-use ahash::RandomState;
-use indexmap::{IndexMap, IndexSet};
+use hashbrown::{HashMap, HashSet};
 use pyo3::prelude::*;
-use pyo3::types::PySet;
 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")]
 pub(crate) fn py_basis_search(
+    py: Python,
     equiv_lib: &mut EquivalenceLibrary,
-    source_basis: &Bound<PySet>,
-    target_basis: &Bound<PySet>,
-) -> PyResult<PyObject> {
-    let new_source_basis: PyResult<IndexSet<(String, u32), RandomState>> =
-        source_basis.iter().map(|item| item.extract()).collect();
-    let new_target_basis: PyResult<IndexSet<String, RandomState>> =
-        target_basis.iter().map(|item| item.extract()).collect();
-    Ok(basis_search(equiv_lib, new_source_basis?, new_target_basis?).into_py(source_basis.py()))
+    source_basis: HashSet<(String, u32)>,
+    target_basis: HashSet<String>,
+) -> PyObject {
+    basis_search(
+        equiv_lib,
+        source_basis
+            .iter()
+            .map(|(name, num_qubits)| (name.as_str(), *num_qubits))
+            .collect(),
+        target_basis.iter().map(|name| name.as_str()).collect(),
+    )
+    .into_py(py)
 }
 
 type BasisTransforms = Vec<(String, u32, SmallVec<[Param; 3]>, CircuitRep)>;
@@ -51,24 +55,20 @@ type BasisTransforms = Vec<(String, u32, SmallVec<[Param; 3]>, CircuitRep)>;
 ///         was found.
 pub(crate) fn basis_search(
     equiv_lib: &mut EquivalenceLibrary,
-    source_basis: IndexSet<(String, u32), RandomState>,
-    target_basis: IndexSet<String, RandomState>,
+    source_basis: HashSet<(&str, u32)>,
+    target_basis: HashSet<&str>,
 ) -> Option<BasisTransforms> {
     // Build the visitor attributes:
-    let mut num_gates_remaining_for_rule: IndexMap<usize, usize, RandomState> = IndexMap::default();
-    let predecessors: IndexMap<(&str, u32), Equivalence, RandomState> = IndexMap::default();
-    let predecessors_cell: RefCell<IndexMap<(&str, u32), Equivalence, RandomState>> =
-        RefCell::new(predecessors);
-    let opt_cost_map: IndexMap<(&str, u32), u32, RandomState> = IndexMap::default();
-    let opt_cost_map_cell: RefCell<IndexMap<(&str, u32), u32, RandomState>> =
-        RefCell::new(opt_cost_map);
+    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 mut basis_transforms: Vec<(String, u32, SmallVec<[Param; 3]>, CircuitRep)> = vec![];
 
     // Initialize visitor attributes:
     initialize_num_gates_remain_for_rule(&equiv_lib.graph, &mut num_gates_remaining_for_rule);
 
     // TODO: Logs
-    let mut source_basis_remain: IndexSet<Key> = source_basis
+    let mut source_basis_remain: HashSet<Key> = source_basis
         .iter()
         .filter_map(|(gate_name, gate_num_qubits)| {
             if !target_basis.contains(gate_name) {
@@ -92,7 +92,7 @@ pub(crate) fn basis_search(
     let target_basis_keys: Vec<Key> = equiv_lib
         .keys()
         .cloned()
-        .filter(|key| target_basis.contains(&key.name))
+        .filter(|key| target_basis.contains(key.name.as_str()))
         .collect();
 
     // Dummy node is inserted in the graph. Which is where the search will start
@@ -123,7 +123,7 @@ pub(crate) fn basis_search(
             }
             let edge_data = edge.weight().as_ref().unwrap();
             let mut cost_tot = 0;
-            let borrowed_cost = opt_cost_map_cell.borrow();
+            let borrowed_cost = opt_cost_map.borrow();
             for instruction in edge_data.rule.circuit.data.iter() {
                 cost_tot += borrowed_cost[&(instruction.op.name(), instruction.op.num_qubits())];
             }
@@ -140,13 +140,13 @@ pub(crate) fn basis_search(
                 DijkstraEvent::Discover(n, score) => {
                     let gate_key = &equiv_lib.graph[n].key;
                     let gate = &(gate_key.name.as_str(), gate_key.num_qubits);
-                    source_basis_remain.swap_remove(gate_key);
-                    let mut borrowed_cost_map = opt_cost_map_cell.borrow_mut();
+                    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_cell.borrow().get(gate) {
+                    if let Some(rule) = predecessors.borrow().get(gate) {
                         // TODO: Logger
                         basis_transforms.push((
                             gate_key.name.to_string(),
@@ -163,7 +163,7 @@ pub(crate) fn basis_search(
                 }
                 DijkstraEvent::EdgeRelaxed(_, target, Some(edata)) => {
                     let gate = &equiv_lib.graph[target].key;
-                    predecessors_cell
+                    predecessors
                         .borrow_mut()
                         .entry((gate.name.as_str(), gate.num_qubits))
                         .and_modify(|value| *value = edata.rule.clone())
@@ -200,7 +200,7 @@ pub(crate) fn basis_search(
 
 fn initialize_num_gates_remain_for_rule(
     graph: &StableDiGraph<NodeData, Option<EdgeData>>,
-    source: &mut IndexMap<usize, usize, RandomState>,
+    source: &mut HashMap<usize, usize>,
 ) {
     let mut save_index = usize::MAX;
     for edge_data in graph.edge_weights().flatten() {