diff --git a/crates/accelerate/src/equivalence.rs b/crates/accelerate/src/equivalence.rs
new file mode 100644
index 000000000000..e389151b6b04
--- /dev/null
+++ b/crates/accelerate/src/equivalence.rs
@@ -0,0 +1,805 @@
+// This code is part of Qiskit.
+//
+// (C) Copyright IBM 2024
+//
+// This code is licensed under the Apache License, Version 2.0. You may
+// obtain a copy of this license in the LICENSE.txt file in the root directory
+// of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
+//
+// Any modifications or derivative works of this code must retain this
+// copyright notice, and modified files need to carry a notice indicating
+// that they have been altered from the originals.
+
+use itertools::Itertools;
+
+use pyo3::exceptions::PyTypeError;
+use qiskit_circuit::parameter_table::ParameterUuid;
+use rustworkx_core::petgraph::csr::IndexType;
+use rustworkx_core::petgraph::stable_graph::StableDiGraph;
+use rustworkx_core::petgraph::visit::IntoEdgeReferences;
+
+use smallvec::SmallVec;
+use std::collections::hash_map::DefaultHasher;
+use std::hash::{Hash, Hasher};
+use std::{error::Error, fmt::Display};
+
+use exceptions::CircuitError;
+
+use ahash::RandomState;
+use indexmap::{IndexMap, IndexSet};
+use pyo3::prelude::*;
+use pyo3::types::{PyDict, PyList, PySet, PyString};
+
+use rustworkx_core::petgraph::{
+    graph::{EdgeIndex, NodeIndex},
+    visit::EdgeRef,
+};
+
+use qiskit_circuit::circuit_data::CircuitData;
+use qiskit_circuit::circuit_instruction::OperationFromPython;
+use qiskit_circuit::imports::{ImportOnceCell, QUANTUM_CIRCUIT};
+use qiskit_circuit::operations::Param;
+use qiskit_circuit::operations::{Operation, OperationRef};
+use qiskit_circuit::packed_instruction::PackedOperation;
+
+mod exceptions {
+    use pyo3::import_exception_bound;
+    import_exception_bound! {qiskit.circuit.exceptions, CircuitError}
+}
+pub static PYDIGRAPH: ImportOnceCell = ImportOnceCell::new("rustworkx", "PyDiGraph");
+
+// Custom Structs
+
+#[pyclass(frozen, sequence, module = "qiskit._accelerate.equivalence")]
+#[derive(Debug, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)]
+pub struct Key {
+    #[pyo3(get)]
+    pub name: String,
+    #[pyo3(get)]
+    pub num_qubits: u32,
+}
+
+#[pymethods]
+impl Key {
+    #[new]
+    #[pyo3(signature = (name, num_qubits))]
+    fn new(name: String, num_qubits: u32) -> Self {
+        Self { name, num_qubits }
+    }
+
+    fn __hash__(&self) -> u64 {
+        let mut hasher = DefaultHasher::new();
+        (self.name.to_string(), self.num_qubits).hash(&mut hasher);
+        hasher.finish()
+    }
+
+    fn __repr__(slf: PyRef<Self>) -> String {
+        slf.to_string()
+    }
+
+    fn __getnewargs__(slf: PyRef<Self>) -> (Bound<PyString>, u32) {
+        (
+            PyString::new_bound(slf.py(), slf.name.as_str()),
+            slf.num_qubits,
+        )
+    }
+
+    // Ord methods for Python
+    fn __lt__(&self, other: &Self) -> bool {
+        self.lt(other)
+    }
+    fn __le__(&self, other: &Self) -> bool {
+        self.le(other)
+    }
+    fn __eq__(&self, other: &Self) -> bool {
+        self.eq(other)
+    }
+    fn __ne__(&self, other: &Self) -> bool {
+        self.ne(other)
+    }
+    fn __ge__(&self, other: &Self) -> bool {
+        self.ge(other)
+    }
+    fn __gt__(&self, other: &Self) -> bool {
+        self.gt(other)
+    }
+}
+impl Key {
+    fn from_operation(operation: &PackedOperation) -> Self {
+        let op_ref: OperationRef = operation.view();
+        Key {
+            name: op_ref.name().to_string(),
+            num_qubits: op_ref.num_qubits(),
+        }
+    }
+}
+
+impl Display for Key {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
+        write!(
+            f,
+            "Key(name=\'{}\', num_qubits={})",
+            self.name, self.num_qubits
+        )
+    }
+}
+
+#[pyclass(frozen, sequence, module = "qiskit._accelerate.equivalence")]
+#[derive(Debug, Clone)]
+pub struct Equivalence {
+    #[pyo3(get)]
+    pub params: SmallVec<[Param; 3]>,
+    #[pyo3(get)]
+    pub circuit: CircuitRep,
+}
+
+#[pymethods]
+impl Equivalence {
+    #[new]
+    #[pyo3(signature = (params, circuit))]
+    fn new(params: SmallVec<[Param; 3]>, circuit: CircuitRep) -> Self {
+        Self { circuit, params }
+    }
+
+    fn __repr__(&self) -> String {
+        self.to_string()
+    }
+
+    fn __eq__(slf: &Bound<Self>, other: &Bound<PyAny>) -> PyResult<bool> {
+        let other_params = other.getattr("params")?;
+        let other_circuit = other.getattr("circuit")?;
+        Ok(other_params.eq(&slf.getattr("params")?)?
+            && other_circuit.eq(&slf.getattr("circuit")?)?)
+    }
+
+    fn __getnewargs__<'py>(
+        slf: &'py Bound<'py, Self>,
+    ) -> PyResult<(Bound<'py, PyAny>, Bound<'py, PyAny>)> {
+        Ok((slf.getattr("params")?, slf.getattr("circuit")?))
+    }
+}
+
+impl Display for Equivalence {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
+        write!(
+            f,
+            "Equivalence(params=[{}], circuit={:?})",
+            self.params
+                .iter()
+                .map(|param| format!("{:?}", param))
+                .format(", "),
+            self.circuit
+        )
+    }
+}
+
+#[pyclass(frozen, sequence, module = "qiskit._accelerate.equivalence")]
+#[derive(Debug, Clone)]
+pub struct NodeData {
+    #[pyo3(get)]
+    pub key: Key,
+    #[pyo3(get)]
+    pub equivs: Vec<Equivalence>,
+}
+
+#[pymethods]
+impl NodeData {
+    #[new]
+    #[pyo3(signature = (key, equivs))]
+    fn new(key: Key, equivs: Vec<Equivalence>) -> Self {
+        Self { key, equivs }
+    }
+
+    fn __repr__(&self) -> String {
+        self.to_string()
+    }
+
+    fn __eq__(slf: &Bound<Self>, other: &Bound<PyAny>) -> PyResult<bool> {
+        Ok(slf.getattr("key")?.eq(other.getattr("key")?)?
+            && slf.getattr("equivs")?.eq(other.getattr("equivs")?)?)
+    }
+
+    fn __getnewargs__<'py>(
+        slf: &'py Bound<'py, Self>,
+    ) -> PyResult<(Bound<'py, PyAny>, Bound<'py, PyAny>)> {
+        Ok((slf.getattr("key")?, slf.getattr("equivs")?))
+    }
+}
+
+impl Display for NodeData {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
+        write!(
+            f,
+            "NodeData(key={}, equivs=[{}])",
+            self.key,
+            self.equivs.iter().format(", ")
+        )
+    }
+}
+
+#[pyclass(frozen, sequence, module = "qiskit._accelerate.equivalence")]
+#[derive(Debug, Clone)]
+pub struct EdgeData {
+    #[pyo3(get)]
+    pub index: usize,
+    #[pyo3(get)]
+    pub num_gates: usize,
+    #[pyo3(get)]
+    pub rule: Equivalence,
+    #[pyo3(get)]
+    pub source: Key,
+}
+
+#[pymethods]
+impl EdgeData {
+    #[new]
+    #[pyo3(signature = (index, num_gates, rule, source))]
+    fn new(index: usize, num_gates: usize, rule: Equivalence, source: Key) -> Self {
+        Self {
+            index,
+            num_gates,
+            rule,
+            source,
+        }
+    }
+
+    fn __repr__(&self) -> String {
+        self.to_string()
+    }
+
+    fn __eq__(slf: &Bound<Self>, other: &Bound<Self>) -> PyResult<bool> {
+        let other_borrowed = other.borrow();
+        let slf_borrowed = slf.borrow();
+        Ok(slf_borrowed.index == other_borrowed.index
+            && slf_borrowed.num_gates == other_borrowed.num_gates
+            && slf_borrowed.source == other_borrowed.source
+            && other.getattr("rule")?.eq(slf.getattr("rule")?)?)
+    }
+
+    fn __getnewargs__(slf: PyRef<Self>) -> (usize, usize, Equivalence, Key) {
+        (
+            slf.index,
+            slf.num_gates,
+            slf.rule.clone(),
+            slf.source.clone(),
+        )
+    }
+}
+
+impl Display for EdgeData {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
+        write!(
+            f,
+            "EdgeData(index={}, num_gates={}, rule={}, source={})",
+            self.index, self.num_gates, self.rule, self.source
+        )
+    }
+}
+
+/// Enum that helps extract the Operation and Parameters on a Gate.
+/// It is highly derivative of `PackedOperation` while also tracking the specific
+/// parameter objects.
+#[derive(Debug, Clone)]
+pub struct GateOper {
+    operation: PackedOperation,
+    params: SmallVec<[Param; 3]>,
+}
+
+impl<'py> FromPyObject<'py> for GateOper {
+    fn extract(ob: &'py PyAny) -> PyResult<Self> {
+        let op_struct: OperationFromPython = ob.extract()?;
+        Ok(Self {
+            operation: op_struct.operation,
+            params: op_struct.params,
+        })
+    }
+}
+
+/// Representation of QuantumCircuit by using an instance of `CircuitData`.]
+///
+/// TODO: Remove this implementation once the `EquivalenceLibrary` is no longer
+/// called from Python, or once the API is able to seamlessly accept instances
+/// of `CircuitData`.
+#[derive(Debug, Clone)]
+pub struct CircuitRep(pub CircuitData);
+
+impl FromPyObject<'_> for CircuitRep {
+    fn extract_bound(ob: &Bound<'_, PyAny>) -> PyResult<Self> {
+        if ob.is_instance(QUANTUM_CIRCUIT.get_bound(ob.py()))? {
+            let data: Bound<PyAny> = ob.getattr("_data")?;
+            let data_downcast: Bound<CircuitData> = data.downcast_into()?;
+            let data_extract: CircuitData = data_downcast.extract()?;
+            Ok(Self(data_extract))
+        } else {
+            Err(PyTypeError::new_err(
+                "Provided object was not an instance of QuantumCircuit",
+            ))
+        }
+    }
+}
+
+impl IntoPy<PyObject> for CircuitRep {
+    fn into_py(self, py: Python<'_>) -> PyObject {
+        QUANTUM_CIRCUIT
+            .get_bound(py)
+            .call_method1("_from_circuit_data", (self.0,))
+            .unwrap()
+            .unbind()
+    }
+}
+
+impl ToPyObject for CircuitRep {
+    fn to_object(&self, py: Python<'_>) -> PyObject {
+        self.clone().into_py(py)
+    }
+}
+
+// Custom Types
+type GraphType = StableDiGraph<NodeData, Option<EdgeData>>;
+type KTIType = IndexMap<Key, NodeIndex, RandomState>;
+
+#[pyclass(
+    subclass,
+    name = "BaseEquivalenceLibrary",
+    module = "qiskit._accelerate.equivalence"
+)]
+#[derive(Debug, Clone)]
+pub struct EquivalenceLibrary {
+    graph: GraphType,
+    key_to_node_index: KTIType,
+    rule_id: usize,
+    _graph: Option<PyObject>,
+}
+
+#[pymethods]
+impl EquivalenceLibrary {
+    /// Create a new equivalence library.
+    ///
+    /// Args:
+    ///     base (Optional[EquivalenceLibrary]):  Base equivalence library to
+    ///         be referenced if an entry is not found in this library.
+    #[new]
+    #[pyo3(signature= (base=None))]
+    fn new(base: Option<&EquivalenceLibrary>) -> Self {
+        if let Some(base) = base {
+            Self {
+                graph: base.graph.clone(),
+                key_to_node_index: base.key_to_node_index.clone(),
+                rule_id: base.rule_id,
+                _graph: None,
+            }
+        } else {
+            Self {
+                graph: GraphType::new(),
+                key_to_node_index: KTIType::default(),
+                rule_id: 0_usize,
+                _graph: None,
+            }
+        }
+    }
+
+    /// Add a new equivalence to the library. Future queries for the Gate
+    /// will include the given circuit, in addition to all existing equivalences
+    /// (including those from base).
+    ///
+    /// Parameterized Gates (those including `qiskit.circuit.Parameters` in their
+    /// `Gate.params`) can be marked equivalent to parameterized circuits,
+    /// provided the parameters match.
+    ///
+    /// Args:
+    ///     gate (Gate): A Gate instance.
+    ///     equivalent_circuit (QuantumCircuit): A circuit equivalently
+    ///         implementing the given Gate.
+    #[pyo3(name = "add_equivalence")]
+    fn py_add_equivalence(
+        &mut self,
+        py: Python,
+        gate: GateOper,
+        equivalent_circuit: CircuitRep,
+    ) -> PyResult<()> {
+        self.add_equivalence(py, &gate.operation, &gate.params, equivalent_circuit)
+    }
+
+    /// Check if a library contains any decompositions for gate.
+    ///
+    /// Args:
+    ///     gate (Gate): A Gate instance.
+    ///
+    /// Returns:
+    ///     Bool: True if gate has a known decomposition in the library.
+    ///         False otherwise.
+    #[pyo3(name = "has_entry")]
+    fn py_has_entry(&self, gate: GateOper) -> bool {
+        self.has_entry(&gate.operation)
+    }
+
+    /// Set the equivalence record for a Gate. Future queries for the Gate
+    /// will return only the circuits provided.
+    ///
+    /// Parameterized Gates (those including `qiskit.circuit.Parameters` in their
+    /// `Gate.params`) can be marked equivalent to parameterized circuits,
+    /// provided the parameters match.
+    ///
+    /// Args:
+    ///     gate (Gate): A Gate instance.
+    ///     entry (List['QuantumCircuit']) : A list of QuantumCircuits, each
+    ///         equivalently implementing the given Gate.
+    #[pyo3(name = "set_entry")]
+    fn py_set_entry(&mut self, py: Python, gate: GateOper, entry: Vec<CircuitRep>) -> PyResult<()> {
+        self.set_entry(py, &gate.operation, &gate.params, entry)
+    }
+
+    /// Gets the set of QuantumCircuits circuits from the library which
+    /// equivalently implement the given Gate.
+    ///
+    /// Parameterized circuits will have their parameters replaced with the
+    /// corresponding entries from Gate.params.
+    ///
+    /// Args:
+    ///     gate (Gate) - Gate: A Gate instance.
+    ///
+    /// Returns:
+    ///     List[QuantumCircuit]: A list of equivalent QuantumCircuits. If empty,
+    ///         library contains no known decompositions of Gate.
+    ///
+    ///         Returned circuits will be ordered according to their insertion in
+    ///         the library, from earliest to latest, from top to base. The
+    ///         ordering of the StandardEquivalenceLibrary will not generally be
+    ///         consistent across Qiskit versions.
+    fn get_entry(&self, py: Python, gate: GateOper) -> PyResult<Py<PyList>> {
+        let key = Key::from_operation(&gate.operation);
+        let query_params = gate.params;
+
+        let bound_equivalencies = self
+            ._get_equivalences(&key)
+            .into_iter()
+            .filter_map(|equivalence| rebind_equiv(py, equivalence, &query_params).ok());
+        let return_list = PyList::empty_bound(py);
+        for equiv in bound_equivalencies {
+            return_list.append(equiv)?;
+        }
+        Ok(return_list.unbind())
+    }
+
+    // TODO: Remove once BasisTranslator is in Rust.
+    #[getter]
+    fn get_graph(&mut self, py: Python) -> PyResult<PyObject> {
+        if let Some(graph) = &self._graph {
+            Ok(graph.clone_ref(py))
+        } else {
+            self._graph = Some(to_pygraph(py, &self.graph)?);
+            Ok(self
+                ._graph
+                .as_ref()
+                .map(|graph| graph.clone_ref(py))
+                .unwrap())
+        }
+    }
+
+    /// Get all the equivalences for the given key
+    pub fn _get_equivalences(&self, key: &Key) -> Vec<Equivalence> {
+        if let Some(key_in) = self.key_to_node_index.get(key) {
+            self.graph[*key_in].equivs.clone()
+        } else {
+            vec![]
+        }
+    }
+
+    #[pyo3(name = "keys")]
+    fn py_keys(slf: PyRef<Self>) -> PyResult<PyObject> {
+        let py_dict = PyDict::new_bound(slf.py());
+        for key in slf.keys() {
+            py_dict.set_item(key.clone().into_py(slf.py()), slf.py().None())?;
+        }
+        Ok(py_dict.as_any().call_method0("keys")?.into())
+    }
+
+    #[pyo3(name = "node_index")]
+    fn py_node_index(&self, key: &Key) -> usize {
+        self.node_index(key).index()
+    }
+
+    fn __getstate__(slf: PyRef<Self>) -> PyResult<Bound<PyDict>> {
+        let ret = PyDict::new_bound(slf.py());
+        ret.set_item("rule_id", slf.rule_id)?;
+        let key_to_usize_node: Bound<PyDict> = PyDict::new_bound(slf.py());
+        for (key, val) in slf.key_to_node_index.iter() {
+            key_to_usize_node.set_item(key.clone().into_py(slf.py()), val.index())?;
+        }
+        ret.set_item("key_to_node_index", key_to_usize_node)?;
+        let graph_nodes: Bound<PyList> = PyList::empty_bound(slf.py());
+        for weight in slf.graph.node_weights() {
+            graph_nodes.append(weight.clone().into_py(slf.py()))?;
+        }
+        ret.set_item("graph_nodes", graph_nodes.unbind())?;
+        let edges = slf.graph.edge_references().map(|edge| {
+            (
+                edge.source().index(),
+                edge.target().index(),
+                edge.weight().clone().into_py(slf.py()),
+            )
+        });
+        let graph_edges = PyList::empty_bound(slf.py());
+        for edge in edges {
+            graph_edges.append(edge)?;
+        }
+        ret.set_item("graph_edges", graph_edges.unbind())?;
+        Ok(ret)
+    }
+
+    fn __setstate__(mut slf: PyRefMut<Self>, state: &Bound<PyDict>) -> PyResult<()> {
+        slf.rule_id = state.get_item("rule_id")?.unwrap().extract()?;
+        let graph_nodes_ref: Bound<PyAny> = state.get_item("graph_nodes")?.unwrap();
+        let graph_nodes: &Bound<PyList> = graph_nodes_ref.downcast()?;
+        let graph_edge_ref: Bound<PyAny> = state.get_item("graph_edges")?.unwrap();
+        let graph_edges: &Bound<PyList> = graph_edge_ref.downcast()?;
+        slf.graph = GraphType::new();
+        for node_weight in graph_nodes {
+            slf.graph.add_node(node_weight.extract()?);
+        }
+        for edge in graph_edges {
+            let (source_node, target_node, edge_weight) = edge.extract()?;
+            slf.graph.add_edge(
+                NodeIndex::new(source_node),
+                NodeIndex::new(target_node),
+                edge_weight,
+            );
+        }
+        slf.key_to_node_index = state
+            .get_item("key_to_node_index")?
+            .unwrap()
+            .extract::<IndexMap<Key, usize>>()?
+            .into_iter()
+            .map(|(key, val)| (key, NodeIndex::new(val)))
+            .collect();
+        slf._graph = None;
+        Ok(())
+    }
+}
+
+// Rust native methods
+impl EquivalenceLibrary {
+    /// Add a new equivalence to the library. Future queries for the Gate
+    /// will include the given circuit, in addition to all existing equivalences
+    /// (including those from base).
+    pub fn add_equivalence(
+        &mut self,
+        py: Python,
+        gate: &PackedOperation,
+        params: &[Param],
+        equivalent_circuit: CircuitRep,
+    ) -> PyResult<()> {
+        raise_if_shape_mismatch(gate, &equivalent_circuit)?;
+        raise_if_param_mismatch(py, params, equivalent_circuit.0.unsorted_parameters(py)?)?;
+        let key: Key = Key::from_operation(gate);
+        let equiv = Equivalence {
+            circuit: equivalent_circuit.clone(),
+            params: params.into(),
+        };
+
+        let target = self.set_default_node(key);
+        if let Some(node) = self.graph.node_weight_mut(target) {
+            node.equivs.push(equiv.clone());
+        }
+        let sources: IndexSet<Key, RandomState> = IndexSet::from_iter(
+            equivalent_circuit
+                .0
+                .iter()
+                .map(|inst| Key::from_operation(&inst.op)),
+        );
+        let edges = Vec::from_iter(sources.iter().map(|source| {
+            (
+                self.set_default_node(source.clone()),
+                target,
+                EdgeData {
+                    index: self.rule_id,
+                    num_gates: sources.len(),
+                    rule: equiv.clone(),
+                    source: source.clone(),
+                },
+            )
+        }));
+        for edge in edges {
+            self.graph.add_edge(edge.0, edge.1, Some(edge.2));
+        }
+        self.rule_id += 1;
+        self._graph = None;
+        Ok(())
+    }
+
+    /// Set the equivalence record for a Gate. Future queries for the Gate
+    /// will return only the circuits provided.
+    pub fn set_entry(
+        &mut self,
+        py: Python,
+        gate: &PackedOperation,
+        params: &[Param],
+        entry: Vec<CircuitRep>,
+    ) -> PyResult<()> {
+        for equiv in entry.iter() {
+            raise_if_shape_mismatch(gate, equiv)?;
+            raise_if_param_mismatch(py, params, equiv.0.unsorted_parameters(py)?)?;
+        }
+        let key = Key::from_operation(gate);
+        let node_index = self.set_default_node(key);
+
+        if let Some(graph_ind) = self.graph.node_weight_mut(node_index) {
+            graph_ind.equivs.clear();
+        }
+
+        let edges: Vec<EdgeIndex> = self
+            .graph
+            .edges_directed(node_index, rustworkx_core::petgraph::Direction::Incoming)
+            .map(|x| x.id())
+            .collect();
+        for edge in edges {
+            self.graph.remove_edge(edge);
+        }
+        for equiv in entry {
+            self.add_equivalence(py, gate, params, equiv)?
+        }
+        self._graph = None;
+        Ok(())
+    }
+
+    /// Rust native equivalent to `EquivalenceLibrary.has_entry()`
+    ///
+    /// Check if a library contains any decompositions for gate.
+    ///
+    /// # Arguments:
+    /// * `operation` OperationType: A Gate instance.
+    ///
+    /// # Returns:
+    /// `bool`: `true` if gate has a known decomposition in the library.
+    ///         `false` otherwise.
+    pub fn has_entry(&self, operation: &PackedOperation) -> bool {
+        let key = Key::from_operation(operation);
+        self.key_to_node_index.contains_key(&key)
+    }
+
+    pub fn keys(&self) -> impl Iterator<Item = &Key> {
+        self.key_to_node_index.keys()
+    }
+
+    /// Create a new node if key not found
+    pub fn set_default_node(&mut self, key: Key) -> NodeIndex {
+        if let Some(value) = self.key_to_node_index.get(&key) {
+            *value
+        } else {
+            let node = self.graph.add_node(NodeData {
+                key: key.clone(),
+                equivs: vec![],
+            });
+            self.key_to_node_index.insert(key, node);
+            node
+        }
+    }
+
+    /// Retrieve the `NodeIndex` that represents a `Key`
+    ///
+    /// # Arguments:
+    /// * `key`: The `Key` to look for.
+    ///
+    /// # Returns:
+    /// `NodeIndex`
+    pub fn node_index(&self, key: &Key) -> NodeIndex {
+        self.key_to_node_index[key]
+    }
+
+    /// Expose an immutable view of the inner graph.
+    pub fn graph(&self) -> &GraphType {
+        &self.graph
+    }
+}
+
+fn raise_if_param_mismatch(
+    py: Python,
+    gate_params: &[Param],
+    circuit_parameters: Bound<PySet>,
+) -> PyResult<()> {
+    let gate_params_obj = PySet::new_bound(
+        py,
+        gate_params
+            .iter()
+            .filter(|param| matches!(param, Param::ParameterExpression(_))),
+    )?;
+    if !gate_params_obj.eq(&circuit_parameters)? {
+        return Err(CircuitError::new_err(format!(
+            "Cannot add equivalence between circuit and gate \
+            of different parameters. Gate params: {:?}. \
+            Circuit params: {:?}.",
+            gate_params, circuit_parameters
+        )));
+    }
+    Ok(())
+}
+
+fn raise_if_shape_mismatch(gate: &PackedOperation, circuit: &CircuitRep) -> PyResult<()> {
+    let op_ref = gate.view();
+    if op_ref.num_qubits() != circuit.0.num_qubits() as u32
+        || op_ref.num_clbits() != circuit.0.num_clbits() as u32
+    {
+        return Err(CircuitError::new_err(format!(
+            "Cannot add equivalence between circuit and gate \
+            of different shapes. Gate: {} qubits and {} clbits. \
+            Circuit: {} qubits and {} clbits.",
+            op_ref.num_qubits(),
+            op_ref.num_clbits(),
+            circuit.0.num_qubits(),
+            circuit.0.num_clbits()
+        )));
+    }
+    Ok(())
+}
+
+fn rebind_equiv(py: Python, equiv: Equivalence, query_params: &[Param]) -> PyResult<CircuitRep> {
+    let (equiv_params, mut equiv_circuit) = (equiv.params, equiv.circuit);
+    let param_mapping: PyResult<IndexMap<ParameterUuid, &Param>> = equiv_params
+        .iter()
+        .zip(query_params.iter())
+        .filter_map(|(param_x, param_y)| match param_x {
+            Param::ParameterExpression(param) => {
+                let param_uuid = ParameterUuid::from_parameter(param.bind(py));
+                Some(param_uuid.map(|uuid| (uuid, param_y)))
+            }
+            _ => None,
+        })
+        .collect();
+    equiv_circuit
+        .0
+        .assign_parameters_from_mapping(py, param_mapping?)?;
+    Ok(equiv_circuit)
+}
+
+// Errors
+
+#[derive(Debug, Clone)]
+pub struct EquivalenceError {
+    pub message: String,
+}
+
+impl EquivalenceError {
+    pub fn new_err(message: String) -> Self {
+        Self { message }
+    }
+}
+
+impl Error for EquivalenceError {}
+
+impl Display for EquivalenceError {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
+        write!(f, "{}", self.message)
+    }
+}
+
+fn to_pygraph<N, E>(py: Python, pet_graph: &StableDiGraph<N, E>) -> PyResult<PyObject>
+where
+    N: IntoPy<PyObject> + Clone,
+    E: IntoPy<PyObject> + Clone,
+{
+    let graph = PYDIGRAPH.get_bound(py).call0()?;
+    let node_weights: Vec<N> = pet_graph.node_weights().cloned().collect();
+    graph.call_method1("add_nodes_from", (node_weights,))?;
+    let edge_weights: Vec<(usize, usize, E)> = pet_graph
+        .edge_references()
+        .map(|edge| {
+            (
+                edge.source().index(),
+                edge.target().index(),
+                edge.weight().clone(),
+            )
+        })
+        .collect();
+    graph.call_method1("add_edges_from", (edge_weights,))?;
+    Ok(graph.unbind())
+}
+
+#[pymodule]
+pub fn equivalence(m: &Bound<'_, PyModule>) -> PyResult<()> {
+    m.add_class::<EquivalenceLibrary>()?;
+    m.add_class::<NodeData>()?;
+    m.add_class::<EdgeData>()?;
+    m.add_class::<Equivalence>()?;
+    m.add_class::<Key>()?;
+    Ok(())
+}
diff --git a/crates/accelerate/src/lib.rs b/crates/accelerate/src/lib.rs
index 9111f932e270..f17fea5bccfe 100644
--- a/crates/accelerate/src/lib.rs
+++ b/crates/accelerate/src/lib.rs
@@ -22,6 +22,7 @@ pub mod commutation_checker;
 pub mod convert_2q_block_matrix;
 pub mod dense_layout;
 pub mod edge_collections;
+pub mod equivalence;
 pub mod error_map;
 pub mod euler_one_qubit_decomposer;
 pub mod filter_op_nodes;
diff --git a/crates/circuit/Cargo.toml b/crates/circuit/Cargo.toml
index ed1f849bbf62..ef79525ab761 100644
--- a/crates/circuit/Cargo.toml
+++ b/crates/circuit/Cargo.toml
@@ -38,4 +38,4 @@ workspace = true
 features = ["union"]
 
 [features]
-cache_pygates = []
+cache_pygates = []
\ No newline at end of file
diff --git a/crates/pyext/src/lib.rs b/crates/pyext/src/lib.rs
index 675cdfb3961d..8b54c535db2f 100644
--- a/crates/pyext/src/lib.rs
+++ b/crates/pyext/src/lib.rs
@@ -35,6 +35,7 @@ fn _accelerate(m: &Bound<PyModule>) -> PyResult<()> {
     add_submodule(m, ::qiskit_accelerate::commutation_checker::commutation_checker, "commutation_checker")?;
     add_submodule(m, ::qiskit_accelerate::convert_2q_block_matrix::convert_2q_block_matrix, "convert_2q_block_matrix")?;
     add_submodule(m, ::qiskit_accelerate::dense_layout::dense_layout, "dense_layout")?;
+    add_submodule(m, ::qiskit_accelerate::equivalence::equivalence, "equivalence")?;
     add_submodule(m, ::qiskit_accelerate::error_map::error_map, "error_map")?;
     add_submodule(m, ::qiskit_accelerate::euler_one_qubit_decomposer::euler_one_qubit_decomposer, "euler_one_qubit_decomposer")?;
     add_submodule(m, ::qiskit_accelerate::filter_op_nodes::filter_op_nodes_mod, "filter_op_nodes")?;
diff --git a/qiskit/__init__.py b/qiskit/__init__.py
index 25137d7a5918..0e9694217cd8 100644
--- a/qiskit/__init__.py
+++ b/qiskit/__init__.py
@@ -56,6 +56,7 @@
 sys.modules["qiskit._accelerate.converters"] = _accelerate.converters
 sys.modules["qiskit._accelerate.convert_2q_block_matrix"] = _accelerate.convert_2q_block_matrix
 sys.modules["qiskit._accelerate.dense_layout"] = _accelerate.dense_layout
+sys.modules["qiskit._accelerate.equivalence"] = _accelerate.equivalence
 sys.modules["qiskit._accelerate.error_map"] = _accelerate.error_map
 sys.modules["qiskit._accelerate.isometry"] = _accelerate.isometry
 sys.modules["qiskit._accelerate.uc_gate"] = _accelerate.uc_gate
diff --git a/qiskit/circuit/equivalence.py b/qiskit/circuit/equivalence.py
index 17912517d244..05441e68e600 100644
--- a/qiskit/circuit/equivalence.py
+++ b/qiskit/circuit/equivalence.py
@@ -12,190 +12,24 @@
 
 """Gate equivalence library."""
 
-import copy
-from collections import namedtuple
-
 from rustworkx.visualization import graphviz_draw
 import rustworkx as rx
 
-from qiskit.exceptions import InvalidFileError
-from .exceptions import CircuitError
-from .parameter import Parameter
-from .parameterexpression import ParameterExpression
 
-Key = namedtuple("Key", ["name", "num_qubits"])
-Equivalence = namedtuple("Equivalence", ["params", "circuit"])  # Ordered to match Gate.params
-NodeData = namedtuple("NodeData", ["key", "equivs"])
-EdgeData = namedtuple("EdgeData", ["index", "num_gates", "rule", "source"])
+from qiskit.exceptions import InvalidFileError
+from qiskit._accelerate.equivalence import (  # pylint: disable=unused-import
+    BaseEquivalenceLibrary,
+    Key,
+    Equivalence,
+    NodeData,
+    EdgeData,
+)
 
 
-class EquivalenceLibrary:
+class EquivalenceLibrary(BaseEquivalenceLibrary):
     """A library providing a one-way mapping of Gates to their equivalent
     implementations as QuantumCircuits."""
 
-    def __init__(self, *, base=None):
-        """Create a new equivalence library.
-
-        Args:
-            base (Optional[EquivalenceLibrary]):  Base equivalence library to
-                be referenced if an entry is not found in this library.
-        """
-        self._base = base
-
-        if base is None:
-            self._graph = rx.PyDiGraph()
-            self._key_to_node_index = {}
-            # Some unique identifier for rules.
-            self._rule_id = 0
-        else:
-            self._graph = base._graph.copy()
-            self._key_to_node_index = copy.deepcopy(base._key_to_node_index)
-            self._rule_id = base._rule_id
-
-    @property
-    def graph(self) -> rx.PyDiGraph:
-        """Return graph representing the equivalence library data.
-
-        This property should be treated as read-only as it provides
-        a reference to the internal state of the :class:`~.EquivalenceLibrary` object.
-        If the graph returned by this property is mutated it could corrupt the
-        the contents of the object. If you need to modify the output ``PyDiGraph``
-        be sure to make a copy prior to any modification.
-
-        Returns:
-            PyDiGraph: A graph object with equivalence data in each node.
-        """
-        return self._graph
-
-    def _set_default_node(self, key):
-        """Create a new node if key not found"""
-        if key not in self._key_to_node_index:
-            self._key_to_node_index[key] = self._graph.add_node(NodeData(key=key, equivs=[]))
-        return self._key_to_node_index[key]
-
-    def add_equivalence(self, gate, equivalent_circuit):
-        """Add a new equivalence to the library. Future queries for the Gate
-        will include the given circuit, in addition to all existing equivalences
-        (including those from base).
-
-        Parameterized Gates (those including `qiskit.circuit.Parameters` in their
-        `Gate.params`) can be marked equivalent to parameterized circuits,
-        provided the parameters match.
-
-        Args:
-            gate (Gate): A Gate instance.
-            equivalent_circuit (QuantumCircuit): A circuit equivalently
-                implementing the given Gate.
-        """
-
-        _raise_if_shape_mismatch(gate, equivalent_circuit)
-        _raise_if_param_mismatch(gate.params, equivalent_circuit.parameters)
-
-        key = Key(name=gate.name, num_qubits=gate.num_qubits)
-        equiv = Equivalence(params=gate.params.copy(), circuit=equivalent_circuit.copy())
-
-        target = self._set_default_node(key)
-        self._graph[target].equivs.append(equiv)
-
-        sources = {
-            Key(name=instruction.operation.name, num_qubits=len(instruction.qubits))
-            for instruction in equivalent_circuit
-        }
-        edges = [
-            (
-                self._set_default_node(source),
-                target,
-                EdgeData(index=self._rule_id, num_gates=len(sources), rule=equiv, source=source),
-            )
-            for source in sources
-        ]
-        self._graph.add_edges_from(edges)
-        self._rule_id += 1
-
-    def has_entry(self, gate):
-        """Check if a library contains any decompositions for gate.
-
-        Args:
-            gate (Gate): A Gate instance.
-
-        Returns:
-            Bool: True if gate has a known decomposition in the library.
-                False otherwise.
-        """
-        key = Key(name=gate.name, num_qubits=gate.num_qubits)
-
-        return key in self._key_to_node_index
-
-    def set_entry(self, gate, entry):
-        """Set the equivalence record for a Gate. Future queries for the Gate
-        will return only the circuits provided.
-
-        Parameterized Gates (those including `qiskit.circuit.Parameters` in their
-        `Gate.params`) can be marked equivalent to parameterized circuits,
-        provided the parameters match.
-
-        Args:
-            gate (Gate): A Gate instance.
-            entry (List['QuantumCircuit']) : A list of QuantumCircuits, each
-                equivalently implementing the given Gate.
-        """
-        for equiv in entry:
-            _raise_if_shape_mismatch(gate, equiv)
-            _raise_if_param_mismatch(gate.params, equiv.parameters)
-
-        node_index = self._set_default_node(Key(name=gate.name, num_qubits=gate.num_qubits))
-        # Remove previous equivalences of this node, leaving in place any later equivalences that
-        # were added that use `gate`.
-        self._graph[node_index].equivs.clear()
-        for parent, child, _ in self._graph.in_edges(node_index):
-            # `child` should always be ourselves, but there might be parallel edges.
-            self._graph.remove_edge(parent, child)
-        for equivalence in entry:
-            self.add_equivalence(gate, equivalence)
-
-    def get_entry(self, gate):
-        """Gets the set of QuantumCircuits circuits from the library which
-        equivalently implement the given Gate.
-
-        Parameterized circuits will have their parameters replaced with the
-        corresponding entries from Gate.params.
-
-        Args:
-            gate (Gate) - Gate: A Gate instance.
-
-        Returns:
-            List[QuantumCircuit]: A list of equivalent QuantumCircuits. If empty,
-                library contains no known decompositions of Gate.
-
-                Returned circuits will be ordered according to their insertion in
-                the library, from earliest to latest, from top to base. The
-                ordering of the StandardEquivalenceLibrary will not generally be
-                consistent across Qiskit versions.
-        """
-        key = Key(name=gate.name, num_qubits=gate.num_qubits)
-        query_params = gate.params
-
-        return [_rebind_equiv(equiv, query_params) for equiv in self._get_equivalences(key)]
-
-    def keys(self):
-        """Return list of keys to key to node index map.
-
-        Returns:
-            List: Keys to the key to node index map.
-        """
-        return self._key_to_node_index.keys()
-
-    def node_index(self, key):
-        """Return node index for a given key.
-
-        Args:
-            key (Key): Key to an equivalence.
-
-        Returns:
-            Int: Index to the node in the graph for the given key.
-        """
-        return self._key_to_node_index[key]
-
     def draw(self, filename=None):
         """Draws the equivalence relations available in the library.
 
@@ -227,12 +61,13 @@ def _build_basis_graph(self):
         graph = rx.PyDiGraph()
 
         node_map = {}
-        for key in self._key_to_node_index:
-            name, num_qubits = key
+        for key in super().keys():
+            name, num_qubits = key.name, key.num_qubits
             equivalences = self._get_equivalences(key)
 
             basis = frozenset([f"{name}/{num_qubits}"])
-            for params, decomp in equivalences:
+            for equivalence in equivalences:
+                params, decomp = equivalence.params, equivalence.circuit
                 decomp_basis = frozenset(
                     f"{name}/{num_qubits}"
                     for name, num_qubits in {
@@ -257,39 +92,3 @@ def _build_basis_graph(self):
                 )
 
         return graph
-
-    def _get_equivalences(self, key):
-        """Get all the equivalences for the given key"""
-        return (
-            self._graph[self._key_to_node_index[key]].equivs
-            if key in self._key_to_node_index
-            else []
-        )
-
-
-def _raise_if_param_mismatch(gate_params, circuit_parameters):
-    gate_parameters = [p for p in gate_params if isinstance(p, ParameterExpression)]
-
-    if set(gate_parameters) != circuit_parameters:
-        raise CircuitError(
-            "Cannot add equivalence between circuit and gate "
-            f"of different parameters. Gate params: {gate_parameters}. "
-            f"Circuit params: {circuit_parameters}."
-        )
-
-
-def _raise_if_shape_mismatch(gate, circuit):
-    if gate.num_qubits != circuit.num_qubits or gate.num_clbits != circuit.num_clbits:
-        raise CircuitError(
-            "Cannot add equivalence between circuit and gate "
-            f"of different shapes. Gate: {gate.num_qubits} qubits and {gate.num_clbits} clbits. "
-            f"Circuit: {circuit.num_qubits} qubits and {circuit.num_clbits} clbits."
-        )
-
-
-def _rebind_equiv(equiv, query_params):
-    equiv_params, equiv_circuit = equiv
-    param_map = {x: y for x, y in zip(equiv_params, query_params) if isinstance(x, Parameter)}
-    equiv = equiv_circuit.assign_parameters(param_map, inplace=False, flat_input=True)
-
-    return equiv
diff --git a/qiskit/transpiler/passes/basis/basis_translator.py b/qiskit/transpiler/passes/basis/basis_translator.py
index 5737385af15b..0fdae973e1b9 100644
--- a/qiskit/transpiler/passes/basis/basis_translator.py
+++ b/qiskit/transpiler/passes/basis/basis_translator.py
@@ -13,6 +13,7 @@
 
 """Translates gates to a target basis using a given equivalence library."""
 
+import random
 import time
 import logging
 
@@ -32,7 +33,7 @@
 )
 from qiskit.dagcircuit import DAGCircuit, DAGOpNode
 from qiskit.converters import circuit_to_dag, dag_to_circuit
-from qiskit.circuit.equivalence import Key, NodeData
+from qiskit.circuit.equivalence import Key, NodeData, Equivalence
 from qiskit.transpiler.basepasses import TransformationPass
 from qiskit.transpiler.exceptions import TranspilerError
 from qiskit.circuit.controlflow import CONTROL_FLOW_OP_NAMES
@@ -559,7 +560,7 @@ def _basis_search(equiv_lib, source_basis, target_basis):
     logger.debug("Begining basis search from %s to %s.", source_basis, target_basis)
 
     source_basis = {
-        (gate_name, gate_num_qubits)
+        Key(gate_name, gate_num_qubits)
         for gate_name, gate_num_qubits in source_basis
         if gate_name not in target_basis
     }
@@ -577,7 +578,12 @@ def _basis_search(equiv_lib, source_basis, target_basis):
 
     # we add a dummy node and connect it with gates in the target basis.
     # we'll start the search from this dummy node.
-    dummy = graph.add_node(NodeData(key="key", equivs=[("dummy starting node", 0)]))
+    dummy = graph.add_node(
+        NodeData(
+            key=Key("".join(chr(random.randint(0, 26) + 97) for _ in range(10)), 0),
+            equivs=[Equivalence([], QuantumCircuit(0, name="dummy starting node"))],
+        )
+    )
 
     try:
         graph.add_edges_from_no_data(
diff --git a/qiskit/transpiler/passes/synthesis/high_level_synthesis.py b/qiskit/transpiler/passes/synthesis/high_level_synthesis.py
index 31036de69645..733d5465fd0a 100644
--- a/qiskit/transpiler/passes/synthesis/high_level_synthesis.py
+++ b/qiskit/transpiler/passes/synthesis/high_level_synthesis.py
@@ -673,7 +673,7 @@ def _definitely_skip_node(
                 or (
                     self._equiv_lib is not None
                     and equivalence.Key(name=node.name, num_qubits=node.num_qubits)
-                    in self._equiv_lib._key_to_node_index
+                    in self._equiv_lib.keys()
                 )
             )
         )
diff --git a/test/python/circuit/test_equivalence.py b/test/python/circuit/test_equivalence.py
index 286941313ae9..0b523f7e5249 100644
--- a/test/python/circuit/test_equivalence.py
+++ b/test/python/circuit/test_equivalence.py
@@ -21,10 +21,10 @@
 from qiskit.circuit import QuantumCircuit, Parameter, Gate
 from qiskit.circuit.library import U2Gate
 from qiskit.circuit.exceptions import CircuitError
+from qiskit.circuit.quantumregister import Qubit
 from qiskit.converters import circuit_to_instruction, circuit_to_gate
-from qiskit.circuit import EquivalenceLibrary
+from qiskit.circuit.equivalence import EquivalenceLibrary, Key, Equivalence, NodeData, EdgeData
 from qiskit.utils import optionals
-from qiskit.circuit.equivalence import Key, Equivalence, NodeData, EdgeData
 from test import QiskitTestCase  # pylint: disable=wrong-import-order
 
 from ..visualization.visualization import QiskitVisualizationTestCase, path_to_diagram_reference
@@ -70,7 +70,7 @@ def test_add_single_entry(self):
         eq_lib = EquivalenceLibrary()
 
         gate = OneQubitZeroParamGate()
-        equiv = QuantumCircuit(1)
+        equiv = QuantumCircuit([Qubit()])
         equiv.h(0)
 
         eq_lib.add_equivalence(gate, equiv)
@@ -86,12 +86,12 @@ def test_add_double_entry(self):
         eq_lib = EquivalenceLibrary()
 
         gate = OneQubitZeroParamGate()
-        first_equiv = QuantumCircuit(1)
+        first_equiv = QuantumCircuit([Qubit()])
         first_equiv.h(0)
 
         eq_lib.add_equivalence(gate, first_equiv)
 
-        second_equiv = QuantumCircuit(1)
+        second_equiv = QuantumCircuit([Qubit()])
         second_equiv.append(U2Gate(0, np.pi), [0])
 
         eq_lib.add_equivalence(gate, second_equiv)
@@ -110,12 +110,12 @@ def test_set_entry(self):
         gates = {key: Gate(key, 1, []) for key in "abcd"}
         target = Gate("target", 1, [])
 
-        old = QuantumCircuit(1)
+        old = QuantumCircuit([Qubit()])
         old.append(gates["a"], [0])
         old.append(gates["b"], [0])
         eq_lib.add_equivalence(target, old)
 
-        outbound = QuantumCircuit(1)
+        outbound = QuantumCircuit([Qubit()])
         outbound.append(target, [0])
         eq_lib.add_equivalence(gates["c"], outbound)
 
@@ -127,7 +127,7 @@ def test_set_entry(self):
         self.assertTrue(eq_lib.graph.has_edge(gate_indices["b"], gate_indices["target"]))
         self.assertTrue(eq_lib.graph.has_edge(gate_indices["target"], gate_indices["c"]))
 
-        new = QuantumCircuit(1)
+        new = QuantumCircuit([Qubit()])
         new.append(gates["d"], [0])
         eq_lib.set_entry(target, [new])
 
@@ -146,12 +146,12 @@ def test_set_entry_parallel_edges(self):
         gates = {key: Gate(key, 1, []) for key in "abcd"}
         target = Gate("target", 1, [])
 
-        old_1 = QuantumCircuit(1, name="a")
+        old_1 = QuantumCircuit([Qubit()], name="a")
         old_1.append(gates["a"], [0])
         old_1.append(gates["b"], [0])
         eq_lib.add_equivalence(target, old_1)
 
-        old_2 = QuantumCircuit(1, name="b")
+        old_2 = QuantumCircuit([Qubit()], name="b")
         old_2.append(gates["b"], [0])
         old_2.append(gates["a"], [0])
         eq_lib.add_equivalence(target, old_2)
@@ -159,13 +159,13 @@ def test_set_entry_parallel_edges(self):
         # This extra rule is so that 'a' still has edges, so we can do an exact isomorphism test.
         # There's not particular requirement for `set_entry` to remove orphan nodes, so we'll just
         # craft a test that doesn't care either way.
-        a_to_b = QuantumCircuit(1)
+        a_to_b = QuantumCircuit([Qubit()])
         a_to_b.append(gates["b"], [0])
         eq_lib.add_equivalence(gates["a"], a_to_b)
 
         self.assertEqual(sorted(eq_lib.get_entry(target), key=lambda qc: qc.name), [old_1, old_2])
 
-        new = QuantumCircuit(1, name="c")
+        new = QuantumCircuit([Qubit()], name="c")
         # No more use of 'a', but re-use 'b' and introduce 'c'.
         new.append(gates["b"], [0])
         new.append(gates["c"], [0])
@@ -204,7 +204,7 @@ def test_has_entry(self):
         eq_lib = EquivalenceLibrary()
 
         gate = OneQubitZeroParamGate()
-        equiv = QuantumCircuit(1)
+        equiv = QuantumCircuit([Qubit()])
         equiv.h(0)
 
         eq_lib.add_equivalence(gate, equiv)
@@ -225,7 +225,7 @@ def test_equivalence_graph(self):
         eq_lib = EquivalenceLibrary()
 
         gate = OneQubitZeroParamGate()
-        first_equiv = QuantumCircuit(1)
+        first_equiv = QuantumCircuit([Qubit()])
         first_equiv.h(0)
         eq_lib.add_equivalence(gate, first_equiv)
 
@@ -282,7 +282,7 @@ def test_get_through_empty_library_to_base(self):
         base = EquivalenceLibrary()
 
         gate = OneQubitZeroParamGate()
-        equiv = QuantumCircuit(1)
+        equiv = QuantumCircuit([Qubit()])
         equiv.h(0)
         base.add_equivalence(gate, equiv)
 
@@ -299,13 +299,13 @@ def test_add_equivalence(self):
         base = EquivalenceLibrary()
 
         gate = OneQubitZeroParamGate()
-        first_equiv = QuantumCircuit(1)
+        first_equiv = QuantumCircuit([Qubit()])
         first_equiv.h(0)
         base.add_equivalence(gate, first_equiv)
 
         eq_lib = EquivalenceLibrary(base=base)
 
-        second_equiv = QuantumCircuit(1)
+        second_equiv = QuantumCircuit([Qubit()])
         second_equiv.append(U2Gate(0, np.pi), [0])
 
         eq_lib.add_equivalence(gate, second_equiv)
@@ -322,13 +322,13 @@ def test_set_entry(self):
         base = EquivalenceLibrary()
 
         gate = OneQubitZeroParamGate()
-        first_equiv = QuantumCircuit(1)
+        first_equiv = QuantumCircuit([Qubit()])
         first_equiv.h(0)
         base.add_equivalence(gate, first_equiv)
 
         eq_lib = EquivalenceLibrary(base=base)
 
-        second_equiv = QuantumCircuit(1)
+        second_equiv = QuantumCircuit([Qubit()])
         second_equiv.append(U2Gate(0, np.pi), [0])
 
         eq_lib.set_entry(gate, [second_equiv])
@@ -344,7 +344,7 @@ def test_has_entry_in_base(self):
         base_eq_lib = EquivalenceLibrary()
 
         gate = OneQubitZeroParamGate()
-        equiv = QuantumCircuit(1)
+        equiv = QuantumCircuit([Qubit()])
         equiv.h(0)
 
         base_eq_lib.add_equivalence(gate, equiv)
@@ -355,7 +355,7 @@ def test_has_entry_in_base(self):
         self.assertTrue(eq_lib.has_entry(OneQubitZeroParamGate()))
 
         gate = OneQubitZeroParamGate()
-        equiv2 = QuantumCircuit(1)
+        equiv2 = QuantumCircuit([Qubit()])
         equiv.append(U2Gate(0, np.pi), [0])
 
         eq_lib.add_equivalence(gate, equiv2)
@@ -383,7 +383,7 @@ def test_raise_if_gate_equiv_parameter_mismatch(self):
         phi = Parameter("phi")
 
         gate = OneQubitOneParamGate(theta)
-        equiv = QuantumCircuit(1)
+        equiv = QuantumCircuit([Qubit()])
         equiv.p(phi, 0)
 
         with self.assertRaises(CircuitError):
@@ -399,7 +399,7 @@ def test_parameter_in_parameter_out(self):
         theta = Parameter("theta")
 
         gate = OneQubitOneParamGate(theta)
-        equiv = QuantumCircuit(1)
+        equiv = QuantumCircuit([Qubit()])
         equiv.p(theta, 0)
 
         eq_lib.add_equivalence(gate, equiv)
@@ -409,7 +409,7 @@ def test_parameter_in_parameter_out(self):
 
         entry = eq_lib.get_entry(gate_phi)
 
-        expected = QuantumCircuit(1)
+        expected = QuantumCircuit([Qubit()])
         expected.p(phi, 0)
 
         self.assertEqual(len(entry), 1)
@@ -423,7 +423,7 @@ def test_partial_parameter_in_parameter_out(self):
         phi = Parameter("phi")
 
         gate = OneQubitTwoParamGate(theta, phi)
-        equiv = QuantumCircuit(1)
+        equiv = QuantumCircuit([Qubit()])
         equiv.u(theta, phi, 0, 0)
 
         eq_lib.add_equivalence(gate, equiv)
@@ -433,7 +433,7 @@ def test_partial_parameter_in_parameter_out(self):
 
         entry = eq_lib.get_entry(gate_partial)
 
-        expected = QuantumCircuit(1)
+        expected = QuantumCircuit([Qubit()])
         expected.u(lam, 1.59, 0, 0)
 
         self.assertEqual(len(entry), 1)
@@ -446,14 +446,14 @@ def test_adding_gate_under_different_parameters(self):
         theta = Parameter("theta")
 
         gate_theta = OneQubitOneParamGate(theta)
-        equiv_theta = QuantumCircuit(1)
+        equiv_theta = QuantumCircuit([Qubit()])
         equiv_theta.p(theta, 0)
 
         eq_lib.add_equivalence(gate_theta, equiv_theta)
 
         phi = Parameter("phi")
         gate_phi = OneQubitOneParamGate(phi)
-        equiv_phi = QuantumCircuit(1)
+        equiv_phi = QuantumCircuit([Qubit()])
         equiv_phi.rz(phi, 0)
 
         eq_lib.add_equivalence(gate_phi, equiv_phi)
@@ -463,10 +463,10 @@ def test_adding_gate_under_different_parameters(self):
 
         entry = eq_lib.get_entry(gate_query)
 
-        first_expected = QuantumCircuit(1)
+        first_expected = QuantumCircuit([Qubit()])
         first_expected.p(lam, 0)
 
-        second_expected = QuantumCircuit(1)
+        second_expected = QuantumCircuit([Qubit()])
         second_expected.rz(lam, 0)
 
         self.assertEqual(len(entry), 2)
@@ -482,13 +482,13 @@ def test_adding_gate_and_partially_specified_gate(self):
 
         # e.g. RGate(theta, phi)
         gate_full = OneQubitTwoParamGate(theta, phi)
-        equiv_full = QuantumCircuit(1)
+        equiv_full = QuantumCircuit([Qubit()])
         equiv_full.append(U2Gate(theta, phi), [0])
 
         eq_lib.add_equivalence(gate_full, equiv_full)
 
         gate_partial = OneQubitTwoParamGate(theta, 0)
-        equiv_partial = QuantumCircuit(1)
+        equiv_partial = QuantumCircuit([Qubit()])
         equiv_partial.rx(theta, 0)
 
         eq_lib.add_equivalence(gate_partial, equiv_partial)
@@ -498,10 +498,10 @@ def test_adding_gate_and_partially_specified_gate(self):
 
         entry = eq_lib.get_entry(gate_query)
 
-        first_expected = QuantumCircuit(1)
+        first_expected = QuantumCircuit([Qubit()])
         first_expected.append(U2Gate(lam, 0), [0])
 
-        second_expected = QuantumCircuit(1)
+        second_expected = QuantumCircuit([Qubit()])
         second_expected.rx(lam, 0)
 
         self.assertEqual(len(entry), 2)
@@ -514,7 +514,7 @@ class TestSessionEquivalenceLibrary(QiskitTestCase):
 
     def test_converter_gate_registration(self):
         """Verify converters register gates in session equivalence library."""
-        qc_gate = QuantumCircuit(2)
+        qc_gate = QuantumCircuit([Qubit() for _ in range(2)])
         qc_gate.h(0)
         qc_gate.cx(0, 1)
 
@@ -522,7 +522,7 @@ def test_converter_gate_registration(self):
 
         bell_gate = circuit_to_gate(qc_gate, equivalence_library=sel)
 
-        qc_inst = QuantumCircuit(2)
+        qc_inst = QuantumCircuit([Qubit() for _ in range(2)])
         qc_inst.h(0)
         qc_inst.cx(0, 1)
 
@@ -539,7 +539,7 @@ def test_converter_gate_registration(self):
 
     def test_gate_decomposition_properties(self):
         """Verify decompositions are accessible via gate properties."""
-        qc = QuantumCircuit(2)
+        qc = QuantumCircuit([Qubit() for _ in range(2)])
         qc.h(0)
         qc.cx(0, 1)
 
@@ -552,7 +552,7 @@ def test_gate_decomposition_properties(self):
         self.assertEqual(len(decomps), 1)
         self.assertEqual(decomps[0], qc)
 
-        qc2 = QuantumCircuit(2)
+        qc2 = QuantumCircuit([Qubit() for _ in range(2)])
         qc2.h([0, 1])
         qc2.cz(0, 1)
         qc2.h(1)
@@ -582,12 +582,12 @@ def test_equivalence_draw(self):
         """Verify EquivalenceLibrary drawing with reference image."""
         sel = EquivalenceLibrary()
         gate = OneQubitZeroParamGate()
-        first_equiv = QuantumCircuit(1)
+        first_equiv = QuantumCircuit([Qubit()])
         first_equiv.h(0)
 
         sel.add_equivalence(gate, first_equiv)
 
-        second_equiv = QuantumCircuit(1)
+        second_equiv = QuantumCircuit([Qubit()])
         second_equiv.append(U2Gate(0, np.pi), [0])
 
         sel.add_equivalence(gate, second_equiv)
diff --git a/test/python/circuit/test_gate_definitions.py b/test/python/circuit/test_gate_definitions.py
index 914ac8bab985..b6f160a644d9 100644
--- a/test/python/circuit/test_gate_definitions.py
+++ b/test/python/circuit/test_gate_definitions.py
@@ -21,6 +21,7 @@
 from qiskit import QuantumCircuit, QuantumRegister
 from qiskit.quantum_info import Operator
 from qiskit.circuit import ParameterVector, Gate, ControlledGate
+from qiskit.circuit.quantumregister import Qubit
 from qiskit.circuit.singleton import SingletonGate, SingletonControlledGate
 from qiskit.circuit.library import standard_gates
 from qiskit.circuit.library import (
@@ -402,11 +403,11 @@ def test_definition_parameters(self, gate_class):
         self.assertGreaterEqual(len(param_entry), 1)
         self.assertGreaterEqual(len(float_entry), 1)
 
-        param_qc = QuantumCircuit(param_gate.num_qubits)
-        float_qc = QuantumCircuit(float_gate.num_qubits)
+        param_qc = QuantumCircuit([Qubit() for _ in range(param_gate.num_qubits)])
+        float_qc = QuantumCircuit([Qubit() for _ in range(float_gate.num_qubits)])
 
-        param_qc.append(param_gate, param_qc.qregs[0])
-        float_qc.append(float_gate, float_qc.qregs[0])
+        param_qc.append(param_gate, param_qc.qubits)
+        float_qc.append(float_gate, float_qc.qubits)
 
         self.assertTrue(any(equiv == param_qc.decompose() for equiv in param_entry))
         self.assertTrue(any(equiv == float_qc.decompose() for equiv in float_entry))