diff --git a/crates/accelerate/src/equivalence.rs b/crates/accelerate/src/equivalence.rs
index dfa338fba45a..5e7da30f108b 100644
--- a/crates/accelerate/src/equivalence.rs
+++ b/crates/accelerate/src/equivalence.rs
@@ -37,7 +37,7 @@ use rustworkx_core::petgraph::{
 
 use qiskit_circuit::circuit_data::CircuitData;
 use qiskit_circuit::circuit_instruction::OperationFromPython;
-use qiskit_circuit::imports::{ImportOnceCell, QUANTUM_CIRCUIT};
+use qiskit_circuit::imports::{PYDIGRAPH, QUANTUM_CIRCUIT};
 use qiskit_circuit::operations::Param;
 use qiskit_circuit::operations::{Operation, OperationRef};
 use qiskit_circuit::packed_instruction::PackedOperation;
@@ -46,7 +46,6 @@ mod exceptions {
     use pyo3::import_exception_bound;
     import_exception_bound! {qiskit.circuit.exceptions, CircuitError}
 }
-pub static PYDIGRAPH: ImportOnceCell = ImportOnceCell::new("rustworkx", "PyDiGraph");
 
 // Custom Structs
 
@@ -605,7 +604,7 @@ impl EquivalenceLibrary {
         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(),
+            circuit: CircuitFromPython(equivalent_circuit.0.copy(py, true, false)?),
             params: params.into(),
         };
 
diff --git a/crates/circuit/src/imports.rs b/crates/circuit/src/imports.rs
index 67ae9f85897f..b56ddb941697 100644
--- a/crates/circuit/src/imports.rs
+++ b/crates/circuit/src/imports.rs
@@ -74,6 +74,7 @@ pub static PARAMETER_EXPRESSION: ImportOnceCell =
     ImportOnceCell::new("qiskit.circuit.parameterexpression", "ParameterExpression");
 pub static PARAMETER_VECTOR: ImportOnceCell =
     ImportOnceCell::new("qiskit.circuit.parametervector", "ParameterVector");
+pub static PYDIGRAPH: ImportOnceCell = ImportOnceCell::new("rustworkx", "PyDiGraph");
 pub static QUANTUM_CIRCUIT: ImportOnceCell =
     ImportOnceCell::new("qiskit.circuit.quantumcircuit", "QuantumCircuit");
 pub static SINGLETON_GATE: ImportOnceCell =
diff --git a/releasenotes/notes/fix-equivalence-copy-205adf94f0380219.yaml b/releasenotes/notes/fix-equivalence-copy-205adf94f0380219.yaml
new file mode 100644
index 000000000000..d99b97967ba5
--- /dev/null
+++ b/releasenotes/notes/fix-equivalence-copy-205adf94f0380219.yaml
@@ -0,0 +1,6 @@
+---
+fixes:
+  - |
+    Add missing calls to `copy` to the :class:`.Parameter` and :class:`.CircuitData` instances
+    being added to an `Equivalence` when calling :meth:`.EquivalenceLibrary.add_equivalence` in
+    :class:`.EquivalenceLibrary`.
diff --git a/test/python/transpiler/test_basis_translator.py b/test/python/transpiler/test_basis_translator.py
index 820c4f241a1e..75907239d836 100644
--- a/test/python/transpiler/test_basis_translator.py
+++ b/test/python/transpiler/test_basis_translator.py
@@ -14,6 +14,7 @@
 """Test the BasisTranslator pass"""
 
 import os
+from pickle import loads, dumps
 
 from numpy import pi
 import scipy
@@ -21,6 +22,7 @@
 from qiskit import QuantumRegister, ClassicalRegister, QuantumCircuit
 from qiskit import transpile
 from qiskit.circuit import Gate, Parameter, EquivalenceLibrary, Qubit, Clbit, Measure
+from qiskit.circuit.equivalence_library import StandardEquivalenceLibrary as std_eq_lib
 from qiskit.circuit.classical import expr, types
 from qiskit.circuit.library import (
     HGate,
@@ -481,6 +483,29 @@ def test_different_bits(self):
         out = BasisTranslator(std_eqlib, basis).run(circuit_to_dag(base))
         self.assertEqual(set(out.count_ops(recurse=True)), basis)
 
+    def test_equivalence_library_serialization(self):
+        """Test correct serialization of the EquivalenceLibrart that the BasisTranslator
+        depends on."""
+        pass_ = BasisTranslator(
+            equivalence_library=std_eq_lib, target_basis=["cx", "id", "rz", "sx", "x"]
+        )
+
+        # Add a parameter a which we will share between circuits
+        param = Parameter("a")
+        gate = RXGate(param)
+
+        qc = QuantumCircuit(1)
+        qc.append(gate, [0])
+
+        # Before serialization, should not fail.
+        first = pass_.run(circuit_to_dag(qc))
+
+        # After serialization, should not fail.
+        pass_ = BasisTranslator(
+            equivalence_library=loads(dumps(std_eq_lib)), target_basis=["cx", "id", "rz", "sx", "x"]
+        )
+        self.assertEqual(first, pass_.run(circuit_to_dag(qc)))
+
 
 class TestUnrollerCompatability(QiskitTestCase):
     """Tests backward compatability with the Unroller pass.