diff --git a/qiskit/quantum_info/operators/symplectic/sparse_pauli_op.py b/qiskit/quantum_info/operators/symplectic/sparse_pauli_op.py
index 34e524348bf8..50b80ec89051 100644
--- a/qiskit/quantum_info/operators/symplectic/sparse_pauli_op.py
+++ b/qiskit/quantum_info/operators/symplectic/sparse_pauli_op.py
@@ -167,7 +167,16 @@ def __init__(
             # move the phase of `pauli_list` to `self._coeffs`
             phase = pauli_list._phase
             count_y = pauli_list._count_y()
-            self._coeffs = np.asarray((-1j) ** (phase - count_y) * coeffs, dtype=coeffs.dtype)
+
+            # Compute exponentiation via integer arithmetic and lookup table to avoid
+            # floating point errors
+            exponent = (phase - count_y) % 4
+            lookup = np.array([1 + 0j, -1j, -1 + 0j, 1j], dtype=coeffs.dtype)
+
+            vals = lookup[exponent]
+            self._coeffs = vals * coeffs
+
+            # Update pauli_list phase
             pauli_list._phase = np.mod(count_y, 4)
             self._pauli_list = pauli_list
 
diff --git a/releasenotes/notes/sparse-pauli-op-heavy-weight-fix-aa822428643d642a.yaml b/releasenotes/notes/sparse-pauli-op-heavy-weight-fix-aa822428643d642a.yaml
new file mode 100644
index 000000000000..d285c6fb6857
--- /dev/null
+++ b/releasenotes/notes/sparse-pauli-op-heavy-weight-fix-aa822428643d642a.yaml
@@ -0,0 +1,7 @@
+fixes:
+  - |
+    Fixed a bug where a initializing :class:`.SparsePauliOp` with a large
+    number of Pauli-``Y`` terms (typically :math:`\geq 100`) and no explicit
+    ``coeffs`` would result in a coefficient close to 1 but with a floating point
+    error. The coefficient is now correctly 1 per default.
+    Fixed `#13522 <https://github.com/Qiskit/qiskit/issues/13522>`__.
\ No newline at end of file
diff --git a/test/python/quantum_info/operators/symplectic/test_sparse_pauli_op.py b/test/python/quantum_info/operators/symplectic/test_sparse_pauli_op.py
index 766011042e43..3602492030c4 100644
--- a/test/python/quantum_info/operators/symplectic/test_sparse_pauli_op.py
+++ b/test/python/quantum_info/operators/symplectic/test_sparse_pauli_op.py
@@ -140,6 +140,11 @@ def test_sparse_pauli_op_init(self):
             coeffs[:] = 0
             self.assertEqual(spp_op, ref_op)
 
+    def test_sparse_pauli_op_init_long_ys(self):
+        """Test heavy-weight SparsePauliOp initialization."""
+        y = SparsePauliOp("Y" * 1000)
+        self.assertEqual(1, y.coeffs[0])
+
 
 @ddt.ddt
 class TestSparsePauliOpConversions(QiskitTestCase):