diff --git a/test/test_iterdatapipe.py b/test/test_iterdatapipe.py
index 632a66613..e8b0ccaaf 100644
--- a/test/test_iterdatapipe.py
+++ b/test/test_iterdatapipe.py
@@ -744,7 +744,7 @@ def test_unzipper_iterdatapipe(self):
         self.assertEqual(list(range(10, 20)), list(dp2))
         self.assertEqual(list(range(20, 30)), list(dp3))
 
-        (dp2,) = source_dp.unzip(sequence_length=3, columns_to_skip=[0, 2])
+        (dp2,) = source_dp.unzip(sequence_length=3, columns_to_skip=[0, 2], buffer_size=0)
         self.assertEqual(list(range(10, 20)), list(dp2))
 
         source_dp = IterableWrapper([(i, i + 10, i + 20, i + 30) for i in range(10)])
@@ -754,15 +754,19 @@ def test_unzipper_iterdatapipe(self):
 
         # Functional Test: one child DataPipe yields all value first, but buffer_size = 5 being too small, raises error
         source_dp = IterableWrapper([(i, i + 10) for i in range(10)])
-        dp1, dp2 = source_dp.unzip(sequence_length=2, buffer_size=5)
+        dp1, dp2 = source_dp.unzip(sequence_length=2, buffer_size=4)
         it1 = iter(dp1)
-        for _ in range(5):
+        for _ in range(4):
             next(it1)
         with self.assertRaises(BufferError):
             next(it1)
         with self.assertRaises(BufferError):
             list(dp2)
 
+        dp1, dp2 = source_dp.unzip(sequence_length=2, buffer_size=4)
+        with self.assertRaises(BufferError):
+            list(dp2)
+
         # Reset Test: DataPipe resets when a new iterator is created, even if this datapipe hasn't been read
         dp1, dp2 = source_dp.unzip(sequence_length=2)
         _ = iter(dp1)
diff --git a/torchdata/datapipes/iter/util/unzipper.py b/torchdata/datapipes/iter/util/unzipper.py
index 320535872..8f841077b 100644
--- a/torchdata/datapipes/iter/util/unzipper.py
+++ b/torchdata/datapipes/iter/util/unzipper.py
@@ -4,7 +4,7 @@
 # This source code is licensed under the BSD-style license found in the
 # LICENSE file in the root directory of this source tree.
 
-from typing import Optional, Sequence, TypeVar
+from typing import List, Optional, Sequence, TypeVar
 
 from torch.utils.data.datapipes.iter.combining import _ChildDataPipe, _ForkerIterDataPipe
 from torchdata.datapipes import functional_datapipe
@@ -65,11 +65,28 @@ def __new__(
             )
 
         # The implementation basically uses Forker but only yields a specific element within the sequence
-        container = _UnZipperIterDataPipe(source_datapipe, sequence_length, buffer_size)  # type: ignore[arg-type]
-        return [_ChildDataPipe(container, i) for i in instance_ids]
+        container = _UnZipperIterDataPipe(source_datapipe, instance_ids, buffer_size)  # type: ignore[arg-type]
+        return [_ChildDataPipe(container, i) for i in range(len(instance_ids))]
 
 
 class _UnZipperIterDataPipe(_ForkerIterDataPipe):
+    def __init__(self, datapipe: IterDataPipe, instance_ids: List[int], buffer_size: int = 1000):
+        super().__init__(datapipe, len(instance_ids), buffer_size)  # type: ignore[arg-type]
+        self.instance_ids = instance_ids
+
     def get_next_element_by_instance(self, instance_id: int):
+        r"""
+        Note:
+            Each element returned from the source datapipe is required to be a sequnce that can
+            be subscribed with a column index
+        """
         for return_val in super().get_next_element_by_instance(instance_id):
-            yield return_val[instance_id]
+            yield return_val[self.instance_ids[instance_id]]
+
+    def __getstate__(self):
+        state = super().__getstate__()
+        return (*state, self.instance_ids)
+
+    def __setstate__(self, state):
+        super().__setstate__(state[:-1])
+        self.instance_ids = state[-1]