diff --git a/.gitignore b/.gitignore
index 8442ab5..8cce041 100644
--- a/.gitignore
+++ b/.gitignore
@@ -3,5 +3,7 @@ env
 __pycache__
 parity_tensor/_version.py
 build
+.idea
+.venv
 .coverage
 coverage.xml
diff --git a/tests/parity_tensor_test.py b/tests/parity_tensor_test.py
new file mode 100644
index 0000000..b98dad5
--- /dev/null
+++ b/tests/parity_tensor_test.py
@@ -0,0 +1,198 @@
+"""
+[#22](https://github.com/hzhangxyz/parity_tensor/issues/22)
+Test essential functions.
+"""
+from typing import Tuple
+
+import torch
+import pytest
+
+from parity_tensor import ParityTensor
+
+
+@pytest.fixture(name="parity_instance_fx",
+                params=[(
+                    ParityTensor((False, False), ((2, 2), (1, 3)), torch.randn([4, 4])),
+                    ParityTensor((False, False), ((2, 2), (1, 3)), torch.randn([4, 4])),
+                ), (
+                    ParityTensor((True, False, True), ((1, 1), (2, 2), (3, 1)), torch.randn([2, 4, 4])),
+                    ParityTensor((True, False, True), ((1, 1), (2, 2), (3, 1)), torch.randn([2, 4, 4])),
+                ),
+                        (ParityTensor((True, True, False, False), ((1, 2), (2, 2), (1, 1), (3, 1)),
+                                      torch.randn([3, 4, 2, 4])), ParityTensor((True, True, False, False), ((1, 2), (2, 2), (1, 1), (3, 1)), torch.randn([3, 4, 2, 4])))])
+def _parity_instance_fixture(request: pytest.FixtureRequest) -> Tuple[ParityTensor, ParityTensor]:
+    """Fixture for a parity tensor instance."""
+    return request.param
+
+
+@pytest.fixture
+def permute_tuple_fx(parity_instance_fx: Tuple[ParityTensor, ParityTensor]) -> Tuple[int, ...]:
+    """Fixture for a permutation tuple."""
+    parity_tensor, _ = parity_instance_fx
+    rank = parity_tensor.tensor.ndim
+    return tuple(reversed(range(rank)))
+
+
+def test_arithmetic(parity_instance_fx: Tuple[ParityTensor, ParityTensor]) -> None:
+    """Test the arithmetic operators on ParityTensor."""
+    parity_tensor, instance = parity_instance_fx
+
+    # Test __pos__ method.
+    print("-" * 5, "Test __pos__ method", "-" * 5)
+    print(parity_tensor)
+    print(+parity_tensor)
+
+    # Test __neg__ method.
+    print("-" * 5, "Test __neg__ method", "-" * 5)
+    print(parity_tensor)
+    print(-parity_tensor)
+
+    # Test __add__ method.
+    print("-" * 5, "Test __add__ method", "-" * 5)
+    print(parity_tensor)
+    print(parity_tensor + 1)
+    print(parity_tensor + instance)
+
+    # Test __radd__ method.
+    print("-" * 5, "Test __radd__ method", "-" * 5)
+    print(parity_tensor)
+    print(1 + parity_tensor)
+    print(instance + parity_tensor)
+
+    # Test __iadd__ method.
+    print("-" * 5, "Test __iadd__ method", "-" * 5)
+    print(parity_tensor)
+    parity_tensor += 1
+    print(parity_tensor)
+    parity_tensor += instance
+    print(parity_tensor)
+
+    # Test __sub__ method.
+    print("-" * 5, "Test __sub__ method", "-" * 5)
+    print(parity_tensor)
+    print(parity_tensor - 1)
+    print(parity_tensor - instance)
+
+    # Test __rsub__ method.
+    print("-" * 5, "Test __rsub__ method", "-" * 5)
+    print(parity_tensor)
+    print(1 - parity_tensor)
+    print(instance - parity_tensor)
+
+    # Test __isub__ method.
+    print("-" * 5, "Test __isub__ method", "-" * 5)
+    print(parity_tensor)
+    parity_tensor -= 1
+    print(parity_tensor)
+    parity_tensor -= instance
+    print(parity_tensor)
+
+    # Test __mul__ method.
+    print("-" * 5, "Test __mul__ method", "-" * 5)
+    print(parity_tensor)
+    print(parity_tensor * 2)
+
+    # Test __rmul__ method.
+    print("-" * 5, "Test __rmul__ method", "-" * 5)
+    print(parity_tensor)
+    print(2 * parity_tensor)
+
+    # Test __imul__ method.
+    print("-" * 5, "Test __imul__ method", "-" * 5)
+    print(parity_tensor)
+    parity_tensor *= 2
+    print(parity_tensor)
+
+    # Test __truediv__ method.
+    print("-" * 5, "Test __truediv__ method", "-" * 5)
+    print(parity_tensor)
+    print(parity_tensor / 2)
+    print(parity_tensor / instance)
+
+    # Test __rtruediv__ method.
+    print("-" * 5, "Test __rtruediv__ method", "-" * 5)
+    print(parity_tensor)
+    print(2 / parity_tensor)
+
+    # Test __itruediv__ method.
+    print("-" * 5, "Test __itruediv__ method", "-" * 5)
+    print(parity_tensor)
+    parity_tensor /= 2
+    print(parity_tensor)
+
+
+def test_masking(parity_instance_fx: tuple[ParityTensor, ParityTensor]) -> None:
+    """Test masking."""
+    parity_tensor, _ = parity_instance_fx
+
+    mask = parity_tensor.mask
+    assert mask.shape == parity_tensor.tensor.shape, "Mask shape mismatch"
+
+    assert mask.dtype == torch.bool, "Mask must be of dtype bool"
+
+    original_tensor = parity_tensor.tensor.clone()
+    update_tensor = parity_tensor.update_mask().tensor
+
+    assert torch.all(update_tensor[~mask] == 0), "Masked-out positions must be zero"
+
+    assert torch.all(update_tensor[mask] == original_tensor[mask]), ("Unmasked positions must be preserved")
+
+    assert parity_tensor.mask is not None, "Mask should be cached after first access"
+
+
+def test_permute(parity_instance_fx: Tuple[ParityTensor, ParityTensor], permute_tuple_fx: Tuple[int, ...]) -> None:
+    """Test permute."""
+    parity_tensor, _ = parity_instance_fx
+    print("Before permute:", parity_tensor)
+    parity_tensor.permute(permute_tuple_fx)
+    print("After permute:", parity_tensor)
+
+
+def test_reverse() -> None:
+    """Test tensor reverse."""
+    assert True
+
+
+def test_split_edge() -> None:
+    """Test split edge."""
+    assert True
+
+
+def test_merge_edge() -> None:
+    """Test merge edge."""
+    assert True
+
+
+def test_contract() -> None:
+    """Test contract."""
+    assert True
+
+
+def test_trace() -> None:
+    """Test trace."""
+    assert True
+
+
+def test_conjugate() -> None:
+    """Test conjugate."""
+    assert True
+
+
+def test_svd() -> None:
+    """Test svd."""
+    assert True
+
+
+def test_qr() -> None:
+    """Test qr."""
+    assert True
+
+
+def test_identity() -> None:
+    """Test identity."""
+    assert True
+
+
+def test_exponential() -> None:
+    """Test tensor exponential."""
+    assert True