def test_ghz_2_3(self):
"""Produces the 3-qubit GHZ state: `1/sqrt(2) * (|000> + |111>)`."""
e_0, e_1 = ket(2, 0), ket(2, 1)
expected_res = (
1 / np.sqrt(2) *
(tensor_list([e_0, e_0, e_0]) + tensor_list([e_1, e_1, e_1])))
res = ghz(2, 3).toarray()
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
def test_w_state_3(self):
"""The 3-qubit W-state."""
e_0, e_1 = ket(2, 0), ket(2, 1)
expected_res = (
1
/ np.sqrt(3)
* (
tensor_list([e_1, e_0, e_0])
+ tensor_list([e_0, e_1, e_0])
+ tensor_list([e_0, e_0, e_1])
)
)
res = w_state(3)
bool_mat = np.isclose(res, expected_res, atol=0.2)
self.assertEqual(np.all(bool_mat), True)
def test_tensor_list_3(self):
"""Test tensor list with three items."""
e_0, e_1 = ket(2, 0), ket(2, 1)
expected_res = np.kron(np.kron(e_0, e_1), e_0)
res = tensor_list([e_0, e_1, e_0])
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
def test_tensor_list_1(self):
"""Test tensor list with one item."""
e_0 = ket(2, 0)
expected_res = e_0
res = tensor_list([e_0])
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
def test_generalized_w_state(self):
"""Generalized 4-qubit W-state."""
e_0, e_1 = ket(2, 0), ket(2, 1)
expected_res = (
1
/ np.sqrt(30)
* (
tensor_list([e_1, e_0, e_0, e_0])
+ 2 * tensor_list([e_0, e_1, e_0, e_0])
+ 3 * tensor_list([e_0, e_0, e_1, e_0])
+ 4 * tensor_list([e_0, e_0, e_0, e_1])
)
)
coeffs = np.array([1, 2, 3, 4]) / np.sqrt(30)
res = w_state(4, coeffs)
bool_mat = np.isclose(res, expected_res, atol=0.2)
self.assertEqual(np.all(bool_mat), True)
def test_ghz_4_7(self):
r"""
The following generates the following GHZ state in `(C^4)^{\otimes 7}`.
`1/sqrt(30) * (|0000000> + 2|1111111> + 3|2222222> + 4|3333333>)`.
"""
e0_4 = np.array([[1], [0], [0], [0]])
e1_4 = np.array([[0], [1], [0], [0]])
e2_4 = np.array([[0], [0], [1], [0]])
e3_4 = np.array([[0], [0], [0], [1]])
expected_res = (
1 / np.sqrt(30) *
(tensor_list([e0_4, e0_4, e0_4, e0_4, e0_4, e0_4, e0_4]) +
2 * tensor_list([e1_4, e1_4, e1_4, e1_4, e1_4, e1_4, e1_4]) +
3 * tensor_list([e2_4, e2_4, e2_4, e2_4, e2_4, e2_4, e2_4]) +
4 * tensor_list([e3_4, e3_4, e3_4, e3_4, e3_4, e3_4, e3_4])))
res = ghz(4, 7, np.array([1, 2, 3, 4]) / np.sqrt(30)).toarray()
bool_mat = np.isclose(res, expected_res)
self.assertEqual(np.all(bool_mat), True)
def test_tensor_list_0(self):
"""Test tensor empty list."""
expected_res = None
res = tensor_list([])
self.assertEqual(res, expected_res)
def pauli(ind: Union[int, str, List[int], List[str]],
is_sparse: bool = False) -> Union[np.ndarray, sparse.csr_matrix]:
r"""
Produce a Pauli operator.
Provides the 2-by-2 Pauli matrix indicated by the value of `ind`. The
variable `ind = 1` gives the Pauli-X operator, `ind = 2` gives the Pauli-Y
operator, `ind =3` gives the Pauli-Z operator, and `ind = 0`gives the
identity operator. Alternatively, `ind` can be set to "I", "X", "Y", or "Z"
(case insensitive) to indicate the Pauli identity, X, Y, or Z operator.
The 2-by-2 Pauli matrices are defined as the following matrices:
.. math::
\begin{equation}
\begin{aligned}
X = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}, \quad
Y = \begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}, \quad
Z = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}, \quad
I = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}.
\end{aligned}
\end{equation}
References:
[1] Wikipedia: Pauli matrices
https://en.wikipedia.org/wiki/Pauli_matrices
:param ind: The index to indicate which Pauli operator to generate.
:param is_sparse: Returns a sparse matrix if set to True and a non-sparse
matrix if set to False.
"""
if isinstance(ind, (int, str)):
if ind in ("x", "X", 1):
pauli_mat = np.array([[0, 1], [1, 0]])
elif ind in ("y", "Y", 2):
pauli_mat = np.array([[0, -1j], [1j, 0]])
elif ind in ("z", "Z", 3):
pauli_mat = np.array([[1, 0], [0, -1]])
else:
pauli_mat = np.identity(2)
if is_sparse:
pauli_mat = sparse.csr_matrix(pauli_mat)
return pauli_mat
num_qubits = len(ind)
pauli_mats = []
for i in range(num_qubits - 1, -1, -1):
pauli_mats.append(pauli(ind[i], is_sparse))
return tensor_list(pauli_mats)