Channel matrix (#4146)

Author: viathor

cirq-core/cirq/__init__.py

@@ -336,8 +336,10 @@
     dirac_notation,
     eye_tensor,
     fidelity,
+    kraus_to_channel_matrix,
     kraus_to_choi,
     one_hot,
+    operation_to_channel_matrix,
     operation_to_choi,
     QUANTUM_STATE_LIKE,
     QuantumState,

cirq-core/cirq/qis/__init__.py

@@ -15,7 +15,9 @@
 """Tools and methods for quantum information science."""
 
 from cirq.qis.channels import (
+    kraus_to_channel_matrix,
     kraus_to_choi,
+    operation_to_channel_matrix,
     operation_to_choi,
 )
 

cirq-core/cirq/qis/channels.py

@@ -29,6 +29,15 @@ def kraus_to_choi(kraus_operators: Sequence[np.ndarray]) -> np.ndarray:
     return c
 
 
+def kraus_to_channel_matrix(kraus_operators: Sequence[np.ndarray]) -> np.ndarray:
+    """Returns the matrix representation of the linear map with given Kraus operators."""
+    d_out, d_in = kraus_operators[0].shape
+    m = np.zeros((d_out * d_out, d_in * d_in), dtype=np.complex128)
+    for k in kraus_operators:
+        m += np.kron(k, k.conj())
+    return m
+
+
 def operation_to_choi(operation: 'protocols.SupportsChannel') -> np.ndarray:
     r"""Returns the unique Choi matrix associated with a superoperator.
 
@@ -49,3 +58,20 @@ def operation_to_choi(operation: 'protocols.SupportsChannel') -> np.ndarray:
         Choi matrix corresponding to operation.
     """
     return kraus_to_choi(protocols.channel(operation))
+
+
+def operation_to_channel_matrix(operation: 'protocols.SupportsChannel') -> np.ndarray:
+    """Returns the matrix representation of a superoperator in standard basis.
+
+    Let E: L(H1) -> L(H2) denote a linear map which takes linear operators on Hilbert space H1
+    to linear operators on Hilbert space H2 and let d1 = dim H1 and d2 = dim H2. Also, let Fij
+    denote an operator whose matrix has one in ith row and jth column and zeros everywhere else.
+    Note that d1-by-d1 operators Fij form a basis of L(H1). Similarly, d2-by-d2 operators Fij
+    form a basis of L(H2). This function returns the matrix of E in these bases.
+
+    Args:
+        operation: Quantum channel.
+    Returns:
+        Matrix representation of operation.
+    """
+    return kraus_to_channel_matrix(protocols.channel(operation))

cirq-core/cirq/qis/channels_test.py

@@ -12,6 +12,8 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 """Tests for channels."""
+from typing import Iterable
+
 import numpy as np
 import pytest
 
@@ -28,32 +30,38 @@ def apply_channel(channel: cirq.SupportsChannel, rho: np.ndarray) -> np.ndarray:
     return out
 
 
-def expected_choi(channel: cirq.SupportsChannel) -> np.ndarray:
+def generate_standard_operator_basis(d_out: int, d_in: int) -> Iterable[np.ndarray]:
+    for i in range(d_out):
+        for j in range(d_in):
+            e_ij = np.zeros((d_out, d_in))
+            e_ij[i, j] = 1
+            yield e_ij
+
+
+def compute_choi(channel: cirq.SupportsChannel) -> np.ndarray:
     ks = cirq.channel(channel)
     d_out, d_in = ks[0].shape
     d = d_in * d_out
     c = np.zeros((d, d), dtype=np.complex128)
-    for i in range(d_in):
-        for j in range(d_in):
-            e_ij = np.zeros((d_in, d_in))
-            e_ij[i, j] = 1
-            c += np.kron(apply_channel(channel, e_ij), e_ij)
+    for e in generate_standard_operator_basis(d_in, d_in):
+        c += np.kron(apply_channel(channel, e), e)
     return c
 
 
+def compute_channel_matrix(channel: cirq.SupportsChannel) -> np.ndarray:
+    ks = cirq.channel(channel)
+    d_out, d_in = ks[0].shape
+    m = np.zeros((d_out * d_out, d_in * d_in), dtype=np.complex128)
+    for k, e_in in enumerate(generate_standard_operator_basis(d_in, d_in)):
+        m[:, k] = np.reshape(apply_channel(channel, e_in), d_out * d_out)
+    return m
+
+
 @pytest.mark.parametrize(
     'kraus_operators, expected_choi',
     (
         ([np.eye(2)], np.array([[1, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [1, 0, 0, 1]])),
-        (
-            [
-                np.eye(2) / 2,
-                np.array([[0, 1], [1, 0]]) / 2,
-                np.array([[0, -1j], [1j, 0]]) / 2,
-                np.diag([1, -1]) / 2,
-            ],
-            np.eye(4) / 2,
-        ),
+        (cirq.channel(cirq.depolarize(0.75)), np.eye(4) / 2),
         (
             [
                 np.array([[1, 0, 0], [0, 0, 1]]) / np.sqrt(2),
@@ -80,14 +88,66 @@ def test_kraus_to_choi(kraus_operators, expected_choi):
     ),
 )
 def test_operation_to_choi(channel):
-    """Verifies that cirq.choi correctly computes the Choi matrix."""
+    """Verifies that cirq.operation_to_choi correctly computes the Choi matrix."""
     n_qubits = cirq.num_qubits(channel)
     actual = cirq.operation_to_choi(channel)
-    expected = expected_choi(channel)
+    expected = compute_choi(channel)
     assert np.isclose(np.trace(actual), 2 ** n_qubits)
     assert np.all(actual == expected)
 
 
-def test_choi_on_completely_dephasing_channel():
-    """Checks that cirq.choi returns the right matrix for the completely dephasing channel."""
+def test_choi_for_completely_dephasing_channel():
+    """Checks cirq.operation_to_choi on the completely dephasing channel."""
     assert np.all(cirq.operation_to_choi(cirq.phase_damp(1)) == np.diag([1, 0, 0, 1]))
+
+
+@pytest.mark.parametrize(
+    'kraus_operators, expected_channel_matrix',
+    (
+        ([np.eye(2)], np.eye(4)),
+        (
+            cirq.channel(cirq.depolarize(0.75)),
+            np.array([[1, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [1, 0, 0, 1]]) / 2,
+        ),
+        (
+            [
+                np.array([[0, 1, 0], [0, 0, 1]]) / np.sqrt(2),
+                np.array([[0, 1, 0], [0, 0, -1]]) / np.sqrt(2),
+            ],
+            np.array(
+                [
+                    [0, 0, 0, 0, 1, 0, 0, 0, 0],
+                    [0, 0, 0, 0, 0, 0, 0, 0, 0],
+                    [0, 0, 0, 0, 0, 0, 0, 0, 0],
+                    [0, 0, 0, 0, 0, 0, 0, 0, 1],
+                ]
+            ),
+        ),
+    ),
+)
+def test_kraus_to_channel_matrix(kraus_operators, expected_channel_matrix):
+    """Verifies that cirq.kraus_to_channel_matrix computes the correct channel matrix."""
+    assert np.allclose(cirq.kraus_to_channel_matrix(kraus_operators), expected_channel_matrix)
+
+
+@pytest.mark.parametrize(
+    'channel',
+    (
+        cirq.I,
+        cirq.X,
+        cirq.CNOT,
+        cirq.depolarize(0.1),
+        cirq.depolarize(0.1, n_qubits=2),
+        cirq.amplitude_damp(0.2),
+    ),
+)
+def test_operation_to_channel_matrix(channel):
+    """Verifies that cirq.channel_matrix correctly computes the channel matrix."""
+    actual = cirq.operation_to_channel_matrix(channel)
+    expected = compute_channel_matrix(channel)
+    assert np.all(actual == expected)
+
+
+def test_channel_matrix_for_completely_dephasing_channel():
+    """Checks cirq.operation_to_channel_matrix on the completely dephasing channel."""
+    assert np.all(cirq.operation_to_channel_matrix(cirq.phase_damp(1)) == np.diag([1, 0, 0, 1]))