Entanglement fidelity of a quantum channel (#4158)

Author: viathor

cirq-core/cirq/__init__.py

@@ -334,6 +334,7 @@
     density_matrix,
     density_matrix_from_state_vector,
     dirac_notation,
+    entanglement_fidelity,
     eye_tensor,
     fidelity,
     kraus_to_channel_matrix,

cirq-core/cirq/qis/__init__.py

@@ -24,6 +24,7 @@
 from cirq.qis.clifford_tableau import CliffordTableau
 
 from cirq.qis.measures import (
+    entanglement_fidelity,
     fidelity,
     von_neumann_entropy,
 )

cirq-core/cirq/qis/measures.py

@@ -18,7 +18,7 @@
 import numpy as np
 import scipy
 
-from cirq import value
+from cirq import protocols, value
 from cirq.qis.states import (
     QuantumState,
     infer_qid_shape,
@@ -273,3 +273,28 @@ def von_neumann_entropy(
     if validate:
         _ = quantum_state(state, qid_shape=qid_shape, copy=False, validate=True, atol=atol)
     return 0.0
+
+
+def entanglement_fidelity(operation: 'cirq.SupportsChannel') -> float:
+    r"""Returns entanglement fidelity of a given quantum channel.
+
+    Entanglement fidelity $F_e$ of a quantum channel $E: L(H) \to L(H)$ is the overlap between
+    the maximally entangled state $|\phi\rangle = \frac{1}{\sqrt{dim H}} \sum_i|i\rangle|i\rangle$
+    and the state obtained by sending one half of $|\phi\rangle$ through the channel $E$, i.e.
+
+        $$
+        F_e = \langle\phi|(E \otimes I)(|\phi\rangle\langle\phi|)|\phi\rangle
+        $$
+
+    where $I: L(H) \to L(H)$ is the identity map.
+
+    Args:
+        operation: Quantum channel whose entanglement fidelity is to be computed.
+    Returns:
+        Entanglement fidelity of the channel represented by operation.
+    """
+    f = 0.0
+    for k in protocols.channel(operation):
+        f += np.abs(np.trace(k)) ** 2
+    n_qubits = protocols.num_qubits(operation)
+    return float(f / 4 ** n_qubits)

cirq-core/cirq/qis/measures_test.py

@@ -222,3 +222,40 @@ def test_von_neumann_entropy():
         )
         == 0
     )
+
+
+@pytest.mark.parametrize(
+    'gate, expected_entanglement_fidelity',
+    (
+        (cirq.I, 1),
+        (cirq.X, 0),
+        (cirq.Y, 0),
+        (cirq.Z, 0),
+        (cirq.S, 1 / 2),
+        (cirq.CNOT, 1 / 4),
+        (cirq.TOFFOLI, 9 / 16),
+    ),
+)
+def test_entanglement_fidelity_of_unitary_channels(gate, expected_entanglement_fidelity):
+    assert np.isclose(cirq.entanglement_fidelity(gate), expected_entanglement_fidelity)
+
+
+@pytest.mark.parametrize('p', (0, 0.1, 0.2, 0.5, 0.8, 0.9, 1))
+@pytest.mark.parametrize(
+    'channel_factory, entanglement_fidelity_formula',
+    (
+        # Each Pauli error turns the maximally entangled state into an orthogonal state, so only
+        # the error-free term, whose pre-factor is 1 - p, contributes to entanglement fidelity.
+        (cirq.depolarize, lambda p: 1 - p),
+        (lambda p: cirq.depolarize(p, n_qubits=2), lambda p: 1 - p),
+        (lambda p: cirq.depolarize(p, n_qubits=3), lambda p: 1 - p),
+        # See e.g. https://quantumcomputing.stackexchange.com/questions/16074 for average fidelity,
+        # then use Horodecki formula F_avg = (N F_e + 1) / (N + 1) to find entanglement fidelity.
+        (cirq.amplitude_damp, lambda gamma: 1 / 2 - gamma / 4 + np.sqrt(1 - gamma) / 2),
+    ),
+)
+def test_entanglement_fidelity_of_noisy_channels(p, channel_factory, entanglement_fidelity_formula):
+    channel = channel_factory(p)
+    actual_entanglement_fidelity = cirq.entanglement_fidelity(channel)
+    expected_entanglement_fidelity = entanglement_fidelity_formula(p)
+    assert np.isclose(actual_entanglement_fidelity, expected_entanglement_fidelity)