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More channel representation conversion tools #4553

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Oct 5, 2021
Merged

More channel representation conversion tools #4553

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3 changes: 3 additions & 0 deletions cirq-core/cirq/__init__.py
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Original file line number Diff line number Diff line change
Expand Up @@ -361,6 +361,7 @@
from cirq.qis import (
bloch_vector_from_state_vector,
choi_to_kraus,
choi_to_superoperator,
CliffordTableau,
density_matrix,
density_matrix_from_state_vector,
Expand All @@ -379,6 +380,8 @@
QuantumState,
quantum_state,
STATE_VECTOR_LIKE,
superoperator_to_choi,
superoperator_to_kraus,
to_valid_density_matrix,
to_valid_state_vector,
validate_density_matrix,
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3 changes: 3 additions & 0 deletions cirq-core/cirq/qis/__init__.py
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Expand Up @@ -16,12 +16,15 @@

from cirq.qis.channels import (
choi_to_kraus,
choi_to_superoperator,
kraus_to_channel_matrix,
kraus_to_choi,
kraus_to_superoperator,
operation_to_channel_matrix,
operation_to_choi,
operation_to_superoperator,
superoperator_to_choi,
superoperator_to_kraus,
)

from cirq.qis.clifford_tableau import CliffordTableau
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32 changes: 32 additions & 0 deletions cirq-core/cirq/qis/channels.py
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Expand Up @@ -75,6 +75,38 @@ def kraus_to_superoperator(kraus_operators: Sequence[np.ndarray]) -> np.ndarray:
return m


def superoperator_to_kraus(superoperator: np.ndarray) -> Sequence[np.ndarray]:
"""Returns a Kraus representation of a channel specified via the superoperator matrix."""
return choi_to_kraus(superoperator_to_choi(superoperator))


def choi_to_superoperator(choi: np.ndarray) -> np.ndarray:
"""Returns the superoperator matrix of a quantum channel specified via the Choi matrix."""
d = int(np.round(np.sqrt(choi.shape[0])))
if choi.shape != (d * d, d * d):
raise ValueError(f"Invalid Choi matrix shape, expected {(d * d, d * d)}, got {choi.shape}")
if not np.allclose(choi, choi.T.conj()):
raise ValueError("Choi matrix must be Hermitian")

c = np.reshape(choi, (d, d, d, d))
s = np.swapaxes(c, 1, 2)
return np.reshape(s, (d * d, d * d))


def superoperator_to_choi(superoperator: np.ndarray) -> np.ndarray:
"""Returns the Choi matrix of a quantum channel specified via the superoperator matrix."""
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do we have documentation anywhere that defines what these representations are? If no: we should beef up these docstrings to give some context for the conversion.

In any case, it could be helpful to give a little note about how "involved" each conversion is. E.g. add a line: "Conversions between superoperator and choi representations only involve axes manipulation"

"Conversions to kraus require an eigendecomposition"

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@viathor viathor Oct 5, 2021 •
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Good point. I'll send thorough updates to docstrings in a separate PR.

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Done in #4554 (both definitions of the mathematical objects and the notes about complexity).

d = int(np.round(np.sqrt(superoperator.shape[0])))
if superoperator.shape != (d * d, d * d):
raise ValueError(
f"Invalid superoperator matrix shape, expected {(d * d, d * d)}, "
f"got {superoperator.shape}"
)

s = np.reshape(superoperator, (d, d, d, d))
c = np.swapaxes(s, 1, 2)
return np.reshape(c, (d * d, d * d))


def operation_to_choi(operation: 'protocols.SupportsKraus') -> np.ndarray:
r"""Returns the unique Choi matrix associated with an operation .

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180 changes: 180 additions & 0 deletions cirq-core/cirq/qis/channels_test.py
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Expand Up @@ -285,6 +285,186 @@ def test_kraus_to_superoperator(kraus_operators, expected_superoperator):
assert np.allclose(cirq.kraus_to_channel_matrix(kraus_operators), expected_superoperator)


@pytest.mark.parametrize(
'superoperator, expected_kraus_operators',
(
(np.eye(4), [np.eye(2)]),
(np.diag([1, -1, -1, 1]), [np.diag([1, -1])]),
(np.diag([1, -1j, 1j, 1]), [np.diag([1, 1j])]),
(
np.array([[1, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [1, 0, 0, 1]]) / 2,
cirq.kraus(cirq.depolarize(0.75)),
),
),
)
def test_superoperator_to_kraus_fixed_values(superoperator, expected_kraus_operators):
"""Verifies that cirq.kraus_to_superoperator computes the correct channel matrix."""
actual_kraus_operators = cirq.superoperator_to_kraus(superoperator)
for i in (0, 1):
for j in (0, 1):
input_rho = np.zeros((2, 2))
input_rho[i, j] = 1
actual_rho = apply_kraus_operators(actual_kraus_operators, input_rho)
expected_rho = apply_kraus_operators(expected_kraus_operators, input_rho)
assert np.allclose(actual_rho, expected_rho)


@pytest.mark.parametrize(
'superoperator',
(
np.eye(4),
np.diag([1, 0, 0, 1]),
np.diag([1, -1j, 1j, 1]),
np.array(
[
[1, 0, 0, 1],
[0, 0, 0, 0],
[0, 0, 0, 0],
[1, 0, 0, 1],
]
),
np.array(
[
[1, 0, 0, 0.8],
[0, 0.36, 0, 0],
[0, 0, 0.36, 0],
[0, 0, 0, 0.64],
],
),
),
)
def test_superoperator_to_kraus_inverse_of_kraus_to_superoperator(superoperator):
"""Verifies that cirq.kraus_to_superoperator(cirq.superoperator_to_kraus(.)) is identity."""
kraus = cirq.superoperator_to_kraus(superoperator)
recovered_superoperator = cirq.kraus_to_superoperator(kraus)
assert np.allclose(recovered_superoperator, superoperator)


@pytest.mark.parametrize(
'choi, error',
(
(np.array([[1, 2, 3], [4, 5, 6]]), "shape"),
(np.eye(2), "shape"),
(
np.array(
[
[0.6, 0.0, -0.1j, 0.1],
[0.0, 0.0, 0.0, 0.0],
[0.1j, 0.0, 0.4, 0.0],
[0.2, 0.0, 0.0, 1.0],
]
),
"Hermitian",
),
),
)
def test_choi_to_superoperator_invalid_input(choi, error):
with pytest.raises(ValueError, match=error):
_ = cirq.choi_to_superoperator(choi)


@pytest.mark.parametrize(
'superoperator, error',
(
(np.array([[1, 2, 3], [4, 5, 6]]), "shape"),
(np.eye(2), "shape"),
),
)
def test_superoperator_to_choi_invalid_input(superoperator, error):
with pytest.raises(ValueError, match=error):
_ = cirq.superoperator_to_choi(superoperator)


@pytest.mark.parametrize(
'superoperator, choi',
(
(
# Identity channel
np.eye(4),
np.array([[1, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [1, 0, 0, 1]]),
),
(
# S gate
np.diag([1, -1j, 1j, 1]),
np.array([[1, 0, 0, -1j], [0, 0, 0, 0], [0, 0, 0, 0], [1j, 0, 0, 1]]),
),
(
# Hadamard
np.array([[1, 1, 1, 1], [1, -1, 1, -1], [1, 1, -1, -1], [1, -1, -1, 1]]) / 2,
np.array([[1, 1, 1, -1], [1, 1, 1, -1], [1, 1, 1, -1], [-1, -1, -1, 1]]) / 2,
),
(
# Completely dephasing channel
np.diag([1, 0, 0, 1]),
np.diag([1, 0, 0, 1]),
),
(
# Amplitude damping channel
np.array(
[
[1, 0, 0, 0.36],
[0, 0.8, 0, 0],
[0, 0, 0.8, 0],
[0, 0, 0, 0.64],
],
),
np.array(
[
[1, 0, 0, 0.8],
[0, 0.36, 0, 0],
[0, 0, 0, 0],
[0.8, 0, 0, 0.64],
],
),
),
(
# Completely depolarizing channel
np.array([[1, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [1, 0, 0, 1]]) / 2,
np.eye(4) / 2,
),
),
)
def test_superoperator_vs_choi_fixed_values(superoperator, choi):
recovered_choi = cirq.superoperator_to_choi(superoperator)
assert np.allclose(recovered_choi, choi)

recovered_superoperator = cirq.choi_to_superoperator(choi)
assert np.allclose(recovered_superoperator, superoperator)


@pytest.mark.parametrize(
'choi',
(
np.eye(4),
np.diag([1, 0, 0, 1]),
np.diag([0.2, 0.3, 0.8, 0.7]),
np.array(
[
[1, 0, 1, 0],
[0, 1, 0, -1],
[1, 0, 1, 0],
[0, -1, 0, 1],
]
),
np.array(
[
[0.8, 0, 0, 0.5],
[0, 0.3, 0, 0],
[0, 0, 0.2, 0],
[0.5, 0, 0, 0.7],
],
),
),
)
def test_choi_to_superoperator_inverse_of_superoperator_to_choi(choi):
superoperator = cirq.choi_to_superoperator(choi)
recovered_choi = cirq.superoperator_to_choi(superoperator)
assert np.allclose(recovered_choi, choi)

recovered_superoperator = cirq.choi_to_superoperator(recovered_choi)
assert np.allclose(recovered_superoperator, superoperator)


@pytest.mark.parametrize(
'channel',
(
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