def test_kron_spreads_values():
u = np.array([[2, 3], [5, 7]])
assert np.allclose(
combinators.kron(np.eye(2), u),
np.array([[2, 3, 0, 0], [5, 7, 0, 0], [0, 0, 2, 3], [0, 0, 5, 7]]))
assert np.allclose(
combinators.kron(u, np.eye(2)),
np.array([[2, 0, 3, 0], [0, 2, 0, 3], [5, 0, 7, 0], [0, 5, 0, 7]]))
assert np.allclose(
combinators.kron(u, u),
np.array([[4, 6, 6, 9], [10, 14, 15, 21], [10, 15, 14, 21],
[25, 35, 35, 49]]))
def test_kron_spreads_values():
u = np.array([[2, 3], [5, 7]])
assert np.allclose(
combinators.kron(np.eye(2), u),
np.array([[2, 3, 0, 0], [5, 7, 0, 0], [0, 0, 2, 3], [0, 0, 5, 7]]))
assert np.allclose(
combinators.kron(u, np.eye(2)),
np.array([[2, 0, 3, 0], [0, 2, 0, 3], [5, 0, 7, 0], [0, 5, 0, 7]]))
assert np.allclose(
combinators.kron(u, u),
np.array([[4, 6, 6, 9], [10, 14, 15, 21], [10, 15, 14, 21],
[25, 35, 35, 49]]))
def test_kron_multiplies_sizes():
assert np.allclose(combinators.kron(), np.eye(1))
assert np.allclose(combinators.kron(np.eye(1)), np.eye(1))
assert np.allclose(combinators.kron(np.eye(2)), np.eye(2))
assert np.allclose(combinators.kron(np.eye(1), np.eye(1)), np.eye(1))
assert np.allclose(combinators.kron(np.eye(1), np.eye(2)), np.eye(2))
assert np.allclose(combinators.kron(np.eye(2), np.eye(3)), np.eye(6))
assert np.allclose(combinators.kron(np.eye(2), np.eye(3), np.eye(4)),
np.eye(24))
def test_kron_multiplies_sizes():
assert np.allclose(combinators.kron(), np.eye(1))
assert np.allclose(combinators.kron(np.eye(1)), np.eye(1))
assert np.allclose(combinators.kron(np.eye(2)), np.eye(2))
assert np.allclose(combinators.kron(np.eye(1), np.eye(1)), np.eye(1))
assert np.allclose(combinators.kron(np.eye(1), np.eye(2)), np.eye(2))
assert np.allclose(combinators.kron(np.eye(2), np.eye(3)), np.eye(6))
assert np.allclose(combinators.kron(np.eye(2), np.eye(3), np.eye(4)),
np.eye(24))
def kron_factor_4x4_to_2x2s(
matrix: np.ndarray,
tolerance: Tolerance = Tolerance.DEFAULT
) -> Tuple[complex, np.ndarray, np.ndarray]:
"""Splits a 4x4 matrix U = kron(A, B) into A, B, and a global factor.
Requires the matrix to be the kronecker product of two 2x2 unitaries.
Requires the matrix to have a non-zero determinant.
Args:
matrix: The 4x4 unitary matrix to factor.
tolerance: Acceptable numeric error thresholds.
Returns:
A scalar factor and a pair of 2x2 unit-determinant matrices. The
kronecker product of all three is equal to the given matrix.
Raises:
ValueError:
The given matrix can't be tensor-factored into 2x2 pieces.
"""
# Use the entry with the largest magnitude as a reference point.
a, b = max(
((i, j) for i in range(4) for j in range(4)),
key=lambda t: abs(matrix[t]))
# Extract sub-factors touching the reference cell.
f1 = np.zeros((2, 2), dtype=np.complex128)
f2 = np.zeros((2, 2), dtype=np.complex128)
for i in range(2):
for j in range(2):
f1[(a >> 1) ^ i, (b >> 1) ^ j] = matrix[a ^ (i << 1), b ^ (j << 1)]
f2[(a & 1) ^ i, (b & 1) ^ j] = matrix[a ^ i, b ^ j]
# Rescale factors to have unit determinants.
f1 /= (np.sqrt(np.linalg.det(f1)) or 1)
f2 /= (np.sqrt(np.linalg.det(f2)) or 1)
# Determine global phase.
g = matrix[a, b] / (f1[a >> 1, b >> 1] * f2[a & 1, b & 1])
if np.real(g) < 0:
f1 *= -1
g = -g
restored = g * combinators.kron(f1, f2)
if np.any(np.isnan(restored)) or not tolerance.all_close(restored, matrix):
raise ValueError("Can't factor into kronecker product.")
return g, f1, f2
def kron_factor_4x4_to_2x2s(
matrix: np.ndarray,
tolerance: Tolerance = Tolerance.DEFAULT
) -> Tuple[complex, np.ndarray, np.ndarray]:
"""Splits a 4x4 matrix U = kron(A, B) into A, B, and a global factor.
Requires the matrix to be the kronecker product of two 2x2 unitaries.
Requires the matrix to have a non-zero determinant.
Args:
matrix: The 4x4 unitary matrix to factor.
tolerance: Acceptable numeric error thresholds.
Returns:
A scalar factor and a pair of 2x2 unit-determinant matrices. The
kronecker product of all three is equal to the given matrix.
Raises:
ValueError:
The given matrix can't be tensor-factored into 2x2 pieces.
"""
# Use the entry with the largest magnitude as a reference point.
a, b = max(
((i, j) for i in range(4) for j in range(4)),
key=lambda t: abs(matrix[t]))
# Extract sub-factors touching the reference cell.
f1 = np.zeros((2, 2), dtype=np.complex128)
f2 = np.zeros((2, 2), dtype=np.complex128)
for i in range(2):
for j in range(2):
f1[(a >> 1) ^ i, (b >> 1) ^ j] = matrix[a ^ (i << 1), b ^ (j << 1)]
f2[(a & 1) ^ i, (b & 1) ^ j] = matrix[a ^ i, b ^ j]
# Rescale factors to have unit determinants.
f1 /= (np.sqrt(np.linalg.det(f1)) or 1)
f2 /= (np.sqrt(np.linalg.det(f2)) or 1)
# Determine global phase.
g = matrix[a, b] / (f1[a >> 1, b >> 1] * f2[a & 1, b & 1])
if np.real(g) < 0:
f1 *= -1
g = -g
restored = g * combinators.kron(f1, f2)
if np.any(np.isnan(restored)) or not tolerance.all_close(restored, matrix):
raise ValueError("Can't factor into kronecker product.")
return g, f1, f2
def so4_to_magic_su2s(
mat: np.ndarray,
tolerance: Tolerance = Tolerance.DEFAULT
) -> Tuple[np.ndarray, np.ndarray]:
"""Finds 2x2 special-unitaries A, B where mat = Mag.H @ kron(A, B) @ Mag.
Mag is the magic basis matrix:
1 0 0 i
0 i 1 0
0 i -1 0 (times sqrt(0.5) to normalize)
1 0 0 -i
Args:
mat: A real 4x4 orthogonal matrix.
tolerance: Per-matrix-entry tolerance on equality.
Returns:
A pair (A, B) of matrices in SU(2) such that Mag.H @ kron(A, B) @ Mag
is approximately equal to the given matrix.
Raises:
ValueError: Bad matrix.
ArithmeticError: Failed to perform the decomposition to desired
tolerance.
"""
if mat.shape != (4, 4) or not predicates.is_special_orthogonal(mat,
tolerance):
raise ValueError('mat must be 4x4 special orthogonal.')
magic = np.array([[1, 0, 0, 1j],
[0, 1j, 1, 0],
[0, 1j, -1, 0],
[1, 0, 0, -1j]]) * np.sqrt(0.5)
ab = combinators.dot(magic, mat, np.conj(magic.T))
_, a, b = kron_factor_4x4_to_2x2s(ab, tolerance)
# Check decomposition against desired tolerance.
reconstructed = combinators.dot(np.conj(magic.T),
combinators.kron(a, b),
magic)
if not tolerance.all_close(reconstructed, mat):
raise ArithmeticError('Failed to decompose to desired tolerance.')
return a, b
def so4_to_magic_su2s(
mat: np.ndarray,
tolerance: Tolerance = Tolerance.DEFAULT
) -> Tuple[np.ndarray, np.ndarray]:
"""Finds 2x2 special-unitaries A, B where mat = Mag.H @ kron(A, B) @ Mag.
Mag is the magic basis matrix:
1 0 0 i
0 i 1 0
0 i -1 0 (times sqrt(0.5) to normalize)
1 0 0 -i
Args:
mat: A real 4x4 orthogonal matrix.
tolerance: Per-matrix-entry tolerance on equality.
Returns:
A pair (A, B) of matrices in SU(2) such that Mag.H @ kron(A, B) @ Mag
is approximately equal to the given matrix.
Raises:
ValueError: Bad matrix.
ArithmeticError: Failed to perform the decomposition to desired
tolerance.
"""
if mat.shape != (4, 4) or not predicates.is_special_orthogonal(mat,
tolerance):
raise ValueError('mat must be 4x4 special orthogonal.')
magic = np.array([[1, 0, 0, 1j],
[0, 1j, 1, 0],
[0, 1j, -1, 0],
[1, 0, 0, -1j]]) * np.sqrt(0.5)
ab = combinators.dot(magic, mat, np.conj(magic.T))
_, a, b = kron_factor_4x4_to_2x2s(ab, tolerance)
# Check decomposition against desired tolerance.
reconstructed = combinators.dot(np.conj(magic.T),
combinators.kron(a, b),
magic)
if not tolerance.all_close(reconstructed, mat):
raise ArithmeticError('Failed to decompose to desired tolerance.')
return a, b
s, m):
m = np.array(m)
s = np.diag(s)
with pytest.raises(ValueError):
diagonalize.diagonalize_real_symmetric_and_sorted_diagonal_matrices(
m, s)
@pytest.mark.parametrize(
'a,b', [
(np.zeros((0, 0)), np.zeros((0, 0))),
(np.eye(2), np.eye(2)),
(np.eye(4), np.eye(4)),
(np.eye(4), np.zeros((4, 4))),
(H, H),
(combinators.kron(np.eye(2), H), combinators.kron(H, np.eye(2))),
(combinators.kron(np.eye(2), Z), combinators.kron(X, np.eye(2))),
] + [random_bi_diagonalizable_pair(2) for _ in range(10)] +
[random_bi_diagonalizable_pair(4) for _ in range(10)] + [
random_bi_diagonalizable_pair(4, d1, d2) for _ in range(10)
for d1 in range(4) for d2 in range(4)
] + [random_bi_diagonalizable_pair(k) for k in range(1, 10)])
def test_bidiagonalize_real_matrix_pair_with_symmetric_products(a, b):
a = np.array(a)
b = np.array(b)
p, q = diagonalize.bidiagonalize_real_matrix_pair_with_symmetric_products(
a, b)
assert_bidiagonalized_by(a, p, q)
assert_bidiagonalized_by(b, p, q)