def so4_to_magic_su2s(
mat: np.ndarray, *, rtol: float = 1e-5, atol: float = 1e-8, check_preconditions: bool = True
) -> 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.
rtol: Per-matrix-entry relative tolerance on equality.
atol: Per-matrix-entry absolute tolerance on equality.
check_preconditions: When set, the code verifies that the given
matrix is from SO(4). Defaults to set.
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.
"""
if check_preconditions:
if mat.shape != (4, 4) or not predicates.is_special_orthogonal(mat, atol=atol, rtol=rtol):
raise ValueError('mat must be 4x4 special orthogonal.')
ab = combinators.dot(MAGIC, mat, MAGIC_CONJ_T)
_, a, b = kron_factor_4x4_to_2x2s(ab)
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.
"""
if mat.shape != (4, 4) or not predicates.is_special_orthogonal(
mat, tolerance):
raise ValueError('mat must be 4x4 special orthogonal.')
ab = combinators.dot(MAGIC, mat, MAGIC_CONJ_T)
_, a, b = kron_factor_4x4_to_2x2s(ab, tolerance)
return a, b
def test_is_special_orthogonal_tolerance():
tol = Tolerance(atol=0.5)
# Pays attention to specified tolerance.
assert predicates.is_special_orthogonal(
np.array([[1, 0], [-0.5, 1]]), tol)
assert not predicates.is_special_orthogonal(
np.array([[1, 0], [-0.6, 1]]), tol)
# Error isn't accumulated across entries, except for determinant factors.
assert predicates.is_special_orthogonal(
np.array([[1.2, 0, 0], [0, 1.2, 0], [0, 0, 1 / 1.2]]), tol)
assert not predicates.is_special_orthogonal(
np.array([[1.2, 0, 0], [0, 1.2, 0], [0, 0, 1.2]]), tol)
assert not predicates.is_special_orthogonal(
np.array([[1.2, 0, 0], [0, 1.3, 0], [0, 0, 1 / 1.2]]), tol)
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
np.linalg.inv(U)
except np.linalg.LinAlgError:
print("Not a valid unitary operation")
# check if it is all real. This lets us use 2 CNOTS
real = True
for row in U:
for element in row:
if np.imag(element) != 0:
real = False
print("Nonreal matrix")
# print("U\n")
# print(U)
print(is_special_orthogonal(U))
print("Determinant = ", np.around(np.linalg.det(U), 2))
mUm = magic_basis @ U @ magic_basis.conjugate().transpose() @ np.kron(
I2, pauli_Z) @ CNOT1 @ CNOT2 @ CNOT1
tuple = kron_factor_4x4_to_2x2s(mUm)
# print(tuple[1])
# print(tuple[2])
print()
print(U)
# print()
# print(np.around(magic_basis.conjugate().transpose() @ U @ magic_basis, 2))
def test_bidiagonalize_unitary_with_special_orthogonals(mat):
p, d, q = diagonalize.bidiagonalize_unitary_with_special_orthogonals(mat)
assert predicates.is_special_orthogonal(p)
assert predicates.is_special_orthogonal(q)
assert np.allclose(p.dot(mat).dot(q), np.diag(d))
assert_bidiagonalized_by(mat, p, q)
def test_is_special_orthogonal():
assert predicates.is_special_orthogonal(np.empty((0, 0)))
assert not predicates.is_special_orthogonal(np.empty((1, 0)))
assert not predicates.is_special_orthogonal(np.empty((0, 1)))
assert predicates.is_special_orthogonal(np.array([[1]]))
assert not predicates.is_special_orthogonal(np.array([[-1]]))
assert not predicates.is_special_orthogonal(np.array([[1j]]))
assert not predicates.is_special_orthogonal(np.array([[5]]))
assert not predicates.is_special_orthogonal(np.array([[3j]]))
assert not predicates.is_special_orthogonal(np.array([[1, 0]]))
assert not predicates.is_special_orthogonal(np.array([[1], [0]]))
assert not predicates.is_special_orthogonal(np.array([[1, 0], [0, -2]]))
assert not predicates.is_special_orthogonal(np.array([[1, 0], [0, -1]]))
assert predicates.is_special_orthogonal(np.array([[-1, 0], [0, -1]]))
assert not predicates.is_special_orthogonal(np.array([[1j, 0], [0, 1]]))
assert not predicates.is_special_orthogonal(np.array([[1, 0], [1, 1]]))
assert not predicates.is_special_orthogonal(np.array([[1, 1], [0, 1]]))
assert not predicates.is_special_orthogonal(np.array([[1, 1], [1, 1]]))
assert not predicates.is_special_orthogonal(np.array([[1, -1], [1, 1]]))
assert predicates.is_special_orthogonal(
np.array([[1, -1], [1, 1]]) * np.sqrt(0.5))
assert not predicates.is_special_orthogonal(
np.array([[1, 1], [1, -1]]) * np.sqrt(0.5))
assert not predicates.is_special_orthogonal(
np.array([[1, 1j], [1j, 1]]) * np.sqrt(0.5))
assert not predicates.is_special_orthogonal(
np.array([[1, -1j], [1j, 1]]) * np.sqrt(0.5))
assert predicates.is_special_orthogonal(
np.array([[1, 1e-11], [0, 1 + 1e-11]]))
