def test_n_equals_9(self):
n = 9
# Obtain a random antisymmetric matrix
rand_mat = numpy.random.randn(2 * n, 2 * n)
antisymmetric_matrix = rand_mat - rand_mat.T
# Get the diagonalizing fermionic unitary
ferm_unitary = diagonalizing_fermionic_unitary(antisymmetric_matrix)
lower_unitary = ferm_unitary[n:]
# Get fermionic Gaussian decomposition of lower_unitary
left_unitary, decomposition, diagonal = (
fermionic_gaussian_decomposition(lower_unitary))
# Check that left_unitary zeroes out the correct entries of
# lower_unitary
product = left_unitary.dot(lower_unitary)
for i in range(n - 1):
for j in range(n - 1 - i):
self.assertAlmostEqual(product[i, j], 0.)
# Compute right_unitary
right_unitary = numpy.eye(2 * n, dtype=complex)
for parallel_set in decomposition:
combined_op = numpy.eye(2 * n, dtype=complex)
for op in parallel_set:
if op == 'pht':
swap_rows(combined_op, n - 1, 2 * n - 1)
else:
i, j, theta, phi = op
c = numpy.cos(theta)
s = numpy.sin(theta)
phase = numpy.exp(1.j * phi)
givens_rotation = numpy.array(
[[c, -phase * s], [s, phase * c]], dtype=complex)
double_givens_rotate(combined_op, givens_rotation, i, j)
right_unitary = combined_op.dot(right_unitary)
# Compute left_unitary * lower_unitary * right_unitary^\dagger
product = left_unitary.dot(lower_unitary.dot(right_unitary.T.conj()))
# Construct the diagonal matrix
diag = numpy.zeros((n, 2 * n), dtype=complex)
diag[range(n), range(n, 2 * n)] = diagonal
# Assert that W and D are the same
for i in numpy.ndindex((n, 2 * n)):
self.assertAlmostEqual(diag[i], product[i])
def test_n_equals_3(self):
n = 3
# Obtain a random antisymmetric matrix
rand_mat = numpy.random.randn(2 * n, 2 * n)
antisymmetric_matrix = rand_mat - rand_mat.T
# Get the diagonalizing fermionic unitary
ferm_unitary = diagonalizing_fermionic_unitary(antisymmetric_matrix)
lower_unitary = ferm_unitary[n:]
lower_left = lower_unitary[:, :n]
lower_right = lower_unitary[:, n:]
# Check that lower_left and lower_right satisfy the constraints
# necessary for the transformed fermionic operators to satisfy
# the fermionic anticommutation relations
constraint_matrix_1 = (lower_left.dot(lower_left.T.conj()) +
lower_right.dot(lower_right.T.conj()))
constraint_matrix_2 = (lower_left.dot(lower_right.T) +
lower_right.dot(lower_left.T))
identity = numpy.eye(n, dtype=complex)
for i in numpy.ndindex((n, n)):
self.assertAlmostEqual(identity[i], constraint_matrix_1[i])
self.assertAlmostEqual(0., constraint_matrix_2[i])
def test_not_antisymmetric(self):
n = 4
ones_mat = numpy.ones((n, n))
with self.assertRaises(ValueError):
ferm_unitary = diagonalizing_fermionic_unitary(ones_mat)
def test_bad_dimensions(self):
n, p = (3, 4)
ones_mat = numpy.ones((n, p))
with self.assertRaises(ValueError):
ferm_unitary = diagonalizing_fermionic_unitary(ones_mat)