def test_isometries_four_by_four(self):
"""Tests that the row (norm) isometry is indeed an isometry for random four by four matrices."""
iden = np.identity(4)
for _ in range(100):
# Get a Hermitian matrix
matrix = np.random.randn(4, 4)
matrix += matrix.conj().T
# Get a QSVE object and compute the row isometry
qsve = QSVE(matrix)
umat = qsve.row_isometry()
vmat = qsve.norm_isometry()
# Make sure U^dagger U = I = V^dagger V
self.assertTrue(np.allclose(umat.conj().T @ umat, iden))
self.assertTrue(np.allclose(vmat.conj().T @ vmat, iden))
# Make sure U^dagger V = V^dagger U = A / ||A||_F
self.assertTrue(
np.allclose(umat.conj().T @ vmat,
matrix / np.linalg.norm(matrix, ord="fro")))
self.assertTrue(
np.allclose(vmat.conj().T @ umat,
matrix / np.linalg.norm(matrix, ord="fro")))
# Make sure rank(U U^dagger) = 4
self.assertEqual(np.linalg.matrix_rank(umat @ umat.conj().T),
len(matrix))
def test_isometries_large_matrix(self):
"""Tests that the row isometry is indeed an isometry for random 16 x 16 matrices."""
iden = np.identity(16)
for _ in range(100):
# Get a random Hermitian matrix
matrix = np.random.randn(16, 16)
matrix += matrix.conj().T
# Get the row isometry
qsve = QSVE(matrix)
umat = qsve.row_isometry()
vmat = qsve.norm_isometry()
# Make sure U^dagger U = I
self.assertTrue(np.allclose(umat.conj().T @ umat, iden))
self.assertTrue(np.allclose(vmat.conj().T @ vmat, iden))
# Make sure U^dagger V = V^dagger U = A / ||A||_F
self.assertTrue(
np.allclose(umat.conj().T @ vmat,
matrix / np.linalg.norm(matrix, ord="fro")))
self.assertTrue(
np.allclose(vmat.conj().T @ umat,
matrix / np.linalg.norm(matrix, ord="fro")))
# Make sure rank(U U^dagger) = 256
self.assertEqual(np.linalg.matrix_rank(vmat @ vmat.conj().T),
len(matrix))
def test_row_isometry_two_by_two(self):
"""Tests that the row isometry is indeed an isometry for random two by two matrices."""
iden = np.identity(2)
for _ in range(100):
# Get a Hermitian matrix
matrix = np.random.randn(2, 2)
matrix += matrix.conj().T
# Create the QSVE object and get the row isometry
qsve = QSVE(matrix)
umat = qsve.row_isometry()
vmat = qsve.norm_isometry()
# Make sure U^dagger U = I
self.assertTrue(np.allclose(umat.conj().T @ umat, iden))
self.assertTrue(np.allclose(vmat.conj().T @ vmat, iden))
# Make sure U^dagger V = V^dagger U = A / ||A||_F
self.assertTrue(
np.allclose(umat.conj().T @ vmat,
matrix / np.linalg.norm(matrix, ord="fro")))
self.assertTrue(
np.allclose(vmat.conj().T @ umat,
matrix / np.linalg.norm(matrix, ord="fro")))
