def test_real_numbers(self):
for m, n in self.test_dimensions:
# Obtain a random real matrix of orthonormal rows
Q = random_unitary_matrix(n, real=True)
Q = Q[:m, :]
# Get Givens decomposition of Q
givens_rotations, V, diagonal = givens_decomposition(Q)
# Compute U
U = numpy.eye(n, dtype=complex)
for parallel_set in givens_rotations:
combined_givens = numpy.eye(n, dtype=complex)
for i, j, theta, phi in reversed(parallel_set):
c = numpy.cos(theta)
s = numpy.sin(theta)
phase = numpy.exp(1.j * phi)
G = numpy.array([[c, -phase * s],
[s, phase * c]], dtype=complex)
givens_rotate(combined_givens, G, i, j)
U = combined_givens.dot(U)
# Compute V * Q * U^\dagger
W = V.dot(Q.dot(U.T.conj()))
# Construct the diagonal matrix
D = numpy.zeros((m, n), dtype=complex)
D[numpy.diag_indices(m)] = diagonal
# Assert that W and D are the same
for i in range(m):
for j in range(n):
self.assertAlmostEqual(D[i, j], W[i, j])
def test_col_eliminate():
"""
Test elimination by rotating in the column space. Left multiplication of
inverse givens
"""
dim = 3
u_generator = numpy.random.random((dim, dim)) + 1j * numpy.random.random(
(dim, dim))
u_generator = u_generator - numpy.conj(u_generator).T
# make sure the generator is actually antihermitian
assert numpy.allclose(-1 * u_generator, numpy.conj(u_generator).T)
unitary = scipy.linalg.expm(u_generator)
# eliminate U[1, 0] by rotation in rows [0, 1] and
# mixing U[1, 0] and U[0, 0]
unitary_original = unitary.copy()
gmat = givens_matrix_elements(unitary[0, 0], unitary[1, 0], which='right')
vec = numpy.array([[unitary[0, 0]], [unitary[1, 0]]])
fullgmat = create_givens(gmat, 0, 1, 3)
zeroed_unitary = fullgmat.dot(unitary)
givens_rotate(unitary, gmat, 0, 1)
assert numpy.isclose(unitary[1, 0], 0.0)
assert numpy.allclose(unitary.real, zeroed_unitary.real)
assert numpy.allclose(unitary.imag, zeroed_unitary.imag)
# eliminate U[2, 0] by rotating columns [0, 1] and
# mixing U[2, 0] and U[2, 1].
unitary = unitary_original.copy()
gmat = givens_matrix_elements(unitary[2, 0], unitary[2, 1], which='left')
vec = numpy.array([[unitary[2, 0]], [unitary[2, 1]]])
assert numpy.isclose((gmat.dot(vec))[0, 0], 0.0)
assert numpy.isclose((vec.T.dot(gmat.T))[0, 0], 0.0)
fullgmat = create_givens(gmat, 0, 1, 3)
zeroed_unitary = unitary.dot(fullgmat.T)
# because col takes g[0, 0] * col_i + g[0, 1].conj() * col_j -> col_i
# this is equivalent ot left multiplication by gmat.T
givens_rotate(unitary, gmat.conj(), 0, 1, which='col')
assert numpy.isclose(zeroed_unitary[2, 0], 0.0)
assert numpy.allclose(unitary, zeroed_unitary)
def test_row_eliminate():
"""
Test elemination of element in U[i, j] by rotating in i-1 and i.
"""
dim = 3
u_generator = numpy.random.random((dim, dim)) + 1j * numpy.random.random(
(dim, dim))
u_generator = u_generator - numpy.conj(u_generator).T
# make sure the generator is actually antihermitian
assert numpy.allclose(-1 * u_generator, numpy.conj(u_generator).T)
unitary = scipy.linalg.expm(u_generator)
# eliminate U[2, 0] by rotating in 1, 2
gmat = givens_matrix_elements(unitary[1, 0], unitary[2, 0], which='right')
givens_rotate(unitary, gmat, 1, 2, which='row')
assert numpy.isclose(unitary[2, 0], 0.0)
# eliminate U[1, 0] by rotating in 0, 1
gmat = givens_matrix_elements(unitary[0, 0], unitary[1, 0], which='right')
givens_rotate(unitary, gmat, 0, 1, which='row')
assert numpy.isclose(unitary[1, 0], 0.0)
# eliminate U[2, 1] by rotating in 1, 2
gmat = givens_matrix_elements(unitary[1, 1], unitary[2, 1], which='right')
givens_rotate(unitary, gmat, 1, 2, which='row')
assert numpy.isclose(unitary[2, 1], 0.0)
def test_bad_input(self):
"""Test bad input."""
with self.assertRaises(ValueError):
v = numpy.random.randn(2)
G = givens_matrix_elements(v[0], v[1])
givens_rotate(v, G, 0, 1, which='a')
def test_main_procedure(self):
for n in self.test_dimensions:
# Obtain a random quadratic Hamiltonian
quadratic_hamiltonian = random_quadratic_hamiltonian(n)
# Get the diagonalizing transformation
transformation_matrix = (
quadratic_hamiltonian.diagonalizing_bogoliubov_transform())
left_block = transformation_matrix[:, :n]
right_block = transformation_matrix[:, n:]
lower_unitary = numpy.empty((n, 2 * n), dtype=complex)
lower_unitary[:, :n] = numpy.conjugate(right_block)
lower_unitary[:, n:] = numpy.conjugate(left_block)
# Get fermionic Gaussian decomposition of lower_unitary
decomposition, left_decomposition, diagonal, left_diagonal = (
fermionic_gaussian_decomposition(lower_unitary))
# Compute left_unitary
left_unitary = numpy.eye(n, dtype=complex)
for parallel_set in left_decomposition:
combined_op = numpy.eye(n, dtype=complex)
for op in reversed(parallel_set):
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)
givens_rotate(combined_op, givens_rotation, i, j)
left_unitary = combined_op.dot(left_unitary)
for i in range(n):
left_unitary[i] *= left_diagonal[i]
left_unitary = left_unitary.T
for i in range(n):
left_unitary[i] *= diagonal[i]
# 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 reversed(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 optimal_givens_decomposition(
qubits: Sequence[cirq.Qid],
unitary: numpy.ndarray) -> Iterable[cirq.Operation]:
r"""
Implement a circuit that provides the unitary that is generated by
single-particle fermion generators
.. math::
U(v) = exp(log(v)_{p,q}(a_{p}^{\dagger}a_{q} - a_{q}^{\dagger}a_{p})
This can be used for implementing an exact single-body basis rotation
Args:
qubits: Sequence of qubits to apply the operations over. The qubits
should be ordered in linear physical order.
unitary:
"""
N = unitary.shape[0]
right_rotations = []
left_rotations = []
for i in range(1, N):
if i % 2 == 1:
for j in range(0, i):
# eliminate U[N - j, i - j] by mixing U[N - j, i - j],
# U[N - j, i - j - 1] by right multiplication
# of a givens rotation matrix in column [i - j, i - j + 1]
gmat = givens_matrix_elements(unitary[N - j - 1, i - j - 1],
unitary[N - j - 1,
i - j - 1 + 1],
which='left')
right_rotations.append((gmat.T, (i - j - 1, i - j)))
givens_rotate(unitary,
gmat.conj(),
i - j - 1,
i - j,
which='col')
else:
for j in range(1, i + 1):
# elimination of U[N + j - i, j] by mixing U[N + j - i, j] and
# U[N + j - i - 1, j] by left multiplication
# of a givens rotation that rotates row space
# [N + j - i - 1, N + j - i
gmat = givens_matrix_elements(unitary[N + j - i - 1 - 1,
j - 1],
unitary[N + j - i - 1, j - 1],
which='right')
left_rotations.append((gmat, (N + j - i - 2, N + j - i - 1)))
givens_rotate(unitary,
gmat,
N + j - i - 2,
N + j - i - 1,
which='row')
new_left_rotations = []
for (left_gmat, (i, j)) in reversed(left_rotations):
phase_matrix = numpy.diag([unitary[i, i], unitary[j, j]])
matrix_to_decompose = left_gmat.conj().T.dot(phase_matrix)
new_givens_matrix = givens_matrix_elements(matrix_to_decompose[1, 0],
matrix_to_decompose[1, 1],
which='left')
new_phase_matrix = matrix_to_decompose.dot(new_givens_matrix.T)
# check if T_{m,n}^{-1}D = D T.
# coverage: ignore
if not numpy.allclose(new_phase_matrix.dot(new_givens_matrix.conj()),
matrix_to_decompose):
raise GivensTranspositionError("Failed to shift the phase matrix "
"from right to left")
# coverage: ignore
unitary[i, i], unitary[j, j] = new_phase_matrix[0,
0], new_phase_matrix[1,
1]
new_left_rotations.append((new_givens_matrix.conj(), (i, j)))
phases = numpy.diag(unitary)
rotations = []
ordered_rotations = []
for (gmat, (i, j)) in list(reversed(new_left_rotations)) + list(
map(lambda x: (x[0].conj().T, x[1]), reversed(right_rotations))):
ordered_rotations.append((gmat, (i, j)))
# if this throws the impossible has happened
# coverage: ignore
if not numpy.isclose(gmat[0, 0].imag, 0.0):
raise GivensMatrixError(
"Givens matrix does not obey our convention that all elements "
"in the first column are real")
if not numpy.isclose(gmat[1, 0].imag, 0.0):
raise GivensMatrixError(
"Givens matrix does not obey our convention that all elements "
"in the first column are real")
# coverage: ignore
theta = numpy.arcsin(numpy.real(gmat[1, 0]))
phi = numpy.angle(gmat[1, 1])
rotations.append((i, j, theta, phi))
for op in reversed(rotations):
i, j, theta, phi = cast(Tuple[int, int, float, float], op)
if not numpy.isclose(phi, 0.0):
yield cirq.Z(qubits[j])**(phi / numpy.pi)
yield Ryxxy(-theta).on(qubits[i], qubits[j])
for idx, phase in enumerate(phases):
yield cirq.Z(qubits[idx])**(numpy.angle(phase) / numpy.pi)