def test_square(self):
m, n = (3, 3)
# Obtain a random matrix of orthonormal rows
Q = random_unitary_matrix(n)
Q = Q[:m, :]
Q = Q[:m, :]
# Get Givens decomposition of Q
givens_rotations, V, diagonal = givens_decomposition(Q)
# There should be no Givens rotations
self.assertEqual(givens_rotations, list())
# Compute V * Q * U^\dagger
W = V.dot(Q)
# 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_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_bad_dimensions(self):
m, n = (3, 2)
# Obtain a random matrix of orthonormal rows
Q = random_unitary_matrix(m)
Q = Q[:m, :]
Q = Q[:m, :n]
with self.assertRaises(ValueError):
_ = givens_decomposition(Q)
def test_identity(self):
n = 3
Q = numpy.eye(n, dtype=complex)
givens_rotations, V, diagonal = givens_decomposition(Q)
# V should be the identity
identity = numpy.eye(n, dtype=complex)
for i in range(n):
for j in range(n):
self.assertAlmostEqual(V[i, j], identity[i, j])
# There should be no Givens rotations
self.assertEqual(givens_rotations, list())
# The diagonal should be ones
for d in diagonal:
self.assertAlmostEqual(d, 1.)
def test_antidiagonal(self):
m, n = (3, 3)
Q = numpy.zeros((m, n), dtype=complex)
Q[0, 2] = 1.
Q[1, 1] = 1.
Q[2, 0] = 1.
givens_rotations, V, diagonal = givens_decomposition(Q)
# There should be no Givens rotations
self.assertEqual(givens_rotations, list())
# VQ should equal the diagonal
VQ = V.dot(Q)
D = numpy.zeros((m, n), dtype=complex)
D[numpy.diag_indices(m)] = diagonal
for i in range(n):
for j in range(n):
self.assertAlmostEqual(VQ[i, j], D[i, j])
def slater_determinant_preparation_circuit(slater_determinant_matrix):
"""Obtain the description of a circuit which prepares a Slater determinant.
The input is an :math:`N_f \\times N` matrix :math:`Q` with orthonormal
rows. Such a matrix describes the Slater determinant
.. math::
b^\dagger_1 \cdots b^\dagger_{N_f} \lvert \\text{vac} \\rangle,
where
.. math::
b^\dagger_j = \sum_{k = 1}^N Q_{jk} a^\dagger_k.
The output is the description of a circuit which prepares this
Slater determinant, up to a global phase.
The starting state which the circuit should be applied to
is a Slater determinant (in the computational basis) with
the first :math:`N_f` orbitals filled.
Args:
slater_determinant_matrix: The matrix :math:`Q` which describes the
Slater determinant to be prepared.
Returns:
circuit_description:
A list of operations describing the circuit. Each operation
is a tuple of elementary operations that can be performed in
parallel. Each elementary operation is a tuple of the form
:math:`(i, j, \\theta, \\varphi)`, indicating a Givens rotation
of modes :math:`i` and :math:`j` by angles :math:`\\theta`
and :math:`\\varphi`.
"""
decomposition, left_unitary, diagonal = givens_decomposition(
slater_determinant_matrix)
circuit_description = list(reversed(decomposition))
return circuit_description
