def _gaussian_basis_change(
qubits: Sequence[cirq.QubitId], transformation_matrix: numpy.ndarray,
initially_occupied_orbitals: Optional[Sequence[int]]) -> cirq.OP_TREE:
n_qubits = len(qubits)
# Rearrange the transformation matrix because the OpenFermion routine
# expects it to describe annihilation operators rather than creation
# operators
left_block = transformation_matrix[:, :n_qubits]
right_block = transformation_matrix[:, n_qubits:]
transformation_matrix = numpy.block(
[numpy.conjugate(right_block),
numpy.conjugate(left_block)])
decomposition, left_decomposition, _, left_diagonal = (
fermionic_gaussian_decomposition(transformation_matrix))
if (initially_occupied_orbitals is not None
and len(initially_occupied_orbitals) == 0):
# Starting with the vacuum state yields additional symmetry
circuit_description = list(reversed(decomposition))
else:
if initially_occupied_orbitals is None:
# The initial state is not a computational basis state so the
# phases left on the diagonal in the Givens decomposition matter
yield (cirq.Rz(rads=numpy.angle(left_diagonal[j])).on(qubits[j])
for j in range(n_qubits))
circuit_description = list(reversed(decomposition +
left_decomposition))
yield _ops_from_givens_rotations_circuit_description(
qubits, circuit_description)
def diagonalizing_circuit(self):
r"""Get a circuit for a unitary that diagonalizes this Hamiltonian
This circuit performs the transformation to a basis in which the
Hamiltonian takes the diagonal form
.. math::
\sum_{j} \varepsilon_j b^\dagger_j b_j + \text{constant}.
Returns
-------
circuit_description (list[tuple]):
A list of operations describing the circuit. Each operation
is a tuple of objects describing elementary operations that
can be performed in parallel. Each elementary operation
is either the string 'pht' indicating a particle-hole
transformation on the last fermionic mode, or 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`.
"""
transformation_matrix = self.diagonalizing_bogoliubov_transform()
if self.conserves_particle_number:
# The Hamiltonian conserves particle number, so we don't need
# to use the most general procedure.
decomposition, _ = givens_decomposition_square(
transformation_matrix)
circuit_description = list(reversed(decomposition))
else:
# The Hamiltonian does not conserve particle number, so we
# need to use the most general procedure.
# Rearrange the transformation matrix because the circuit
# generation routine expects it to describe annihilation
# operators rather than creation operators.
left_block = transformation_matrix[:, :self.n_qubits]
right_block = transformation_matrix[:, self.n_qubits:]
# Can't use numpy.block because that requires numpy>=1.13.0
new_transformation_matrix = numpy.empty(
(self.n_qubits, 2 * self.n_qubits), dtype=complex)
new_transformation_matrix[:, :self.n_qubits] = numpy.conjugate(
right_block)
new_transformation_matrix[:, self.n_qubits:] = numpy.conjugate(
left_block)
# Get the circuit description
decomposition, left_decomposition, _, _ = (
fermionic_gaussian_decomposition(new_transformation_matrix))
# need to use left_diagonal too
circuit_description = list(reversed(
decomposition + left_decomposition))
return circuit_description
def diagonalizing_circuit(self):
"""Get a circuit for a unitary that diagonalizes this Hamiltonian
This circuit performs the transformation to a basis in which the
Hamiltonian takes the diagonal form
.. math::
\sum_{j} \\varepsilon_j b^\dagger_j b_j + \\text{constant}.
Returns
-------
circuit_description (list[tuple]):
A list of operations describing the circuit. Each operation
is a tuple of objects describing elementary operations that
can be performed in parallel. Each elementary operation
is either the string 'pht' indicating a particle-hole
transformation on the last fermionic mode, or 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`.
"""
diagonalizing_unitary = self.diagonalizing_bogoliubov_transform()
if self.conserves_particle_number:
# The Hamiltonian conserves particle number, so we don't need
# to use the most general procedure.
decomposition, diagonal = givens_decomposition_square(
diagonalizing_unitary)
circuit_description = list(reversed(decomposition))
else:
# The Hamiltonian does not conserve particle number, so we
# need to use the most general procedure.
# Get the unitary rows which represent the Gaussian unitary
gaussian_unitary_matrix = diagonalizing_unitary[self.n_qubits:]
# Get the circuit description
decomposition, left_decomposition, diagonal, left_diagonal = (
fermionic_gaussian_decomposition(gaussian_unitary_matrix))
# need to use left_diagonal too
circuit_description = list(
reversed(decomposition + left_decomposition))
return circuit_description
def test_bad_constraints(self):
n = 3
ones_mat = numpy.ones((n, 2 * n))
with self.assertRaises(ValueError):
_ = fermionic_gaussian_decomposition(ones_mat)
def test_bad_dimensions(self):
n, p = (3, 7)
rand_mat = numpy.random.randn(n, p)
with self.assertRaises(ValueError):
_ = fermionic_gaussian_decomposition(rand_mat)
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 gaussian_state_preparation_circuit(
quadratic_hamiltonian, occupied_orbitals=None):
"""Obtain the description of a circuit which prepares a fermionic Gaussian
state.
Fermionic Gaussian states can be regarded as eigenstates of quadratic
Hamiltonians. If the Hamiltonian conserves particle number, then these are
just Slater determinants. See arXiv:1711.05395 for a detailed description
of how this procedure works.
The circuit description is returned as a sequence of elementary
operations; operations that can be performed in parallel are grouped
together. Each elementary operation is either
- the string 'pht', indicating the particle-hole transformation
on the last fermionic mode, which is the operator :math:`\mathcal{B}`
such that
.. math::
\\begin{align}
\mathcal{B} a_N \mathcal{B}^\dagger &= a_N^\dagger,\\\\
\mathcal{B} a_j \mathcal{B}^\dagger &= a_j, \\quad
j = 1, \ldots, N-1,
\end{align}
or
- a tuple :math:`(i, j, \\theta, \\varphi)`, indicating the operation
.. math::
\exp[i \\varphi a_j^\dagger a_j]
\exp[\\theta (a_i^\dagger a_j - a_j^\dagger a_i)],
a Givens rotation of modes :math:`i` and :math:`j` by angles
:math:`\\theta` and :math:`\\varphi`.
Args:
quadratic_hamiltonian(QuadraticHamiltonian):
The Hamiltonian whose eigenstate is desired.
occupied_orbitals(list):
A list of integers representing the indices of the occupied
orbitals in the desired Gaussian state. If this is None
(the default), then it is assumed that the ground state is
desired, i.e., the orbitals with negative energies are filled.
Returns
-------
circuit_description (list[tuple]):
A list of operations describing the circuit. Each operation
is a tuple of objects describing elementary operations that
can be performed in parallel. Each elementary operation
is either the string 'pht', indicating a particle-hole
transformation on the last fermionic mode, or 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`.
start_orbitals (list):
The occupied orbitals to start with. This describes the
initial state that the circuit should be applied to: it should
be a Slater determinant (in the computational basis) with these
orbitals filled.
"""
if not isinstance(quadratic_hamiltonian, QuadraticHamiltonian):
raise ValueError('Input must be an instance of QuadraticHamiltonian.')
if quadratic_hamiltonian.conserves_particle_number:
# The Hamiltonian conserves particle number, so we don't need
# to use the most general procedure.
hermitian_matrix = quadratic_hamiltonian.combined_hermitian_part
energies, diagonalizing_unitary = numpy.linalg.eigh(hermitian_matrix)
if occupied_orbitals is None:
# The ground state is desired, so we fill the orbitals that have
# negative energy
num_negative_energies = numpy.count_nonzero(
energies < -EQ_TOLERANCE)
occupied_orbitals = range(num_negative_energies)
# Get the unitary rows which represent the Slater determinant
slater_determinant_matrix = diagonalizing_unitary.T[occupied_orbitals]
# Get the circuit description
circuit_description = slater_determinant_preparation_circuit(
slater_determinant_matrix)
start_orbitals = range(len(occupied_orbitals))
else:
# The Hamiltonian does not conserve particle number, so we
# need to use the most general procedure.
diagonalizing_unitary = (
quadratic_hamiltonian.diagonalizing_bogoliubov_transform())
# Get the unitary rows which represent the Gaussian unitary
gaussian_unitary_matrix = diagonalizing_unitary[
quadratic_hamiltonian.n_qubits:]
# Get the circuit description
decomposition, left_decomposition, diagonal, left_diagonal = (
fermionic_gaussian_decomposition(gaussian_unitary_matrix))
if occupied_orbitals is None:
# The ground state is desired, so the circuit should be applied
# to the vaccuum state
start_orbitals = []
circuit_description = list(reversed(decomposition))
else:
start_orbitals = occupied_orbitals
# The circuit won't be applied to the ground state, so we need to
# use left_decomposition
circuit_description = list(reversed(
decomposition + left_decomposition))
return circuit_description, start_orbitals
def gaussian_state_preparation_circuit(quadratic_hamiltonian,
occupied_orbitals=None,
spin_sector=None):
r"""Obtain the description of a circuit which prepares a fermionic Gaussian
state.
Fermionic Gaussian states can be regarded as eigenstates of quadratic
Hamiltonians. If the Hamiltonian conserves particle number, then these are
just Slater determinants. See arXiv:1711.05395 for a detailed description
of how this procedure works.
The circuit description is returned as a sequence of elementary
operations; operations that can be performed in parallel are grouped
together. Each elementary operation is either
- the string 'pht', indicating the particle-hole transformation
on the last fermionic mode, which is the operator :math:`\mathcal{B}`
such that
.. math::
\begin{align}
\mathcal{B} a_N \mathcal{B}^\dagger &= a_N^\dagger,\\
\mathcal{B} a_j \mathcal{B}^\dagger &= a_j, \quad
j = 1, \ldots, N-1,
\end{align}
or
- a tuple :math:`(i, j, \theta, \varphi)`, indicating the operation
.. math::
\exp[i \varphi a_j^\dagger a_j]
\exp[\theta (a_i^\dagger a_j - a_j^\dagger a_i)],
a Givens rotation of modes :math:`i` and :math:`j` by angles
:math:`\theta` and :math:`\varphi`.
Args:
quadratic_hamiltonian(QuadraticHamiltonian):
The Hamiltonian whose eigenstate is desired.
occupied_orbitals(list):
A list of integers representing the indices of the occupied
orbitals in the desired Gaussian state. If this is None
(the default), then it is assumed that the ground state is
desired, i.e., the orbitals with negative energies are filled.
spin_sector (optional str): An optional integer specifying
a spin sector to restrict to: 0 for spin-up and 1 for
spin-down. If specified, the returned circuit acts on modes
indexed by spatial indices (rather than spin indices).
Should only be specified if the Hamiltonian
includes a spin degree of freedom and spin-up modes
do not interact with spin-down modes.
Returns
-------
circuit_description (list[tuple]):
A list of operations describing the circuit. Each operation
is a tuple of objects describing elementary operations that
can be performed in parallel. Each elementary operation
is either the string 'pht', indicating a particle-hole
transformation on the last fermionic mode, or 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`.
start_orbitals (list):
The occupied orbitals to start with. This describes the
initial state that the circuit should be applied to: it should
be a Slater determinant (in the computational basis) with these
orbitals filled.
"""
if not isinstance(quadratic_hamiltonian, QuadraticHamiltonian):
raise ValueError('Input must be an instance of QuadraticHamiltonian.')
orbital_energies, transformation_matrix, _ = (
quadratic_hamiltonian.diagonalizing_bogoliubov_transform(
spin_sector=spin_sector))
if quadratic_hamiltonian.conserves_particle_number:
if occupied_orbitals is None:
# The ground state is desired, so we fill the orbitals that have
# negative energy
occupied_orbitals = numpy.where(orbital_energies < 0.0)[0]
# Get the unitary rows which represent the Slater determinant
slater_determinant_matrix = transformation_matrix[occupied_orbitals]
# Get the circuit description
circuit_description = slater_determinant_preparation_circuit(
slater_determinant_matrix)
start_orbitals = range(len(occupied_orbitals))
else:
# TODO implement this
if spin_sector is not None:
raise NotImplementedError(
'Specifying spin sector for non-particle-conserving '
'Hamiltonians is not yet supported.')
# Rearrange the transformation matrix because the circuit generation
# routine expects it to describe annihilation operators rather than
# creation operators
n_qubits = quadratic_hamiltonian.n_qubits
left_block = transformation_matrix[:, :n_qubits]
right_block = transformation_matrix[:, n_qubits:]
# Can't use numpy.block because that requires numpy>=1.13.0
new_transformation_matrix = numpy.empty((n_qubits, 2 * n_qubits),
dtype=complex)
new_transformation_matrix[:, :n_qubits] = numpy.conjugate(right_block)
new_transformation_matrix[:, n_qubits:] = numpy.conjugate(left_block)
# Get the circuit description
decomposition, left_decomposition, diagonal, left_diagonal = (
fermionic_gaussian_decomposition(new_transformation_matrix))
if occupied_orbitals is None:
# The ground state is desired, so the circuit should be applied
# to the vaccuum state
start_orbitals = []
circuit_description = list(reversed(decomposition))
else:
start_orbitals = occupied_orbitals
# The circuit won't be applied to the ground state, so we need to
# use left_decomposition
circuit_description = list(
reversed(decomposition + left_decomposition))
return circuit_description, start_orbitals
def test_bad_constraints(self):
n = 3
ones_mat = numpy.ones((n, 2 * n))
with self.assertRaises(ValueError):
decomposition, left_unitary, antidiagonal = (
fermionic_gaussian_decomposition(ones_mat))
def test_bad_dimensions(self):
n, p = (3, 7)
rand_mat = numpy.random.randn(n, p)
with self.assertRaises(ValueError):
decomposition, left_unitary, antidiagonal = (
fermionic_gaussian_decomposition(rand_mat))
