def jordan_wigner_dual_basis_hamiltonian(grid, geometry=None, spinless=False,
include_constant=False):
"""Return the dual basis Hamiltonian as QubitOperator.
Args:
grid (Grid): The discretization to use.
geometry: A list of tuples giving the coordinates of each atom.
example is [('H', (0, 0, 0)), ('H', (0, 0, 0.7414))].
Distances in atomic units. Use atomic symbols to specify atoms.
spinless (bool): Whether to use the spinless model or not.
include_constant (bool): Whether to include the Madelung constant.
Returns:
hamiltonian (QubitOperator)
"""
jellium_op = jordan_wigner_dual_basis_jellium(
grid, spinless, include_constant)
if geometry is None:
return jellium_op
for item in geometry:
if len(item[1]) != grid.dimensions:
raise ValueError("Invalid geometry coordinate.")
if item[0] not in periodic_hash_table:
raise ValueError("Invalid nuclear element.")
n_orbitals = grid.num_points()
volume = grid.volume_scale()
if spinless:
n_qubits = n_orbitals
else:
n_qubits = 2 * n_orbitals
prefactor = -2 * numpy.pi / volume
external_potential = QubitOperator()
for k_indices in grid.all_points_indices():
momenta = momentum_vector(k_indices, grid)
momenta_squared = momenta.dot(momenta)
if momenta_squared < EQ_TOLERANCE:
continue
for p in range(n_qubits):
index_p = grid_indices(p, grid, spinless)
coordinate_p = position_vector(index_p, grid)
for nuclear_term in geometry:
coordinate_j = numpy.array(nuclear_term[1], float)
exp_index = 1.0j * momenta.dot(coordinate_j - coordinate_p)
coefficient = (prefactor / momenta_squared *
periodic_hash_table[nuclear_term[0]] *
numpy.exp(exp_index))
external_potential += (QubitOperator((), coefficient) -
QubitOperator(((p, 'Z'),), coefficient))
return jellium_op + external_potential
def plane_wave_u_operator(n_dimensions, grid_length, length_scale,
nuclear_charges, spinless):
"""Return the external potential operator in plane wave basis.
Args:
n_dimensions: An int giving the number of dimensions for the model.
grid_length: Int, the number of points in one dimension of the grid.
length_scale: Float, the real space length of a box dimension.
nuclear_charges: 3D int array, the nuclear charges.
spinless: Bool, whether to use the spinless model or not.
Returns:
operator: An instance of the FermionOperator class.
"""
n_points = grid_length**n_dimensions
volume = length_scale**float(n_dimensions)
prefactor = -4.0 * numpy.pi / volume
operator = None
if spinless:
spins = [None]
else:
spins = [0, 1]
for grid_indices_p in itertools.product(range(grid_length),
repeat=n_dimensions):
for grid_indices_q in itertools.product(range(grid_length),
repeat=n_dimensions):
shift = grid_length // 2
grid_indices_p_q = [
(grid_indices_p[i] - grid_indices_q[i] + shift) % grid_length
for i in range(n_dimensions)
]
momenta_p_q = momentum_vector(grid_indices_p_q, grid_length,
length_scale)
momenta_p_q_squared = momenta_p_q.dot(momenta_p_q)
if momenta_p_q_squared < EQ_TOLERANCE:
continue
for grid_indices_j in itertools.product(range(grid_length),
repeat=n_dimensions):
coordinate_j = position_vector(grid_indices_j, grid_length,
length_scale)
exp_index = 1.0j * momenta_p_q.dot(coordinate_j)
coefficient = prefactor / momenta_p_q_squared * \
nuclear_charges[grid_indices_j] * numpy.exp(exp_index)
for spin in spins:
orbital_p = orbital_id(grid_length, grid_indices_p, spin)
orbital_q = orbital_id(grid_length, grid_indices_q, spin)
operators = ((orbital_p, 1), (orbital_q, 0))
if operator is None:
operator = FermionOperator(operators, coefficient)
else:
operator += FermionOperator(operators, coefficient)
return operator
def test_momentum_vector(self):
grid = Grid(dimensions=1, length=3, scale=2. * numpy.pi)
test_output = [momentum_vector(i, grid) for i in range(grid.length)]
correct_output = [-1., 0, 1.]
self.assertEqual(correct_output, test_output)
grid = Grid(dimensions=1, length=11, scale=2. * numpy.pi)
for i in range(grid.length):
self.assertAlmostEqual(-momentum_vector(i, grid),
momentum_vector(grid.length - i - 1, grid))
# Test in 2D.
grid = Grid(dimensions=2, length=3, scale=2. * numpy.pi)
test_input = []
test_output = []
for i in range(3):
for j in range(3):
test_input += [(i, j)]
test_output += [momentum_vector((i, j), grid)]
correct_output = numpy.array([[-1, -1], [-1, 0], [-1, 1], [0, -1],
[0, 0], [0, 1], [1, -1], [1, 0], [1, 1]])
self.assertAlmostEqual(0., numpy.amax(test_output - correct_output))
def plane_wave_external_potential(grid, geometry, spinless):
"""Return the external potential operator in plane wave basis.
Args:
grid (Grid): The discretization to use.
geometry: A list of tuples giving the coordinates of each atom.
example is [('H', (0, 0, 0)), ('H', (0, 0, 0.7414))].
Distances in atomic units. Use atomic symbols to specify atoms.
spinless: Bool, whether to use the spinless model or not.
Returns:
FermionOperator: The plane wave operator.
"""
prefactor = -4.0 * numpy.pi / grid.volume_scale()
operator = None
if spinless:
spins = [None]
else:
spins = [0, 1]
for indices_p in grid.all_points_indices():
for indices_q in grid.all_points_indices():
shift = grid.length // 2
grid_indices_p_q = [
(indices_p[i] - indices_q[i] + shift) % grid.length
for i in range(grid.dimensions)]
momenta_p_q = momentum_vector(grid_indices_p_q, grid)
momenta_p_q_squared = momenta_p_q.dot(momenta_p_q)
if momenta_p_q_squared < EQ_TOLERANCE:
continue
for nuclear_term in geometry:
coordinate_j = numpy.array(nuclear_term[1])
exp_index = 1.0j * momenta_p_q.dot(coordinate_j)
coefficient = (prefactor / momenta_p_q_squared *
periodic_hash_table[nuclear_term[0]] *
numpy.exp(exp_index))
for spin in spins:
orbital_p = orbital_id(grid, indices_p, spin)
orbital_q = orbital_id(grid, indices_q, spin)
operators = ((orbital_p, 1), (orbital_q, 0))
if operator is None:
operator = FermionOperator(operators, coefficient)
else:
operator += FermionOperator(operators, coefficient)
return operator
def dual_basis_external_potential(grid, geometry, spinless):
"""Return the external potential in the dual basis of arXiv:1706.00023.
Args:
grid (Grid): The discretization to use.
geometry: A list of tuples giving the coordinates of each atom.
example is [('H', (0, 0, 0)), ('H', (0, 0, 0.7414))].
Distances in atomic units. Use atomic symbols to specify atoms.
spinless (bool): Whether to use the spinless model or not.
Returns:
FermionOperator: The dual basis operator.
"""
prefactor = -4.0 * numpy.pi / grid.volume_scale()
operator = None
if spinless:
spins = [None]
else:
spins = [0, 1]
for pos_indices in grid.all_points_indices():
coordinate_p = position_vector(pos_indices, grid)
for nuclear_term in geometry:
coordinate_j = numpy.array(nuclear_term[1], float)
for momenta_indices in grid.all_points_indices():
momenta = momentum_vector(momenta_indices, grid)
momenta_squared = momenta.dot(momenta)
if momenta_squared < EQ_TOLERANCE:
continue
exp_index = 1.0j * momenta.dot(coordinate_j - coordinate_p)
coefficient = (prefactor / momenta_squared *
periodic_hash_table[nuclear_term[0]] *
numpy.exp(exp_index))
for spin_p in spins:
orbital_p = orbital_id(grid, pos_indices, spin_p)
operators = ((orbital_p, 1), (orbital_p, 0))
if operator is None:
operator = FermionOperator(operators, coefficient)
else:
operator += FermionOperator(operators, coefficient)
return operator
def inverse_fourier_transform(hamiltonian, n_dimensions, grid_length,
length_scale, spinless):
"""Apply Fourier tranform to change hamiltonian in plane wave dual basis.
.. math::
a^\dagger_v = \sqrt{1/N} \sum_m {c^\dagger_m \exp(i k_v r_m)}
a_v = \sqrt{1/N} \sum_m {c_m \exp(-i k_v r_m)}
Args:
hamiltonian: The hamiltonian in plane wave dual basis.
n_dimensions: An int giving the number of dimensions for the model.
grid_length: Int, the number of points in one dimension of the grid.
length_scale: Float, the real space length of a box dimension.
spinless: Bool, whether to use the spinless model or not.
Returns:
hamiltonian_t: An instance of the FermionOperator class.
"""
hamiltonian_t = None
for term in hamiltonian.terms:
transformed_term = None
for ladder_operator in term:
position_indices = grid_indices(ladder_operator[0], n_dimensions,
grid_length, spinless)
position_vec = position_vector(position_indices, grid_length,
length_scale)
new_basis = None
for momentum_indices in itertools.product(range(grid_length),
repeat=n_dimensions):
momentum_vec = momentum_vector(momentum_indices, grid_length,
length_scale)
if spinless:
spin = None
else:
spin = ladder_operator[0] % 2
orbital = orbital_id(grid_length, momentum_indices, spin)
exp_index = -1.0j * numpy.dot(position_vec, momentum_vec)
if ladder_operator[1] == 1:
exp_index *= -1.0
element = FermionOperator(((orbital, ladder_operator[1]), ),
numpy.exp(exp_index))
if new_basis is None:
new_basis = element
else:
new_basis += element
new_basis *= numpy.sqrt(1.0 / float(grid_length**n_dimensions))
if transformed_term is None:
transformed_term = new_basis
else:
transformed_term *= new_basis
if transformed_term is None:
continue
# Coefficient.
transformed_term *= hamiltonian.terms[term]
if hamiltonian_t is None:
hamiltonian_t = transformed_term
else:
hamiltonian_t += transformed_term
return hamiltonian_t
def test_coefficients(self):
# Test that the coefficients post-JW transform are as claimed in paper.
grid = Grid(dimensions=2, length=3, scale=2.)
spinless = 1
n_orbitals = grid.num_points()
n_qubits = (2**(1 - spinless)) * n_orbitals
volume = grid.volume_scale()
# Kinetic operator.
kinetic = dual_basis_kinetic(grid, spinless)
qubit_kinetic = jordan_wigner(kinetic)
# Potential operator.
potential = dual_basis_potential(grid, spinless)
qubit_potential = jordan_wigner(potential)
# Check identity.
identity = tuple()
kinetic_coefficient = qubit_kinetic.terms[identity]
potential_coefficient = qubit_potential.terms[identity]
paper_kinetic_coefficient = 0.
paper_potential_coefficient = 0.
for indices in grid.all_points_indices():
momenta = momentum_vector(indices, grid)
paper_kinetic_coefficient += float(n_qubits) * momenta.dot(
momenta) / float(4. * n_orbitals)
if momenta.any():
potential_contribution = -numpy.pi * float(n_qubits) / float(
2. * momenta.dot(momenta) * volume)
paper_potential_coefficient += potential_contribution
self.assertAlmostEqual(kinetic_coefficient, paper_kinetic_coefficient)
self.assertAlmostEqual(potential_coefficient,
paper_potential_coefficient)
# Check Zp.
for p in range(n_qubits):
zp = ((p, 'Z'), )
kinetic_coefficient = qubit_kinetic.terms[zp]
potential_coefficient = qubit_potential.terms[zp]
paper_kinetic_coefficient = 0.
paper_potential_coefficient = 0.
for indices in grid.all_points_indices():
momenta = momentum_vector(indices, grid)
paper_kinetic_coefficient -= momenta.dot(momenta) / float(
4. * n_orbitals)
if momenta.any():
potential_contribution = numpy.pi / float(
momenta.dot(momenta) * volume)
paper_potential_coefficient += potential_contribution
self.assertAlmostEqual(kinetic_coefficient,
paper_kinetic_coefficient)
self.assertAlmostEqual(potential_coefficient,
paper_potential_coefficient)
# Check Zp Zq.
if spinless:
spins = [None]
else:
spins = [0, 1]
for indices_a in grid.all_points_indices():
for indices_b in grid.all_points_indices():
paper_kinetic_coefficient = 0.
paper_potential_coefficient = 0.
position_a = position_vector(indices_a, grid)
position_b = position_vector(indices_b, grid)
differences = position_b - position_a
for spin_a in spins:
for spin_b in spins:
p = orbital_id(grid, indices_a, spin_a)
q = orbital_id(grid, indices_b, spin_b)
if p == q:
continue
zpzq = ((min(p, q), 'Z'), (max(p, q), 'Z'))
if zpzq in qubit_potential.terms:
potential_coefficient = qubit_potential.terms[zpzq]
for indices_c in grid.all_points_indices():
momenta = momentum_vector(indices_c, grid)
if momenta.any():
potential_contribution = numpy.pi * numpy.cos(
differences.dot(momenta)) / float(
momenta.dot(momenta) * volume)
paper_potential_coefficient += (
potential_contribution)
self.assertAlmostEqual(potential_coefficient,
paper_potential_coefficient)
