def test_nonperiodic_external_potential_integration(self):
# Compute potential energy operator in momentum and position space.
# Non-periodic test.
'''
Key: Spatial dimension.
Value: List of geometries.
'''
geometry_sets = {
2: [[('H', (0., 0.)), ('H', (0.8, 0.))],
[('H', (0.1, 0.))]],
3: [[('H', (0., 0., 0.)), ('H', (0.8, 0., 0.))],
[('H', (0.1, 0., 0.))]]
}
# [[spatial dimension, fieldline dimension]]
dims = [[2, 2], [2, 3], [3, 3]]
scale = 8 * 1.
spinless = True
for dim in dims:
# If dim[0] == 2, get range(2, 4).
# If dim[0] == 3, get range(2, 3).
for length in range(2, 6 - dim[0]):
for geometry in geometry_sets[dim[0]]:
grid = Grid(dimensions=dim[0], scale=scale, length=length)
period_cutoff = grid.volume_scale() ** (1. / grid.dimensions)
momentum_external_potential = plane_wave_external_potential(grid, geometry, spinless, e_cutoff=None,
non_periodic=True, period_cutoff=period_cutoff,
fieldlines=dim[1])
position_external_potential = dual_basis_external_potential(grid, geometry, spinless,
non_periodic=True, period_cutoff=period_cutoff,
fieldlines=dim[1])
# Confirm they are Hermitian
momentum_external_potential_operator = (
get_sparse_operator(momentum_external_potential))
self.assertTrue(is_hermitian(momentum_external_potential_operator))
position_external_potential_operator = (
get_sparse_operator(position_external_potential))
self.assertTrue(is_hermitian(position_external_potential_operator))
# Diagonalize.
jw_momentum = jordan_wigner(momentum_external_potential)
jw_position = jordan_wigner(position_external_potential)
momentum_spectrum = eigenspectrum(jw_momentum)
position_spectrum = eigenspectrum(jw_position)
# Confirm spectra are the same.
difference = numpy.amax(
numpy.absolute(momentum_spectrum - position_spectrum))
self.assertAlmostEqual(difference, 0.)
def test_nonperiodic_potential_integration(self):
# Compute potential energy operator in momentum and position space.
# Non-periodic test.
spinless = True
# [[spatial dimension, fieldline dimension]]
dims = [[2, 2], [2, 3], [3, 3]]
for dim in dims:
# If dim[0] == 2, get range(2, 4).
# If dim[0] == 3, get range(2, 3).
for length in range(2, 6 - dim[0]):
grid = Grid(dimensions=dim[0], length=length, scale=1.)
period_cutoff = grid.volume_scale()**(1. /
grid.dimensions) / 2.
momentum_potential = plane_wave_potential(
grid,
spinless=spinless,
e_cutoff=None,
non_periodic=True,
period_cutoff=period_cutoff,
fieldlines=dim[1])
position_potential = dual_basis_potential(
grid,
spinless=spinless,
non_periodic=True,
period_cutoff=period_cutoff,
fieldlines=dim[1])
# Confirm they are Hermitian
momentum_potential_operator = (
get_sparse_operator(momentum_potential))
self.assertTrue(is_hermitian(momentum_potential_operator))
position_potential_operator = (
get_sparse_operator(position_potential))
self.assertTrue(is_hermitian(position_potential_operator))
# Diagonalize and confirm the same energy.
jw_momentum = jordan_wigner(momentum_potential)
jw_position = jordan_wigner(position_potential)
momentum_spectrum = eigenspectrum(jw_momentum)
position_spectrum = eigenspectrum(jw_position)
# Confirm spectra are the same.
difference = numpy.amax(
numpy.absolute(momentum_spectrum - position_spectrum))
self.assertAlmostEqual(difference, 0.)
def test_properties(self):
g = Grid(dimensions=2, length=3, scale=5.0)
self.assertEqual(g.num_points, 9)
self.assertEqual(g.volume_scale(), 25)
self.assertEqual(list(g.all_points_indices()), [
(0, 0),
(0, 1),
(0, 2),
(1, 0),
(1, 1),
(1, 2),
(2, 0),
(2, 1),
(2, 2),
])
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 = grid.momentum_vector(indices)
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 = grid.momentum_vector(indices)
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]
for indices_a in grid.all_points_indices():
for indices_b in grid.all_points_indices():
potential_coefficient = 0.
paper_kinetic_coefficient = 0.
paper_potential_coefficient = 0.
position_a = grid.position_vector(indices_a)
position_b = grid.position_vector(indices_b)
differences = position_b - position_a
for spin_a in spins:
for spin_b in spins:
p = grid.orbital_id(indices_a, spin_a)
q = grid.orbital_id(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 = grid.momentum_vector(indices_c)
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)
def test_plane_wave_period_cutoff(self):
# TODO: After figuring out the correct formula for period cutoff for
# dual basis, change period_cutoff to default, and change
# h_1 to also accept period_cutoff for real integration test.
'''
Key: Spatial dimension.
Value: List of geometries.
'''
geometry_sets = {
2: [[('H', (0., 0.)), ('H', (0.8, 0.))],
[('H', (0.1, 0.))]],
3: [[('H', (0., 0., 0.)), ('H', (0.8, 0., 0.))],
[('H', (0.1, 0., 0.))]]
}
# [[spatial dimension, fieldline dimension]]
dims = [[2, 2], [2, 3], [3, 3]]
spinless_set = [True, False]
scale = 8 * 1.
for dim in dims:
for geometry in geometry_sets[dim[0]]:
for spinless in spinless_set:
grid = Grid(dimensions=dim[0], scale=scale, length=2)
period_cutoff = grid.volume_scale() ** (1. / grid.dimensions)
h_1 = plane_wave_hamiltonian(grid, geometry, spinless=True, plane_wave=True, include_constant=False,
e_cutoff=None, fieldlines=dim[1])
jw_1 = jordan_wigner(h_1)
spectrum_1 = eigenspectrum(jw_1)
h_2 = plane_wave_hamiltonian(grid, geometry, spinless=True, plane_wave=True, include_constant=False,
e_cutoff=None, non_periodic=True, period_cutoff=period_cutoff,
fieldlines=dim[1])
jw_2 = jordan_wigner(h_2)
spectrum_2 = eigenspectrum(jw_2)
max_diff = numpy.amax(numpy.absolute(spectrum_1 - spectrum_2))
# Checks if non-periodic and periodic cases are different.
self.assertGreater(max_diff, 0.)
# TODO: This is only for code coverage. Remove after having real
# integration test.
momentum_hamiltonian = plane_wave_hamiltonian(grid, geometry, spinless=True, plane_wave=True,
include_constant=False, e_cutoff=None,
non_periodic=True, period_cutoff=period_cutoff,
fieldlines=dim[1])
position_hamiltonian = plane_wave_hamiltonian(grid, geometry, spinless=True, plane_wave=False,
include_constant=False, e_cutoff=None,
non_periodic=True, period_cutoff=period_cutoff,
fieldlines=dim[1])
# Confirm they are Hermitian
momentum_hamiltonian_operator = (
get_sparse_operator(momentum_hamiltonian))
self.assertTrue(is_hermitian(momentum_hamiltonian_operator))
position_hamiltonian_operator = (
get_sparse_operator(position_hamiltonian))
self.assertTrue(is_hermitian(position_hamiltonian_operator))
# Diagonalize.
jw_momentum = jordan_wigner(momentum_hamiltonian)
jw_position = jordan_wigner(position_hamiltonian)
momentum_spectrum = eigenspectrum(jw_momentum)
position_spectrum = eigenspectrum(jw_position)
# Confirm spectra are the same.
difference = numpy.amax(
numpy.absolute(momentum_spectrum - position_spectrum))
self.assertAlmostEqual(difference, 0.)
def test_plane_wave_period_cutoff(self):
# TODO: After figuring out the correct formula for period cutoff for
# dual basis, change period_cutoff to default, and change
# hamiltonian_1 to a real jellium_model for real integration test.
spinless = True
# [[spatial dimension, fieldline dimension]]
dims = [[2, 2], [2, 3], [3, 3]]
for dim in dims:
# If dim[0] == 2, get range(2, 4).
# If dim[0] == 3, get range(2, 3).
for length in range(2, 6 - dim[0]):
grid = Grid(dimensions=dim[0], length=length, scale=1.0)
period_cutoff = grid.volume_scale()**(1. / grid.dimensions)
hamiltonian_1 = jellium_model(grid,
spinless=spinless,
plane_wave=True,
include_constant=False,
fieldlines=dim[1])
jw_1 = jordan_wigner(hamiltonian_1)
spectrum_1 = eigenspectrum(jw_1)
hamiltonian_2 = jellium_model(grid,
spinless=spinless,
plane_wave=True,
include_constant=False,
e_cutoff=None,
non_periodic=True,
period_cutoff=period_cutoff,
fieldlines=dim[1])
jw_2 = jordan_wigner(hamiltonian_2)
spectrum_2 = eigenspectrum(jw_2)
max_diff = numpy.amax(numpy.absolute(spectrum_1 - spectrum_2))
# Checks if non-periodic and periodic cases are different.
self.assertGreater(max_diff, 0.)
# TODO: This is only for code coverage. Remove after having real
# integration test.
momentum_hamiltonian = jellium_model(
grid,
spinless=spinless,
plane_wave=True,
include_constant=False,
e_cutoff=None,
non_periodic=True,
period_cutoff=period_cutoff,
fieldlines=dim[1])
position_hamiltonian = jellium_model(
grid,
spinless=spinless,
plane_wave=False,
include_constant=False,
e_cutoff=None,
non_periodic=True,
period_cutoff=period_cutoff,
fieldlines=dim[1])
# Confirm they are Hermitian
momentum_hamiltonian_operator = (
get_sparse_operator(momentum_hamiltonian))
self.assertTrue(is_hermitian(momentum_hamiltonian_operator))
position_hamiltonian_operator = (
get_sparse_operator(position_hamiltonian))
self.assertTrue(is_hermitian(position_hamiltonian_operator))
# Diagonalize.
jw_momentum = jordan_wigner(momentum_hamiltonian)
jw_position = jordan_wigner(position_hamiltonian)
momentum_spectrum = eigenspectrum(jw_momentum)
position_spectrum = eigenspectrum(jw_position)
# Confirm spectra are the same.
difference = numpy.amax(
numpy.absolute(momentum_spectrum - position_spectrum))
self.assertAlmostEqual(difference, 0.)
def test_coefficients(self):
# Test that the coefficients post-JW transform are as claimed in paper.
spinless = True
non_periodics = [False, True]
# [[spatial dimension, fieldline dimension]].
dims = [[2, 2], [2, 3], [3, 3]]
# Reference length scale where the 2D Coulomb potential is zero.
R0 = 1e8
for dim in dims:
# If dim[0] == 2, get range(2, 4).
# If dim[0] == 3, get range(2, 3).
for length in range(2, 6 - dim[0]):
for non_periodic in non_periodics:
grid = Grid(dimensions=dim[0], length=length, scale=2.)
n_orbitals = grid.num_points
n_qubits = (2**(1 - spinless)) * n_orbitals
volume = grid.volume_scale()
period_cutoff = None
if non_periodic:
period_cutoff = volume**(1. / grid.dimensions)
fieldlines = dim[1]
# Kinetic operator.
kinetic = dual_basis_kinetic(grid, spinless)
qubit_kinetic = jordan_wigner(kinetic)
# Potential operator.
potential = dual_basis_potential(
grid,
spinless,
non_periodic=non_periodic,
period_cutoff=period_cutoff,
fieldlines=fieldlines)
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 = grid.momentum_vector(indices)
momenta_squared = momenta.dot(momenta)
paper_kinetic_coefficient += (float(n_qubits) *
momenta_squared /
float(4. * n_orbitals))
if momenta.any():
potential_contribution = 0.
# 3D case.
if grid.dimensions == 3:
potential_contribution = -(
numpy.pi * float(n_qubits) /
float(2. * momenta_squared * volume))
# If non-periodic.
if non_periodic:
correction = 1.0 - numpy.cos(
period_cutoff *
numpy.sqrt(momenta_squared))
potential_contribution *= correction
# 2D case.
elif grid.dimensions == 2:
V_nu = 0.
# 2D Coulomb potential.
if fieldlines == 2:
# If non-periodic.
if non_periodic:
Dkv = period_cutoff * numpy.sqrt(
momenta_squared)
V_nu = (
2. * numpy.pi / momenta_squared *
(Dkv *
numpy.log(R0 / period_cutoff) *
scipy.special.jv(1, Dkv) -
scipy.special.jv(0, Dkv)))
# If periodic.
else:
var1 = 4. / momenta_squared
var2 = 0.25 * momenta_squared
V_nu = 0.5 * numpy.complex128(
mpmath.meijerg([[1., 1.5, 2.], []],
[[1.5], []], var1) -
mpmath.meijerg(
[[-0.5, 0., 0.], []],
[[-0.5, 0.], [-1.]], var2))
# 3D Coulomb potential.
elif fieldlines == 3:
# If non-periodic.
if non_periodic:
var = (-0.25 * period_cutoff**2 *
momenta_squared)
V_nu = numpy.complex128(
2 * numpy.pi * period_cutoff *
mpmath.hyp1f2(0.5, 1., 1.5, var))
# If periodic.
else:
V_nu = (2 * numpy.pi /
numpy.sqrt(momenta_squared))
potential_contribution = (-float(n_qubits) /
(8. * volume) * V_nu)
paper_potential_coefficient += potential_contribution
# Check if coefficients are equal.
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 = grid.momentum_vector(indices)
momenta_squared = momenta.dot(momenta)
paper_kinetic_coefficient -= (
momenta_squared / float(4. * n_orbitals))
if momenta.any():
potential_contribution = 0.
# 3D case.
if grid.dimensions == 3:
potential_contribution = (
numpy.pi /
float(momenta_squared * volume))
# If non-periodic.
if non_periodic:
correction = 1.0 - numpy.cos(
period_cutoff *
numpy.sqrt(momenta_squared))
potential_contribution *= correction
# 2D case.
elif grid.dimensions == 2:
V_nu = 0.
# 2D Coulomb potential.
if fieldlines == 2:
# If non-periodic.
if non_periodic:
Dkv = period_cutoff * numpy.sqrt(
momenta_squared)
V_nu = (2. * numpy.pi /
momenta_squared *
(Dkv * numpy.log(
R0 / period_cutoff) *
scipy.special.jv(1, Dkv) -
scipy.special.jv(0, Dkv)))
# If periodic.
else:
var1 = 4. / momenta_squared
var2 = 0.25 * momenta_squared
V_nu = 0.5 * numpy.complex128(
mpmath.meijerg(
[[1., 1.5, 2.], []],
[[1.5], []], var1) -
mpmath.meijerg(
[[-0.5, 0., 0.], []],
[[-0.5, 0.], [-1.]], var2))
# 3D Coulomb potential.
elif fieldlines == 3:
# If non-periodic.
if non_periodic:
var = (-0.25 * period_cutoff**2 *
momenta_squared)
V_nu = numpy.complex128(
2 * numpy.pi * period_cutoff *
mpmath.hyp1f2(
0.5, 1., 1.5, var))
# If periodic.
else:
V_nu = (
2 * numpy.pi /
numpy.sqrt(momenta_squared))
potential_contribution = (1. /
(4. * volume) *
V_nu)
paper_potential_coefficient += potential_contribution
# Check if coefficients are equal.
self.assertAlmostEqual(kinetic_coefficient,
paper_kinetic_coefficient)
self.assertAlmostEqual(potential_coefficient,
paper_potential_coefficient)
# Check Zp Zq.
if spinless:
spins = [None]
for indices_a in grid.all_points_indices():
for indices_b in grid.all_points_indices():
potential_coefficient = 0.
paper_kinetic_coefficient = 0.
paper_potential_coefficient = 0.
position_a = grid.position_vector(indices_a)
position_b = grid.position_vector(indices_b)
differences = position_b - position_a
for spin_a in spins:
for spin_b in spins:
p = grid.orbital_id(indices_a, spin_a)
q = grid.orbital_id(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 = grid.momentum_vector(
indices_c)
momenta_squared = momenta.dot(momenta)
if momenta.any():
potential_contribution = 0.
# 3D case.
if grid.dimensions == 3:
potential_contribution = (
numpy.pi * numpy.cos(
differences.dot(
momenta)) /
float(momenta_squared *
volume))
# If non-periodic.
if non_periodic:
correction = 1.0 - numpy.cos(
period_cutoff * numpy.
sqrt(momenta_squared))
potential_contribution *= correction
# 2D case.
elif grid.dimensions == 2:
V_nu = 0.
# 2D Coulomb potential.
if fieldlines == 2:
# If non-periodic.
if non_periodic:
Dkv = (
period_cutoff *
numpy.sqrt(
momenta_squared
))
V_nu = (
2. * numpy.pi /
momenta_squared *
(Dkv * numpy.log(
R0 /
period_cutoff)
* scipy.special.
jv(1, Dkv) -
scipy.special.jv(
0, Dkv)))
# If periodic.
else:
var1 = 4. / momenta_squared
var2 = 0.25 * momenta_squared
V_nu = 0.5 * numpy.complex128(
mpmath.meijerg([[
1., 1.5, 2.
], []], [[1.5], []
], var1) -
mpmath.meijerg([[
-0.5, 0., 0.
], []], [[
-0.5, 0.
], [-1.]], var2))
# 3D Coulomb potential.
elif fieldlines == 3:
# If non-periodic.
if non_periodic:
var = -0.25 * period_cutoff**2 * momenta_squared
V_nu = numpy.complex128(
2 * numpy.pi *
period_cutoff *
mpmath.hyp1f2(
0.5, 1., 1.5,
var))
# If periodic.
else:
V_nu = 2 * numpy.pi / numpy.sqrt(
momenta_squared)
potential_contribution = (
numpy.cos(
differences.dot(
momenta)) /
(4. * volume) * V_nu)
paper_potential_coefficient += potential_contribution
# Check if coefficients are equal.
self.assertAlmostEqual(
potential_coefficient,
paper_potential_coefficient)
