def setUp(self):
self.n_qubits = 5
self.constant = 1.7
self.chemical_potential = 2.
# Obtain random Hermitian and antisymmetric matrices
self.hermitian_mat = random_hermitian_matrix(self.n_qubits)
self.antisymmetric_mat = random_antisymmetric_matrix(self.n_qubits)
self.combined_hermitian = (
self.hermitian_mat -
self.chemical_potential * numpy.eye(self.n_qubits))
# Initialize a particle-number-conserving Hamiltonian
self.quad_ham_pc = QuadraticHamiltonian(
self.constant, self.hermitian_mat)
# Initialize a non-particle-number-conserving Hamiltonian
self.quad_ham_npc = QuadraticHamiltonian(
self.constant, self.hermitian_mat, self.antisymmetric_mat,
self.chemical_potential)
# Initialize the sparse operators and get their ground energies
self.quad_ham_pc_sparse = get_sparse_operator(self.quad_ham_pc)
self.quad_ham_npc_sparse = get_sparse_operator(self.quad_ham_npc)
self.pc_ground_energy, self.pc_ground_state = get_ground_state(
self.quad_ham_pc_sparse)
self.npc_ground_energy, self.npc_ground_state = get_ground_state(
self.quad_ham_npc_sparse)
def setUp(self):
self.n_qubits = 5
self.constant = 1.7
self.chemical_potential = 2.
# Obtain random Hermitian and antisymmetric matrices
rand_mat_A = numpy.random.randn(self.n_qubits, self.n_qubits)
rand_mat_B = numpy.random.randn(self.n_qubits, self.n_qubits)
rand_mat = rand_mat_A + 1.j * rand_mat_B
self.hermitian_mat = rand_mat + rand_mat.T.conj()
rand_mat_A = numpy.random.randn(self.n_qubits, self.n_qubits)
rand_mat_B = numpy.random.randn(self.n_qubits, self.n_qubits)
rand_mat = rand_mat_A + 1.j * rand_mat_B
self.antisymmetric_mat = rand_mat - rand_mat.T
self.combined_hermitian = (
self.hermitian_mat -
self.chemical_potential * numpy.eye(self.n_qubits))
# Initialize a particle-number-conserving Hamiltonian
self.quad_ham_pc = QuadraticHamiltonian(self.constant,
self.hermitian_mat)
# Initialize a non-particle-number-conserving Hamiltonian
self.quad_ham_npc = QuadraticHamiltonian(self.constant,
self.hermitian_mat,
self.antisymmetric_mat,
self.chemical_potential)
def setUp(self):
self.n_qubits = 5
self.constant = 1.7
self.chemical_potential = 2.
# Obtain random Hermitian and antisymmetric matrices
self.hermitian_mat = random_hermitian_matrix(self.n_qubits)
self.antisymmetric_mat = random_antisymmetric_matrix(self.n_qubits)
# Initialize a non-particle-number-conserving Hamiltonian
self.quad_ham_npc = QuadraticHamiltonian(
self.constant, self.hermitian_mat, self.antisymmetric_mat,
self.chemical_potential)
def random_quadratic_hamiltonian(n_orbitals,
conserves_particle_number=False,
real=False,
expand_spin=False,
seed=None):
"""Generate a random instance of QuadraticHamiltonian.
Args:
n_orbitals(int): the number of orbitals
conserves_particle_number(bool): whether the returned Hamiltonian
should conserve particle number
real(bool): whether to use only real numbers
expand_spin: Whether to expand each orbital symmetrically into two
spin orbitals. Note that if this option is set to True, then
the total number of orbitals will be doubled.
Returns:
QuadraticHamiltonian
"""
if seed is not None:
numpy.random.seed(seed)
constant = numpy.random.randn()
chemical_potential = numpy.random.randn()
hermitian_mat = random_hermitian_matrix(n_orbitals, real)
if conserves_particle_number:
antisymmetric_mat = None
else:
antisymmetric_mat = random_antisymmetric_matrix(n_orbitals, real)
if expand_spin:
hermitian_mat = numpy.kron(hermitian_mat, numpy.eye(2))
if antisymmetric_mat is not None:
antisymmetric_mat = numpy.kron(antisymmetric_mat, numpy.eye(2))
return QuadraticHamiltonian(hermitian_mat, antisymmetric_mat, constant,
chemical_potential)
def random_quadratic_hamiltonian(n_qubits,
conserves_particle_number=False,
real=False):
"""Generate a random instance of QuadraticHamiltonian.
Args:
n_qubits(int): the number of qubits
conserves_particle_number(bool): whether the returned Hamiltonian
should conserve particle number
real(bool): whether to use only real numbers
Returns:
QuadraticHamiltonian
"""
constant = numpy.random.randn()
chemical_potential = numpy.random.randn()
hermitian_mat = random_hermitian_matrix(n_qubits, real)
if conserves_particle_number:
antisymmetric_mat = None
else:
antisymmetric_mat = random_antisymmetric_matrix(n_qubits, real)
return QuadraticHamiltonian(hermitian_mat, antisymmetric_mat, constant,
chemical_potential)
def get_quadratic_hamiltonian(fermion_operator,
chemical_potential=0.,
n_qubits=None):
"""Convert a quadratic fermionic operator to QuadraticHamiltonian.
This function should only be called on fermionic operators which
consist of only a_p^\dagger a_q, a_p^\dagger a_q^\dagger, and a_p a_q
terms.
Returns:
quadratic_hamiltonian: An instance of the QuadraticHamiltonian class.
Raises:
TypeError: Input must be a FermionOperator.
TypeError: FermionOperator does not map to QuadraticHamiltonian.
Warning:
Even assuming that each creation or annihilation operator appears
at most a constant number of times in the original operator, the
runtime of this method is exponential in the number of qubits.
"""
if not isinstance(fermion_operator, FermionOperator):
raise TypeError('Input must be a FermionOperator.')
if n_qubits is None:
n_qubits = count_qubits(fermion_operator)
if n_qubits < count_qubits(fermion_operator):
raise ValueError('Invalid number of qubits specified.')
# Normal order the terms and initialize.
fermion_operator = normal_ordered(fermion_operator)
constant = 0.
combined_hermitian_part = numpy.zeros((n_qubits, n_qubits), complex)
antisymmetric_part = numpy.zeros((n_qubits, n_qubits), complex)
# Loop through terms and assign to matrix.
for term in fermion_operator.terms:
coefficient = fermion_operator.terms[term]
# Ignore this term if the coefficient is zero
if abs(coefficient) < EQ_TOLERANCE:
continue
if len(term) == 0:
# Constant term
constant = coefficient
elif len(term) == 2:
ladder_type = [operator[1] for operator in term]
p, q = [operator[0] for operator in term]
if ladder_type == [1, 0]:
combined_hermitian_part[p, q] = coefficient
elif ladder_type == [1, 1]:
# Need to check that the corresponding [0, 0] term is present
conjugate_term = ((p, 0), (q, 0))
if conjugate_term not in fermion_operator.terms:
raise QuadraticHamiltonianError(
'FermionOperator does not map '
'to QuadraticHamiltonian (not Hermitian).')
else:
matching_coefficient = -fermion_operator.terms[
conjugate_term].conjugate()
discrepancy = abs(coefficient - matching_coefficient)
if discrepancy > EQ_TOLERANCE:
raise QuadraticHamiltonianError(
'FermionOperator does not map '
'to QuadraticHamiltonian (not Hermitian).')
antisymmetric_part[p, q] += .5 * coefficient
antisymmetric_part[q, p] -= .5 * coefficient
else:
# ladder_type == [0, 0]
# Need to check that the corresponding [1, 1] term is present
conjugate_term = ((p, 1), (q, 1))
if conjugate_term not in fermion_operator.terms:
raise QuadraticHamiltonianError(
'FermionOperator does not map '
'to QuadraticHamiltonian (not Hermitian).')
else:
matching_coefficient = -fermion_operator.terms[
conjugate_term].conjugate()
discrepancy = abs(coefficient - matching_coefficient)
if discrepancy > EQ_TOLERANCE:
raise QuadraticHamiltonianError(
'FermionOperator does not map '
'to QuadraticHamiltonian (not Hermitian).')
antisymmetric_part[p, q] -= .5 * coefficient.conjugate()
antisymmetric_part[q, p] += .5 * coefficient.conjugate()
else:
# Operator contains non-quadratic terms
raise QuadraticHamiltonianError('FermionOperator does not map '
'to QuadraticHamiltonian '
'(contains non-quadratic terms).')
# Compute Hermitian part
hermitian_part = (combined_hermitian_part +
chemical_potential * numpy.eye(n_qubits))
# Check that the operator is Hermitian
difference = hermitian_part - hermitian_part.T.conj()
discrepancy = numpy.max(numpy.abs(difference))
if discrepancy > EQ_TOLERANCE:
raise QuadraticHamiltonianError(
'FermionOperator does not map '
'to QuadraticHamiltonian (not Hermitian).')
# Form QuadraticHamiltonian and return.
discrepancy = numpy.max(numpy.abs(antisymmetric_part))
if discrepancy < EQ_TOLERANCE:
# Hamiltonian conserves particle number
quadratic_hamiltonian = QuadraticHamiltonian(
constant, hermitian_part, chemical_potential=chemical_potential)
else:
# Hamiltonian does not conserve particle number
quadratic_hamiltonian = QuadraticHamiltonian(constant, hermitian_part,
antisymmetric_part,
chemical_potential)
return quadratic_hamiltonian
def get_quadratic_hamiltonian(fermion_operator,
chemical_potential=0.,
n_qubits=None,
ignore_incompatible_terms=False):
r"""Convert a quadratic fermionic operator to QuadraticHamiltonian.
Args:
fermion_operator(FermionOperator): The operator to convert.
chemical_potential(float): A chemical potential to include in
the returned operator
n_qubits(int): Optionally specify the total number of qubits in the
system
ignore_incompatible_terms(bool): This flag determines the behavior
of this method when it encounters terms that are not quadratic
that is, terms that are not of the form a^\dagger_p a_q.
If set to True, this method will simply ignore those terms.
If False, then this method will raise an error if it encounters
such a term. The default setting is False.
Returns:
quadratic_hamiltonian: An instance of the QuadraticHamiltonian class.
Raises:
TypeError: Input must be a FermionOperator.
TypeError: FermionOperator does not map to QuadraticHamiltonian.
Warning:
Even assuming that each creation or annihilation operator appears
at most a constant number of times in the original operator, the
runtime of this method is exponential in the number of qubits.
"""
if not isinstance(fermion_operator, FermionOperator):
raise TypeError('Input must be a FermionOperator.')
if n_qubits is None:
n_qubits = count_qubits(fermion_operator)
if n_qubits < count_qubits(fermion_operator):
raise ValueError('Invalid number of qubits specified.')
# Normal order the terms and initialize.
fermion_operator = normal_ordered(fermion_operator)
constant = 0.
combined_hermitian_part = numpy.zeros((n_qubits, n_qubits), complex)
antisymmetric_part = numpy.zeros((n_qubits, n_qubits), complex)
# Loop through terms and assign to matrix.
for term in fermion_operator.terms:
coefficient = fermion_operator.terms[term]
# Ignore this term if the coefficient is zero
if abs(coefficient) < EQ_TOLERANCE:
continue
if len(term) == 0:
# Constant term
constant = coefficient
elif len(term) == 2:
ladder_type = [operator[1] for operator in term]
p, q = [operator[0] for operator in term]
if ladder_type == [1, 0]:
combined_hermitian_part[p, q] = coefficient
elif ladder_type == [1, 1]:
# Need to check that the corresponding [0, 0] term is present
conjugate_term = ((p, 0), (q, 0))
if conjugate_term not in fermion_operator.terms:
raise QuadraticHamiltonianError(
'FermionOperator does not map '
'to QuadraticHamiltonian (not Hermitian).')
else:
matching_coefficient = -fermion_operator.terms[
conjugate_term].conjugate()
discrepancy = abs(coefficient - matching_coefficient)
if discrepancy > EQ_TOLERANCE:
raise QuadraticHamiltonianError(
'FermionOperator does not map '
'to QuadraticHamiltonian (not Hermitian).')
antisymmetric_part[p, q] += .5 * coefficient
antisymmetric_part[q, p] -= .5 * coefficient
else:
# ladder_type == [0, 0]
# Need to check that the corresponding [1, 1] term is present
conjugate_term = ((p, 1), (q, 1))
if conjugate_term not in fermion_operator.terms:
raise QuadraticHamiltonianError(
'FermionOperator does not map '
'to QuadraticHamiltonian (not Hermitian).')
else:
matching_coefficient = -fermion_operator.terms[
conjugate_term].conjugate()
discrepancy = abs(coefficient - matching_coefficient)
if discrepancy > EQ_TOLERANCE:
raise QuadraticHamiltonianError(
'FermionOperator does not map '
'to QuadraticHamiltonian (not Hermitian).')
antisymmetric_part[p, q] -= .5 * coefficient.conjugate()
antisymmetric_part[q, p] += .5 * coefficient.conjugate()
elif not ignore_incompatible_terms:
# Operator contains non-quadratic terms
raise QuadraticHamiltonianError('FermionOperator does not map '
'to QuadraticHamiltonian '
'(contains non-quadratic terms).')
# Compute Hermitian part
hermitian_part = (combined_hermitian_part +
chemical_potential * numpy.eye(n_qubits))
# Check that the operator is Hermitian
if not is_hermitian(hermitian_part):
raise QuadraticHamiltonianError(
'FermionOperator does not map '
'to QuadraticHamiltonian (not Hermitian).')
# Form QuadraticHamiltonian and return.
discrepancy = numpy.max(numpy.abs(antisymmetric_part))
if discrepancy < EQ_TOLERANCE:
# Hamiltonian conserves particle number
quadratic_hamiltonian = QuadraticHamiltonian(
hermitian_part,
constant=constant,
chemical_potential=chemical_potential)
else:
# Hamiltonian does not conserve particle number
quadratic_hamiltonian = QuadraticHamiltonian(hermitian_part,
antisymmetric_part,
constant,
chemical_potential)
return quadratic_hamiltonian
class QuadraticHamiltoniansTest(unittest.TestCase):
def setUp(self):
self.n_qubits = 5
self.constant = 1.7
self.chemical_potential = 2.
# Obtain random Hermitian and antisymmetric matrices
self.hermitian_mat = random_hermitian_matrix(self.n_qubits)
self.antisymmetric_mat = random_antisymmetric_matrix(self.n_qubits)
self.combined_hermitian = (
self.hermitian_mat -
self.chemical_potential * numpy.eye(self.n_qubits))
# Initialize a particle-number-conserving Hamiltonian
self.quad_ham_pc = QuadraticHamiltonian(
self.constant, self.hermitian_mat)
# Initialize a non-particle-number-conserving Hamiltonian
self.quad_ham_npc = QuadraticHamiltonian(
self.constant, self.hermitian_mat, self.antisymmetric_mat,
self.chemical_potential)
# Initialize the sparse operators and get their ground energies
self.quad_ham_pc_sparse = get_sparse_operator(self.quad_ham_pc)
self.quad_ham_npc_sparse = get_sparse_operator(self.quad_ham_npc)
self.pc_ground_energy, self.pc_ground_state = get_ground_state(
self.quad_ham_pc_sparse)
self.npc_ground_energy, self.npc_ground_state = get_ground_state(
self.quad_ham_npc_sparse)
def test_combined_hermitian_part(self):
"""Test getting the combined Hermitian part."""
combined_hermitian_part = self.quad_ham_pc.combined_hermitian_part
for i in numpy.ndindex(combined_hermitian_part.shape):
self.assertAlmostEqual(self.hermitian_mat[i],
combined_hermitian_part[i])
combined_hermitian_part = self.quad_ham_npc.combined_hermitian_part
for i in numpy.ndindex(combined_hermitian_part.shape):
self.assertAlmostEqual(self.combined_hermitian[i],
combined_hermitian_part[i])
def test_hermitian_part(self):
"""Test getting the Hermitian part."""
hermitian_part = self.quad_ham_pc.hermitian_part
for i in numpy.ndindex(hermitian_part.shape):
self.assertAlmostEqual(self.hermitian_mat[i], hermitian_part[i])
hermitian_part = self.quad_ham_npc.hermitian_part
for i in numpy.ndindex(hermitian_part.shape):
self.assertAlmostEqual(self.hermitian_mat[i], hermitian_part[i])
def test_antisymmetric_part(self):
"""Test getting the antisymmetric part."""
antisymmetric_part = self.quad_ham_pc.antisymmetric_part
for i in numpy.ndindex(antisymmetric_part.shape):
self.assertAlmostEqual(0., antisymmetric_part[i])
antisymmetric_part = self.quad_ham_npc.antisymmetric_part
for i in numpy.ndindex(antisymmetric_part.shape):
self.assertAlmostEqual(self.antisymmetric_mat[i],
antisymmetric_part[i])
def test_conserves_particle_number(self):
"""Test checking whether Hamiltonian conserves particle number."""
self.assertTrue(self.quad_ham_pc.conserves_particle_number)
self.assertFalse(self.quad_ham_npc.conserves_particle_number)
def test_add_chemical_potential(self):
"""Test adding a chemical potential."""
self.quad_ham_npc.add_chemical_potential(2.4)
combined_hermitian_part = self.quad_ham_npc.combined_hermitian_part
hermitian_part = self.quad_ham_npc.hermitian_part
want_combined = (self.combined_hermitian -
2.4 * numpy.eye(self.n_qubits))
want_hermitian = self.hermitian_mat
for i in numpy.ndindex(combined_hermitian_part.shape):
self.assertAlmostEqual(combined_hermitian_part[i],
want_combined[i])
for i in numpy.ndindex(hermitian_part.shape):
self.assertAlmostEqual(hermitian_part[i], want_hermitian[i])
self.assertAlmostEqual(2.4 + self.chemical_potential,
self.quad_ham_npc.chemical_potential)
def test_orbital_energies(self):
"""Test getting the orbital energies."""
# Test the particle-number-conserving case
orbital_energies, constant = self.quad_ham_pc.orbital_energies()
# Test the ground energy
energy = numpy.sum(
orbital_energies[orbital_energies < -EQ_TOLERANCE]) + constant
self.assertAlmostEqual(energy, self.pc_ground_energy)
# Test the non-particle-number-conserving case
orbital_energies, constant = self.quad_ham_npc.orbital_energies()
# Test the ground energy
energy = constant
self.assertAlmostEqual(energy, self.npc_ground_energy)
def test_ground_energy(self):
"""Test getting the ground energy."""
# Test particle-number-conserving case
energy = self.quad_ham_pc.ground_energy()
self.assertAlmostEqual(energy, self.pc_ground_energy)
# Test non-particle-number-conserving case
energy = self.quad_ham_npc.ground_energy()
self.assertAlmostEqual(energy, self.npc_ground_energy)
def test_majorana_form(self):
"""Test getting the Majorana form."""
majorana_matrix, majorana_constant = self.quad_ham_npc.majorana_form()
# Convert the Majorana form to a FermionOperator
majorana_op = FermionOperator((), majorana_constant)
for i in range(2 * self.n_qubits):
if i < self.n_qubits:
left_op = majorana_operator((i, 1))
else:
left_op = majorana_operator((i - self.n_qubits, 0))
for j in range(2 * self.n_qubits):
if j < self.n_qubits:
right_op = majorana_operator((j, 1), majorana_matrix[i, j])
else:
right_op = majorana_operator((j - self.n_qubits, 0),
majorana_matrix[i, j])
majorana_op += .5j * left_op * right_op
# Get FermionOperator for original Hamiltonian
fermion_operator = normal_ordered(
get_fermion_operator(self.quad_ham_npc))
self.assertTrue(
normal_ordered(majorana_op).isclose(fermion_operator))
def test_diagonalizing_bogoliubov_transform(self):
"""Test getting the diagonalizing Bogoliubov transformation."""
hermitian_part = self.quad_ham_npc.combined_hermitian_part
antisymmetric_part = self.quad_ham_npc.antisymmetric_part
block_matrix = numpy.zeros((2 * self.n_qubits, 2 * self.n_qubits),
dtype=complex)
block_matrix[:self.n_qubits, :self.n_qubits] = antisymmetric_part
block_matrix[:self.n_qubits, self.n_qubits:] = hermitian_part
block_matrix[self.n_qubits:, :self.n_qubits] = -hermitian_part.conj()
block_matrix[self.n_qubits:, self.n_qubits:] = (
-antisymmetric_part.conj())
ferm_unitary = self.quad_ham_npc.diagonalizing_bogoliubov_transform()
# Check that the transformation is diagonalizing
majorana_matrix, majorana_constant = self.quad_ham_npc.majorana_form()
canonical, orthogonal = antisymmetric_canonical_form(majorana_matrix)
diagonalized = ferm_unitary.conj().dot(
block_matrix.dot(ferm_unitary.T.conj()))
for i in numpy.ndindex((2 * self.n_qubits, 2 * self.n_qubits)):
self.assertAlmostEqual(diagonalized[i], canonical[i])
lower_unitary = ferm_unitary[self.n_qubits:]
lower_left = lower_unitary[:, :self.n_qubits]
lower_right = lower_unitary[:, self.n_qubits:]
# Check that lower_left and lower_right satisfy the constraints
# necessary for the transformed fermionic operators to satisfy
# the fermionic anticommutation relations
constraint_matrix_1 = (lower_left.dot(lower_left.T.conj()) +
lower_right.dot(lower_right.T.conj()))
constraint_matrix_2 = (lower_left.dot(lower_right.T) +
lower_right.dot(lower_left.T))
identity = numpy.eye(self.n_qubits, dtype=complex)
for i in numpy.ndindex((self.n_qubits, self.n_qubits)):
self.assertAlmostEqual(identity[i], constraint_matrix_1[i])
self.assertAlmostEqual(0., constraint_matrix_2[i])
class QuadraticHamiltoniansTest(unittest.TestCase):
def setUp(self):
self.n_qubits = 5
self.constant = 1.7
self.chemical_potential = 2.
# Obtain random Hermitian and antisymmetric matrices
rand_mat_A = numpy.random.randn(self.n_qubits, self.n_qubits)
rand_mat_B = numpy.random.randn(self.n_qubits, self.n_qubits)
rand_mat = rand_mat_A + 1.j * rand_mat_B
self.hermitian_mat = rand_mat + rand_mat.T.conj()
rand_mat_A = numpy.random.randn(self.n_qubits, self.n_qubits)
rand_mat_B = numpy.random.randn(self.n_qubits, self.n_qubits)
rand_mat = rand_mat_A + 1.j * rand_mat_B
self.antisymmetric_mat = rand_mat - rand_mat.T
self.combined_hermitian = (
self.hermitian_mat -
self.chemical_potential * numpy.eye(self.n_qubits))
# Initialize a particle-number-conserving Hamiltonian
self.quad_ham_pc = QuadraticHamiltonian(self.constant,
self.hermitian_mat)
# Initialize a non-particle-number-conserving Hamiltonian
self.quad_ham_npc = QuadraticHamiltonian(self.constant,
self.hermitian_mat,
self.antisymmetric_mat,
self.chemical_potential)
def test_combined_hermitian_part(self):
"""Test getting the combined Hermitian part."""
combined_hermitian_part = self.quad_ham_pc.combined_hermitian_part()
for i in numpy.ndindex(combined_hermitian_part.shape):
self.assertAlmostEqual(self.hermitian_mat[i],
combined_hermitian_part[i])
combined_hermitian_part = self.quad_ham_npc.combined_hermitian_part()
for i in numpy.ndindex(combined_hermitian_part.shape):
self.assertAlmostEqual(self.combined_hermitian[i],
combined_hermitian_part[i])
def test_hermitian_part(self):
"""Test getting the Hermitian part."""
hermitian_part = self.quad_ham_pc.hermitian_part()
for i in numpy.ndindex(hermitian_part.shape):
self.assertAlmostEqual(self.hermitian_mat[i], hermitian_part[i])
hermitian_part = self.quad_ham_npc.hermitian_part()
for i in numpy.ndindex(hermitian_part.shape):
self.assertAlmostEqual(self.hermitian_mat[i], hermitian_part[i])
def test_antisymmetric_part(self):
"""Test getting the antisymmetric part."""
antisymmetric_part = self.quad_ham_pc.antisymmetric_part()
for i in numpy.ndindex(antisymmetric_part.shape):
self.assertAlmostEqual(0., antisymmetric_part[i])
antisymmetric_part = self.quad_ham_npc.antisymmetric_part()
for i in numpy.ndindex(antisymmetric_part.shape):
self.assertAlmostEqual(self.antisymmetric_mat[i],
antisymmetric_part[i])
def test_conserves_particle_number(self):
"""Test checking whether Hamiltonian conserves particle number."""
self.assertTrue(self.quad_ham_pc.conserves_particle_number())
self.assertFalse(self.quad_ham_npc.conserves_particle_number())
def test_majorana_form(self):
"""Test getting the Majorana form."""
majorana_matrix, majorana_constant = self.quad_ham_npc.majorana_form()
# Convert the Majorana form to a FermionOperator
majorana_op = FermionOperator((), majorana_constant)
for i in range(2 * self.n_qubits):
if i < self.n_qubits:
left_op = majorana_operator((i, 1))
else:
left_op = majorana_operator((i - self.n_qubits, 0))
for j in range(2 * self.n_qubits):
if j < self.n_qubits:
right_op = majorana_operator((j, 1), majorana_matrix[i, j])
else:
right_op = majorana_operator((j - self.n_qubits, 0),
majorana_matrix[i, j])
majorana_op += .5j * left_op * right_op
# Get FermionOperator for original Hamiltonian
fermion_operator = normal_ordered(
get_fermion_operator(self.quad_ham_npc))
self.assertTrue(normal_ordered(majorana_op).isclose(fermion_operator))
class QuadraticHamiltoniansTest(unittest.TestCase):
def setUp(self):
self.n_qubits = 5
self.constant = 1.7
self.chemical_potential = 2.
# Obtain random Hermitian and antisymmetric matrices
self.hermitian_mat = random_hermitian_matrix(self.n_qubits)
self.antisymmetric_mat = random_antisymmetric_matrix(self.n_qubits)
self.combined_hermitian = (
self.hermitian_mat -
self.chemical_potential * numpy.eye(self.n_qubits))
# Initialize a particle-number-conserving Hamiltonian
self.quad_ham_pc = QuadraticHamiltonian(
self.constant, self.hermitian_mat)
# Initialize a non-particle-number-conserving Hamiltonian
self.quad_ham_npc = QuadraticHamiltonian(
self.constant, self.hermitian_mat, self.antisymmetric_mat,
self.chemical_potential)
def test_combined_hermitian_part(self):
"""Test getting the combined Hermitian part."""
combined_hermitian_part = self.quad_ham_pc.combined_hermitian_part
for i in numpy.ndindex(combined_hermitian_part.shape):
self.assertAlmostEqual(self.hermitian_mat[i],
combined_hermitian_part[i])
combined_hermitian_part = self.quad_ham_npc.combined_hermitian_part
for i in numpy.ndindex(combined_hermitian_part.shape):
self.assertAlmostEqual(self.combined_hermitian[i],
combined_hermitian_part[i])
def test_hermitian_part(self):
"""Test getting the Hermitian part."""
hermitian_part = self.quad_ham_pc.hermitian_part
for i in numpy.ndindex(hermitian_part.shape):
self.assertAlmostEqual(self.hermitian_mat[i], hermitian_part[i])
hermitian_part = self.quad_ham_npc.hermitian_part
for i in numpy.ndindex(hermitian_part.shape):
self.assertAlmostEqual(self.hermitian_mat[i], hermitian_part[i])
def test_antisymmetric_part(self):
"""Test getting the antisymmetric part."""
antisymmetric_part = self.quad_ham_pc.antisymmetric_part
for i in numpy.ndindex(antisymmetric_part.shape):
self.assertAlmostEqual(0., antisymmetric_part[i])
antisymmetric_part = self.quad_ham_npc.antisymmetric_part
for i in numpy.ndindex(antisymmetric_part.shape):
self.assertAlmostEqual(self.antisymmetric_mat[i],
antisymmetric_part[i])
def test_conserves_particle_number(self):
"""Test checking whether Hamiltonian conserves particle number."""
self.assertTrue(self.quad_ham_pc.conserves_particle_number)
self.assertFalse(self.quad_ham_npc.conserves_particle_number)
def test_add_chemical_potential(self):
"""Test adding a chemical potential."""
self.quad_ham_npc.add_chemical_potential(2.4)
combined_hermitian_part = self.quad_ham_npc.combined_hermitian_part
hermitian_part = self.quad_ham_npc.hermitian_part
want_combined = (self.combined_hermitian -
2.4 * numpy.eye(self.n_qubits))
want_hermitian = self.hermitian_mat
for i in numpy.ndindex(combined_hermitian_part.shape):
self.assertAlmostEqual(combined_hermitian_part[i],
want_combined[i])
for i in numpy.ndindex(hermitian_part.shape):
self.assertAlmostEqual(hermitian_part[i], want_hermitian[i])
self.assertAlmostEqual(2.4 + self.chemical_potential,
self.quad_ham_npc.chemical_potential)
def test_orbital_energies(self):
"""Test getting the orbital energies."""
# Test the particle-number-conserving case
orbital_energies, constant = self.quad_ham_pc.orbital_energies()
# Test the ground energy
energy = numpy.sum(
orbital_energies[orbital_energies < -EQ_TOLERANCE]) + constant
quad_ham_pc_sparse = get_sparse_operator(self.quad_ham_pc)
ground_energy, ground_state = get_ground_state(quad_ham_pc_sparse)
self.assertAlmostEqual(energy, ground_energy)
# Test the non-particle-number-conserving case
orbital_energies, constant = self.quad_ham_npc.orbital_energies()
# Test the ground energy
energy = constant
quad_ham_npc_sparse = get_sparse_operator(self.quad_ham_npc)
ground_energy, ground_state = get_ground_state(quad_ham_npc_sparse)
self.assertAlmostEqual(energy, ground_energy)
def test_majorana_form(self):
"""Test getting the Majorana form."""
majorana_matrix, majorana_constant = self.quad_ham_npc.majorana_form()
# Convert the Majorana form to a FermionOperator
majorana_op = FermionOperator((), majorana_constant)
for i in range(2 * self.n_qubits):
if i < self.n_qubits:
left_op = majorana_operator((i, 1))
else:
left_op = majorana_operator((i - self.n_qubits, 0))
for j in range(2 * self.n_qubits):
if j < self.n_qubits:
right_op = majorana_operator((j, 1), majorana_matrix[i, j])
else:
right_op = majorana_operator((j - self.n_qubits, 0),
majorana_matrix[i, j])
majorana_op += .5j * left_op * right_op
# Get FermionOperator for original Hamiltonian
fermion_operator = normal_ordered(
get_fermion_operator(self.quad_ham_npc))
self.assertTrue(
normal_ordered(majorana_op).isclose(fermion_operator))
class DiagonalizingFermionicUnitaryTest(unittest.TestCase):
def setUp(self):
self.n_qubits = 5
self.constant = 1.7
self.chemical_potential = 2.
# Obtain random Hermitian and antisymmetric matrices
self.hermitian_mat = random_hermitian_matrix(self.n_qubits)
self.antisymmetric_mat = random_antisymmetric_matrix(self.n_qubits)
# Initialize a non-particle-number-conserving Hamiltonian
self.quad_ham_npc = QuadraticHamiltonian(
self.constant, self.hermitian_mat, self.antisymmetric_mat,
self.chemical_potential)
def test_diagonalizes_quadratic_hamiltonian(self):
"""Test that the unitary returned indeed diagonalizes a
quadratic Hamiltonian."""
hermitian_part = self.quad_ham_npc.combined_hermitian_part
antisymmetric_part = self.quad_ham_npc.antisymmetric_part
block_matrix = numpy.zeros((2 * self.n_qubits, 2 * self.n_qubits),
dtype=complex)
block_matrix[:self.n_qubits, :self.n_qubits] = antisymmetric_part
block_matrix[:self.n_qubits, self.n_qubits:] = hermitian_part
block_matrix[self.n_qubits:, :self.n_qubits] = -hermitian_part.conj()
block_matrix[self.n_qubits:, self.n_qubits:] = (
-antisymmetric_part.conj())
majorana_matrix, majorana_constant = self.quad_ham_npc.majorana_form()
canonical, orthogonal = antisymmetric_canonical_form(majorana_matrix)
ferm_unitary = diagonalizing_fermionic_unitary(majorana_matrix)
diagonalized = ferm_unitary.conj().dot(
block_matrix.dot(ferm_unitary.T.conj()))
for i in numpy.ndindex((2 * self.n_qubits, 2 * self.n_qubits)):
self.assertAlmostEqual(diagonalized[i], canonical[i])
def test_bad_dimensions(self):
n, p = (3, 4)
ones_mat = numpy.ones((n, p))
with self.assertRaises(ValueError):
ferm_unitary = diagonalizing_fermionic_unitary(ones_mat)
def test_not_antisymmetric(self):
n = 4
ones_mat = numpy.ones((n, n))
with self.assertRaises(ValueError):
ferm_unitary = diagonalizing_fermionic_unitary(ones_mat)
def test_n_equals_3(self):
n = 3
# Obtain a random antisymmetric matrix
rand_mat = numpy.random.randn(2 * n, 2 * n)
antisymmetric_matrix = rand_mat - rand_mat.T
# Get the diagonalizing fermionic unitary
ferm_unitary = diagonalizing_fermionic_unitary(antisymmetric_matrix)
lower_unitary = ferm_unitary[n:]
lower_left = lower_unitary[:, :n]
lower_right = lower_unitary[:, n:]
# Check that lower_left and lower_right satisfy the constraints
# necessary for the transformed fermionic operators to satisfy
# the fermionic anticommutation relations
constraint_matrix_1 = (lower_left.dot(lower_left.T.conj()) +
lower_right.dot(lower_right.T.conj()))
constraint_matrix_2 = (lower_left.dot(lower_right.T) +
lower_right.dot(lower_left.T))
identity = numpy.eye(n, dtype=complex)
for i in numpy.ndindex((n, n)):
self.assertAlmostEqual(identity[i], constraint_matrix_1[i])
self.assertAlmostEqual(0., constraint_matrix_2[i])
