def test_bk_jw_number_operators(self):
# Check if a number operator has the same spectrum in both
# JW and BK representations
n_qubits = 2
n1 = number_operator(n_qubits, 0)
n2 = number_operator(n_qubits, 1)
n = n1 + n2
jw_n = jordan_wigner(n)
bk_n = bravyi_kitaev(n)
# Diagonalize and make sure the spectra are the same.
jw_spectrum = eigenspectrum(jw_n)
bk_spectrum = eigenspectrum(bk_n)
self.assertAlmostEqual(
0., numpy.amax(numpy.absolute(jw_spectrum - bk_spectrum)))
def test_jordan_wigner_transm_op(self):
n = number_operator(self.n_qubits)
n_jw = jordan_wigner(n)
self.assertEqual(self.n_qubits + 1, len(n_jw.terms))
self.assertEqual(self.n_qubits / 2., n_jw.terms[()])
for qubit in range(self.n_qubits):
operators = ((qubit, 'Z'), )
self.assertEqual(n_jw.terms[operators], -0.5)
def test_bk_jw_number_operator(self):
# Check if number operator has the same spectrum in both
# BK and JW representations
n = number_operator(1, 0)
jw_n = jordan_wigner(n)
bk_n = bravyi_kitaev(n)
# Diagonalize and make sure the spectra are the same.
jw_spectrum = eigenspectrum(jw_n)
bk_spectrum = eigenspectrum(bk_n)
self.assertAlmostEqual(
0., numpy.amax(numpy.absolute(jw_spectrum - bk_spectrum)))
def test_bk_jw_number_operator_scaled(self):
# Check if number operator has the same spectrum in both
# JW and BK representations
n_qubits = 1
n = number_operator(n_qubits, 0, coefficient=2) # eigenspectrum (0,2)
jw_n = jordan_wigner(n)
bk_n = bravyi_kitaev(n)
# Diagonalize and make sure the spectra are the same.
jw_spectrum = eigenspectrum(jw_n)
bk_spectrum = eigenspectrum(bk_n)
self.assertAlmostEqual(
0., numpy.amax(numpy.absolute(jw_spectrum - bk_spectrum)))
def test_jw_restrict_jellium_ground_state_integration(self):
n_qubits = 4
grid = Grid(dimensions=1, length=n_qubits, scale=1.0)
jellium_hamiltonian = jordan_wigner_sparse(
jellium_model(grid, spinless=False))
# 2 * n_qubits because of spin
number_sparse = jordan_wigner_sparse(number_operator(2 * n_qubits))
restricted_number = jw_number_restrict_operator(number_sparse, 2)
restricted_jellium_hamiltonian = jw_number_restrict_operator(
jellium_hamiltonian, 2)
energy, ground_state = get_ground_state(restricted_jellium_hamiltonian)
number_expectation = expectation(restricted_number, ground_state)
self.assertAlmostEqual(number_expectation, 2)
def test_jw_restrict_operator(self):
"""Test the scheme for restricting JW encoded operators to number"""
# Make a Hamiltonian that cares mostly about number of electrons
n_qubits = 6
target_electrons = 3
penalty_const = 100.
number_sparse = jordan_wigner_sparse(number_operator(n_qubits))
bias_sparse = jordan_wigner_sparse(
sum([
FermionOperator(((i, 1), (i, 0)), 1.0) for i in range(n_qubits)
], FermionOperator()))
hamiltonian_sparse = penalty_const * (
number_sparse -
target_electrons * scipy.sparse.identity(2**n_qubits)
).dot(number_sparse - target_electrons *
scipy.sparse.identity(2**n_qubits)) + bias_sparse
restricted_hamiltonian = jw_number_restrict_operator(
hamiltonian_sparse, target_electrons, n_qubits)
true_eigvals, _ = eigh(hamiltonian_sparse.A)
test_eigvals, _ = eigh(restricted_hamiltonian.A)
self.assertAlmostEqual(norm(true_eigvals[:20] - test_eigvals[:20]),
0.0)
def test_transm_number(self):
n = number_operator(self.n_qubits, 3)
n_jw = jordan_wigner(n)
self.assertEqual(n_jw.terms[((3, 'Z'), )], -0.5)
self.assertEqual(n_jw.terms[()], 0.5)
self.assertEqual(len(n_jw.terms), 2)
def reverse_jordan_wigner(qubit_operator, n_qubits=None):
"""Transforms a QubitOperator into a FermionOperator using the
Jordan-Wigner transform.
Operators are mapped as follows:
Z_j -> I - 2 a^\dagger_j a_j
X_j -> (a^\dagger_j + a_j) Z_{j-1} Z_{j-2} .. Z_0
Y_j -> i (a^\dagger_j - a_j) Z_{j-1} Z_{j-2} .. Z_0
Args:
qubit_operator: the QubitOperator to be transformed.
n_qubits: the number of qubits term acts on. If not set, defaults
to the maximum qubit number acted on by term.
Returns:
transformed_term: An instance of the FermionOperator class.
Raises:
TypeError: Input must be a QubitOperator.
TypeError: Invalid number of qubits specified.
TypeError: Pauli operators must be X, Y or Z.
"""
if not isinstance(qubit_operator, QubitOperator):
raise TypeError('Input must be a QubitOperator.')
if n_qubits is None:
n_qubits = count_qubits(qubit_operator)
if n_qubits < count_qubits(qubit_operator):
raise TypeError('Invalid number of qubits specified')
# Loop through terms.
transformed_operator = FermionOperator()
for term in qubit_operator.terms:
transformed_term = FermionOperator(())
if term:
working_term = QubitOperator(term)
pauli_operator = term[-1]
while pauli_operator is not None:
# Handle Pauli Z.
if pauli_operator[1] == 'Z':
transformed_pauli = FermionOperator(
()) + number_operator(n_qubits, pauli_operator[0], -2.)
# Handle Pauli X and Y.
else:
raising_term = FermionOperator(((pauli_operator[0], 1), ))
lowering_term = FermionOperator(((pauli_operator[0], 0), ))
if pauli_operator[1] == 'Y':
raising_term *= 1.j
lowering_term *= -1.j
elif pauli_operator[1] != 'X':
raise TypeError('Pauli operators must be X, Y, or Z')
transformed_pauli = raising_term + lowering_term
# Account for the phase terms.
for j in reversed(range(pauli_operator[0])):
z_term = QubitOperator(((j, 'Z'), ))
working_term = z_term * working_term
term_key = list(working_term.terms)[0]
transformed_pauli *= working_term.terms[term_key]
working_term.terms[list(working_term.terms)[0]] = 1.
# Get next non-identity operator acting below 'working_qubit'.
assert len(working_term.terms) == 1
working_qubit = pauli_operator[0] - 1
for working_operator in reversed(list(working_term.terms)[0]):
if working_operator[0] <= working_qubit:
pauli_operator = working_operator
break
else:
pauli_operator = None
# Multiply term by transformed operator.
transformed_term *= transformed_pauli
# Account for overall coefficient
transformed_term *= qubit_operator.terms[term]
transformed_operator += transformed_term
return transformed_operator
def fermi_hubbard(x_dimension, y_dimension, tunneling, coulomb,
chemical_potential=None, magnetic_field=None,
periodic=True, spinless=False):
"""Return symbolic representation of a Fermi-Hubbard Hamiltonian.
Args:
x_dimension: An integer giving the number of sites in width.
y_dimension: An integer giving the number of sites in height.
tunneling: A float giving the tunneling amplitude.
coulomb: A float giving the attractive local interaction strength.
chemical_potential: An optional float giving the potential of each
site. Default value is None.
magnetic_field: An optional float giving a magnetic field at each
site. Default value is None.
periodic: If True, add periodic boundary conditions.
spinless: An optional Boolean. If False, each site has spin up
orbitals and spin down orbitals. If True, return a spinless
Fermi-Hubbard model.
verbose: An optional Boolean. If True, print all second quantized
terms.
Returns:
hubbard_model: An instance of the FermionOperator class.
"""
# Initialize fermion operator class.
n_sites = x_dimension * y_dimension
if spinless:
n_spin_orbitals = n_sites
else:
n_spin_orbitals = 2 * n_sites
hubbard_model = FermionOperator((), 0.0)
# Loop through sites and add terms.
for site in range(n_sites):
# Add chemical potential and magnetic field terms.
if chemical_potential and spinless:
x_index = site % x_dimension
y_index = (site - 1) // x_dimension
sign = (-1.) ** (x_index + y_index)
coefficient = sign * chemical_potential
hubbard_model += number_operator(
n_spin_orbitals, site, coefficient)
if chemical_potential and not spinless:
coefficient = -1. * chemical_potential
hubbard_model += number_operator(
n_spin_orbitals, up(site), coefficient)
hubbard_model += number_operator(
n_spin_orbitals, down(site), coefficient)
if magnetic_field and not spinless:
coefficient = magnetic_field
hubbard_model += number_operator(
n_spin_orbitals, up(site), -coefficient)
hubbard_model += number_operator(
n_spin_orbitals, down(site), coefficient)
# Add local pair interaction terms.
if not spinless:
operators = ((up(site), 1), (up(site), 0),
(down(site), 1), (down(site), 0))
hubbard_model += FermionOperator(operators, coulomb)
# Index coupled orbitals.
right_neighbor = site + 1
bottom_neighbor = site + x_dimension
# Account periodic boundaries.
if periodic:
if (x_dimension > 2) and ((site + 1) % x_dimension == 0):
right_neighbor -= (x_dimension - 1)
if (y_dimension > 2) and (site + x_dimension + 1 > n_sites):
bottom_neighbor -= (x_dimension - 1) * y_dimension
# Add transition to neighbor on right.
if (site + 1) % x_dimension or (periodic and x_dimension > 2):
if spinless:
# Add Coulomb term.
operators = ((site, 1), (site, 0),
(right_neighbor, 1), (right_neighbor, 0))
hubbard_model += FermionOperator(operators, coulomb)
# Add hopping term.
operators = ((site, 1), (right_neighbor, 0))
hopping_term = FermionOperator(operators, -tunneling)
hubbard_model += hopping_term
hubbard_model += hermitian_conjugated(hopping_term)
else:
# Add hopping term.
operators = ((up(site), 1), (up(right_neighbor), 0))
hopping_term = FermionOperator(operators, -tunneling)
hubbard_model += hopping_term
hubbard_model += hermitian_conjugated(hopping_term)
operators = ((down(site), 1), (down(right_neighbor), 0))
hopping_term = FermionOperator(operators, -tunneling)
hubbard_model += hopping_term
hubbard_model += hermitian_conjugated(hopping_term)
# Add transition to neighbor below.
if site + x_dimension + 1 <= n_sites or (periodic and y_dimension > 2):
if spinless:
# Add Coulomb term.
operators = ((site, 1), (site, 0),
(bottom_neighbor, 1), (bottom_neighbor, 0))
hubbard_model += FermionOperator(operators, coulomb)
# Add hopping term.
operators = ((site, 1), (bottom_neighbor, 0))
hopping_term = FermionOperator(operators, -tunneling)
hubbard_model += hopping_term
hubbard_model += hermitian_conjugated(hopping_term)
else:
# Add hopping term.
operators = ((up(site), 1), (up(bottom_neighbor), 0))
hopping_term = FermionOperator(operators, -tunneling)
hubbard_model += hopping_term
hubbard_model += hermitian_conjugated(hopping_term)
operators = ((down(site), 1), (down(bottom_neighbor), 0))
hopping_term = FermionOperator(operators, -tunneling)
hubbard_model += hopping_term
hubbard_model += hermitian_conjugated(hopping_term)
# Return.
return hubbard_model