def pauli_terms_for_tpdm_aa(spatial_dim, transform=jordan_wigner):
"""
Generate a set of pauli operators to measure to evaluate the alpha-alpha
block of the 2-RDM
Given a dictionary of expected values of Pauli terms, populate the
alpha-beta block of the 2-RDM
:param Int spatial_dim: Dimension of the spatial-orbital basis.
:param transform: fermion-to-qubit transform from OpenFermion
:return: List of PauliTerms that corresponds to set of pauli terms to
measure to construct the 2-RDM.
"""
# build basis lookup table
bas_aa = {}
cnt_aa = 0
for p, q in product(range(spatial_dim), repeat=2):
if q < p:
bas_aa[(p, q)] = cnt_aa
cnt_aa += 1
pauli_terms_to_measure = []
pauli_to_rdm = {}
for p, q, r, s in product(range(spatial_dim), repeat=4):
if p < q and s < r:
# generator 1/sqrt(2) * (a_{p, \alpha}^{\dagger}
# a_{q, \alpha}^{\dagger} -
# a_{p, \beta}^{\dagger}a_{q, \beta}^{\dagger})
spin_adapted_term = FermionOperator(
((2 * p, 1), (2 * q, 1), (2 * r, 0), (2 * s, 0))) - \
FermionOperator(((2 * p, 1), (2 * q, 1),
(2 * s, 0), (2 * r, 0))) - \
FermionOperator(((2 * q, 1), (2 * p, 1),
(2 * r, 0), (2 * s, 0))) + \
FermionOperator(((2 * q, 1), (2 * p, 1),
(2 * s, 0), (2 * r, 0)))
spin_adapted_term *= 0.5
tpdm_element_as_pauli = remove_imaginary_terms(
qubitop_to_pyquilpauli(transform(spin_adapted_term)))
for term in tpdm_element_as_pauli:
pauli_terms_to_measure.append(term)
for term in pauli_terms_to_measure:
# convert term into numerically order pauli tensor term
pauli_tensor_list = sorted(list(term.operations_as_set()),
key=lambda x: x[0])
rev_order_pauli_tensor_list = list(
map(lambda x: (x[1], x[0]), pauli_tensor_list))
pauli_term = PauliTerm.from_list(rev_order_pauli_tensor_list,
coefficient=term.coefficient)
if pauli_term.id() not in pauli_to_rdm.keys():
pauli_to_rdm[pauli_term.id()] = pauli_term
else:
if (abs(pauli_to_rdm[pauli_term.id()].coefficient) < abs(
pauli_term.coefficient)):
pauli_to_rdm[pauli_term.id()] = pauli_term
return list(pauli_to_rdm.values())
def pauli_to_tpdm_bb(spatial_dim, pauli_to_coeff, transform=jordan_wigner):
"""
Populate the beta-beta block of the 2-RDM
:param Int spatial_dim: spatial basis set rank
:param pauli_to_coeff: a map between the Pauli term label to the expected
value.
:param transform: Openfermion fermion-to-qubit transform
:return: the 2-RDM alpha-beta block of the 2-RDM
"""
aa_dim = int(spatial_dim * (spatial_dim - 1) / 2)
d2_aa = np.zeros((aa_dim, aa_dim), dtype=complex)
# build basis lookup table
bas_aa = {}
cnt_aa = 0
for p, q in product(range(spatial_dim), repeat=2):
if p < q:
bas_aa[(p, q)] = cnt_aa
cnt_aa += 1
for p, q, r, s in product(range(spatial_dim), repeat=4):
if p < q and s < r:
# generator 1/sqrt(2) * (a_{p, \alpha}^{\dagger}
# a_{q, \alpha}^{\dagger} -
# a_{p, \beta}^{\dagger}a_{q, \beta}^{\dagger})
spin_adapted_term = FermionOperator(((2 * p + 1, 1), (2 * q + 1, 1),
(2 * r + 1, 0),
(2 * s + 1, 0))) - \
FermionOperator(((2 * p + 1, 1), (2 * q + 1, 1),
(2 * s + 1, 0),
(2 * r + 1, 0))) - \
FermionOperator(((2 * q + 1, 1), (2 * p + 1, 1),
(2 * r + 1, 0),
(2 * s + 1, 0))) + \
FermionOperator(((2 * q + 1, 1), (2 * p + 1, 1),
(2 * s + 1, 0),
(2 * r + 1, 0)))
spin_adapted_term *= 0.5
tpdm_element_as_pauli = remove_imaginary_terms(
qubitop_to_pyquilpauli(transform(spin_adapted_term)))
for term in tpdm_element_as_pauli:
pauli_tensor_list = sorted(list(term.operations_as_set()),
key=lambda x: x[0])
rev_order_pauli_tensor_list = list(
map(lambda x: (x[1], x[0]), pauli_tensor_list))
pauli_term = PauliTerm.from_list(rev_order_pauli_tensor_list,
coefficient=term.coefficient)
try:
d2_aa[bas_aa[(p, q)], bas_aa[(s, r)]] += pauli_to_coeff[
pauli_term.id()] * \
pauli_term.coefficient
except KeyError:
raise Warning("key was not in the coeff matrix.")
return d2_aa
def pauli_terms_for_tpdm_ab(spatial_dim, transform=jordan_wigner):
"""
Build the alpha-beta block of the 2-RDM
Note: OpenFermion ordering is used. The 2-RDM(alpha-beta) block
is spatial_dim**2 linear size. (DOI: 10.1021/acs.jctc.6b00190)
:param spatial_dim: rank of spatial orbitals in the basis set.
:param transform: type of fermion-to-qubit transformation.
:return: list of Pauli terms to measure required to construct a the alpha-
beta block.
"""
# build basis look up table
bas_ab = {}
cnt_ab = 0
# iterate over spatial orbital indices
for p, q in product(range(spatial_dim), repeat=2):
bas_ab[(p, q)] = cnt_ab
cnt_ab += 1
pauli_terms_to_measure = []
pauli_to_rdm = {}
for p, q, r, s in product(range(spatial_dim), repeat=4):
spin_adapted_term = FermionOperator(
((2 * p, 1), (2 * q + 1, 1), (2 * r + 1, 0), (2 * s, 0)))
tpdm_element_as_pauli = remove_imaginary_terms(
qubitop_to_pyquilpauli(transform(spin_adapted_term)))
for term in tpdm_element_as_pauli:
pauli_terms_to_measure.append(term)
for term in pauli_terms_to_measure:
# convert term into numerically order pauli tensor term
pauli_tensor_list = sorted(list(term.operations_as_set()),
key=lambda x: x[0])
rev_order_pauli_tensor_list = list(
map(lambda x: (x[1], x[0]), pauli_tensor_list))
pauli_term = PauliTerm.from_list(rev_order_pauli_tensor_list,
coefficient=term.coefficient)
if pauli_term.id() not in pauli_to_rdm.keys():
pauli_to_rdm[pauli_term.id()] = pauli_term
else:
if (abs(pauli_to_rdm[pauli_term.id()].coefficient) < abs(
pauli_term.coefficient)):
pauli_to_rdm[pauli_term.id()] = pauli_term
return list(pauli_to_rdm.values())
def pauli_to_tpdm_ab(spatial_dim, pauli_to_coeff, transform=jordan_wigner):
"""
Populate the alpha-beta block of the 2-RDM
Given a dictionary of expected values of Pauli terms, populate the
alpha-beta block of the 2-RDM
:param Int spatial_dim: spatial basis set rank
:param pauli_to_coeff: a map between the Pauli term label to the expected
value.
:param transform: Openfermion fermion-to-qubit transform
:return: the 2-RDM alpha-beta block of the 2-RDM
"""
d2_ab = np.zeros((spatial_dim**2, spatial_dim**2), dtype=complex)
# build basis look up table
bas_ab = {}
cnt_ab = 0
# iterate over spatial orbital indices
for p, q in product(range(spatial_dim), repeat=2):
bas_ab[(p, q)] = cnt_ab
cnt_ab += 1
for p, q, r, s in product(range(spatial_dim), repeat=4):
spin_adapted_term = FermionOperator(
((2 * p, 1), (2 * q + 1, 1), (2 * r + 1, 0), (2 * s, 0)))
tpdm_element_as_pauli = remove_imaginary_terms(
qubitop_to_pyquilpauli(transform(spin_adapted_term)))
for term in tpdm_element_as_pauli:
pauli_tensor_list = sorted(list(term.operations_as_set()),
key=lambda x: x[0])
rev_order_pauli_tensor_list = list(
map(lambda x: (x[1], x[0]), pauli_tensor_list))
pauli_term = PauliTerm.from_list(rev_order_pauli_tensor_list,
coefficient=term.coefficient)
try:
d2_ab[bas_ab[(p, q)], bas_ab[(s, r)]] += pauli_to_coeff[
pauli_term.id()] * \
pauli_term.coefficient
except KeyError:
raise Warning("key was not in the coeff matrix.")
return d2_ab
def pauli_to_tpdm(dim, pauli_to_coeff, transform=jordan_wigner):
"""
Construct the 2-RDM from expected values of Pauli operators
Construct the 2-RDM by looping over the 2-RDM and loading up the
coefficients for the expected values of each transformed Pauli operator.
We assume the fermionic ladder operators are transformed via Jordan-Wigner.
This constraint can be relaxed later.
We don't check that the `pauli_expected` dictionary contains an
informationally complete set of expected values. This is useful for
testing if under sampling the 2-RDM is okay if a projection technique is
included in the calculation.
:param Int dim: spin-orbital basis dimension
:param Dict pauli_to_coeff: a map from pauli term ID's to
:param func transform: optional argument defining how to transform
fermionic operators into Pauli operators
"""
tpdm = np.zeros((dim, dim, dim, dim), dtype=complex)
for p, q, r, s in product(range(dim), repeat=4):
if p != q and r != s:
tpdm_element = FermionOperator(((p, 1), (q, 1), (r, 0), (s, 0)))
tpdm_element_as_pauli = remove_imaginary_terms(
qubitop_to_pyquilpauli(transform(tpdm_element)))
for term in tpdm_element_as_pauli:
pauli_tensor_list = sorted(list(term.operations_as_set()),
key=lambda x: x[0])
rev_order_pauli_tensor_list = list(
map(lambda x: (x[1], x[0]), pauli_tensor_list))
pauli_term = PauliTerm.from_list(rev_order_pauli_tensor_list,
coefficient=term.coefficient)
try:
tpdm[p, q, r, s] += pauli_to_coeff[pauli_term.id()] * \
pauli_term.coefficient
except KeyError:
raise Warning("key was not in the coeff matrix.")
return tpdm
def test_imaginary_removal():
"""
remove terms with imaginary coefficients from a pauli sum
"""
test_term = 0.25 * sX(1) * sZ(2) * sX(3) + 0.25j * sX(1) * sZ(2) * sY(3)
test_term += -0.25j * sY(1) * sZ(2) * sX(3) + 0.25 * sY(1) * sZ(2) * sY(3)
true_term = 0.25 * sX(1) * sZ(2) * sX(3) + 0.25 * sY(1) * sZ(2) * sY(3)
assert remove_imaginary_terms(test_term) == true_term
test_term = (0.25 + 1j) * sX(0) * sZ(2) + 1j * sZ(2)
# is_identity in pyquil apparently thinks zero is identity
assert remove_imaginary_terms(test_term) == 0.25 * sX(0) * sZ(2)
test_term = 0.25 * sX(0) * sZ(2) + 1j * sZ(2)
assert remove_imaginary_terms(test_term) == PauliSum([0.25 * sX(0) * sZ(2)])
with pytest.raises(TypeError):
remove_imaginary_terms(5)
with pytest.raises(TypeError):
remove_imaginary_terms(sX(0))
def pauli_terms_for_tpdm(dim, transform=jordan_wigner):
"""
Generate a set of pauli operators to measure to evaluate the 2-RDM
:param Int dim: Dimension of the spin-orbital basis.
:param transform: fermion-to-qubit transform from OpenFermion
:return: List of PauliTerms that corresponds to set of pauli terms to
measure to construct the 2-RDM.
"""
# first make a map between pauli terms and elements of the 2-RDM
pauli_to_rdm = {}
pauli_terms_to_measure = []
for p, q, r, s in product(range(dim), repeat=4):
if p != q and r != s:
tpdm_element = FermionOperator(((p, 1), (q, 1), (r, 0), (s, 0)))
tpdm_element_as_pauli = remove_imaginary_terms(
qubitop_to_pyquilpauli(transform(tpdm_element)))
for term in tpdm_element_as_pauli:
pauli_terms_to_measure.append(term)
for term in pauli_terms_to_measure:
# convert term into numerically order pauli tensor term
pauli_tensor_list = sorted(list(term.operations_as_set()),
key=lambda x: x[0])
rev_order_pauli_tensor_list = list(
map(lambda x: (x[1], x[0]), pauli_tensor_list))
pauli_term = PauliTerm.from_list(rev_order_pauli_tensor_list,
coefficient=term.coefficient)
if pauli_term.id() not in pauli_to_rdm.keys():
pauli_to_rdm[pauli_term.id()] = pauli_term
else:
if (abs(pauli_to_rdm[pauli_term.id()].coefficient) < abs(
pauli_term.coefficient)):
pauli_to_rdm[pauli_term.id()] = pauli_term
return list(pauli_to_rdm.values())