def test_no_intersection_between_first_and_other_terms_is_trivial(self):
self.assertTrue(
trivially_double_commutes_dual_basis_using_term_info(
indices_alpha=set([3, 2]),
indices_beta=set([1, 4]),
indices_alpha_prime=set([4, 5]),
is_hopping_operator_alpha=False,
is_hopping_operator_beta=True,
is_hopping_operator_alpha_prime=False,
jellium_only=True))
def test_alpha_is_hopping_operator_others_number_trivial_commutation(self):
self.assertTrue(
trivially_double_commutes_dual_basis_using_term_info(
indices_alpha=set([1, 2]),
indices_beta=set([3, 4]),
indices_alpha_prime=set([2, 3]),
is_hopping_operator_alpha=True,
is_hopping_operator_beta=False,
is_hopping_operator_alpha_prime=False,
jellium_only=True))
def test_single_intersection_in_first_commutator_nontrivial(self):
self.assertFalse(
trivially_double_commutes_dual_basis_using_term_info(
indices_alpha=set([3, 2]),
indices_beta=set([3, 4]),
indices_alpha_prime=set([4, 5]),
is_hopping_operator_alpha=False,
is_hopping_operator_beta=True,
is_hopping_operator_alpha_prime=False,
jellium_only=True))
def low_depth_second_order_trotter_error_operator(terms,
indices=None,
is_hopping_operator=None,
jellium_only=False,
verbose=False):
"""Determine the difference between the exact generator of unitary
evolution and the approximate generator given by the second-order
Trotter-Suzuki expansion.
Args:
terms: a list of FermionOperators in the Hamiltonian in the
order in which they will be simulated.
indices: a set of indices the terms act on in the same order as terms.
is_hopping_operator: a list of whether each term is a hopping operator.
jellium_only: Whether the terms are from the jellium Hamiltonian only,
rather than the full dual basis Hamiltonian (i.e. whether
c_i = c for all number operators i^ i, or whether they
depend on i as is possible in the general case).
verbose: Whether to print percentage progress.
Returns:
The difference between the true and effective generators of time
evolution for a single Trotter step.
Notes: follows Equation 9 of Poulin et al.'s work in "The Trotter Step
Size Required for Accurate Quantum Simulation of Quantum Chemistry",
applied to the "stagger"-based Trotter step for detailed in
Kivlichan et al., "Quantum Simulation of Electronic Structure with
Linear Depth and Connectivity", arxiv:1711.04789.
"""
more_info = bool(indices)
n_terms = len(terms)
if verbose:
import time
start = time.time()
error_operator = FermionOperator.zero()
for beta in range(n_terms):
if verbose and beta % (n_terms // 30) == 0:
print('%4.3f percent done in' % ((float(beta) / n_terms)**3 * 100),
time.time() - start)
for alpha in range(beta + 1):
for alpha_prime in range(beta):
# If we have pre-computed info on indices, use it to determine
# trivial double commutation.
if more_info:
if (not trivially_double_commutes_dual_basis_using_term_info( # pylint: disable=C
indices[alpha], indices[beta],
indices[alpha_prime], is_hopping_operator[alpha],
is_hopping_operator[beta],
is_hopping_operator[alpha_prime], jellium_only)):
# Determine the result of the double commutator.
double_com = double_commutator(
terms[alpha], terms[beta], terms[alpha_prime],
indices[beta], indices[alpha_prime],
is_hopping_operator[beta],
is_hopping_operator[alpha_prime])
if alpha == beta:
double_com /= 2.0
error_operator += double_com
# If we don't have more info, check for trivial double
# commutation using the terms directly.
elif not trivially_double_commutes_dual_basis(
terms[alpha], terms[beta], terms[alpha_prime]):
double_com = double_commutator(terms[alpha], terms[beta],
terms[alpha_prime])
if alpha == beta:
double_com /= 2.0
error_operator += double_com
error_operator /= 12.0
return error_operator