def clifford_bottoms(c_top):
r"""
This function yields the next set of nq paulis that mutually
commute, and anti-commute with selected elements from c_top. The intent
behind this function is to begin with a list of mutually commuting Paulis,
and ifnd an associated set that share the same commutation relations as
:math:`\left[ X_1,\ldots,X_n \range]` and
:math:`\left[ Z_1,\ldots,Z_n \range]`. This allows the input and output of
this function can be thought of as the :math:`X` and :math:`Z` outputs of a
Clifford operator.
:param c_top: list of :class:`qecc.Pauli` objects, which mutually commute.
"""
nq = len(c_top)
possible_zs = []
for jj in range(nq):
applicable_centralizer_gens = PauliList(
*(c_top[:jj] + c_top[jj + 1:])).centralizer_gens()
possible_zs.append(
filter(lambda a: com(a, c_top[jj]) == 1,
from_generators(applicable_centralizer_gens)))
for possible_set in product(*possible_zs):
if all(
imap(
lambda twolist: pred.commutes_with(twolist[0])(twolist[1]),
combinations(possible_set, 2))):
yield possible_set
def ns_mod_s(*stab_gens):
r"""
Given the generators of a stabilizer group :math:`S`, returns an iterator
that yields all elements of the set :math:`\text{N}(S)\\S`.
:param qecc.Pauli stab_gens: Generators of :math:`S`.
:returns: An iterator such that ``list(ns_mod_s(*stab_gens))`` contains
each element of :math:`\text{N}(S)\\S`.
"""
nq = len(stab_gens[0])
return ifilter(
pred.commutes_with(*stab_gens)
& ~pred.in_group_generated_by(*stab_gens), pauli_group(nq))
def ns_mod_s(*stab_gens):
r"""
Given the generators of a stabilizer group :math:`S`, returns an iterator
that yields all elements of the set :math:`\text{N}(S)\\S`.
:param qecc.Pauli stab_gens: Generators of :math:`S`.
:returns: An iterator such that ``list(ns_mod_s(*stab_gens))`` contains
each element of :math:`\text{N}(S)\\S`.
"""
nq = len(stab_gens[0])
return ifilter(
pred.commutes_with(*stab_gens) & ~pred.in_group_generated_by(*stab_gens),
pauli_group(nq)
)
def clifford_bottoms(c_top):
r"""
This function yields the next set of nq paulis that mutually
commute, and anti-commute with selected elements from c_top. The intent
behind this function is to begin with a list of mutually commuting Paulis,
and ifnd an associated set that share the same commutation relations as
:math:`\left[ X_1,\ldots,X_n \range]` and
:math:`\left[ Z_1,\ldots,Z_n \range]`. This allows the input and output of
this function can be thought of as the :math:`X` and :math:`Z` outputs of a
Clifford operator.
:param c_top: list of :class:`qecc.Pauli` objects, which mutually commute.
"""
nq=len(c_top)
possible_zs=[]
for jj in range(nq):
applicable_centralizer_gens=PauliList(*(c_top[:jj]+c_top[jj+1:])).centralizer_gens()
possible_zs.append(filter(lambda a:com(a,c_top[jj])==1,from_generators(applicable_centralizer_gens)))
for possible_set in product(*possible_zs):
if all(imap(lambda twolist: pred.commutes_with(twolist[0])(twolist[1]),combinations(possible_set,2))):
yield possible_set
