def in_group_generated_by(*paulis):
"""
Returns a predicate that selects Pauli operators in the group generated by
a given list of generators.
"""
# Warning: This is inefficient for large groups!
paulis = list(map(pc.ensure_pauli, paulis))
return PauliMembershipPredicate(pc.from_generators(paulis), ignore_phase=True)
def in_group_generated_by(*paulis):
"""
Returns a predicate that selects Pauli operators in the group generated by
a given list of generators.
"""
# Warning: This is inefficient for large groups!
paulis = list(map(pc.ensure_pauli, paulis))
return PauliMembershipPredicate(pc.from_generators(paulis),
ignore_phase=True)
def logical_pauli_group(self, incl_identity=True):
r"""
Iterator onto the group :math:`\text{N}(S) / S`, where :math:`S` is
the stabilizer group describing this code. Each member of the group
is specified by a coset representative drawn from the respective
elements of :math:`\text{N}(S) / S`. These representatives are
chosen to be the logical :math:`X` and :math:`Z` operators specified
as properties of this instance.
:param bool incl_identity: If ``False``, the identity coset :math:`S`
is excluded from this iterator.
:yields: A representative for each element of :math:`\text{N}(S) / S`.
"""
return p.from_generators(self.logical_xs + self.logical_zs, incl_identity=incl_identity)
def logical_pauli_group(self, incl_identity=True):
r"""
Iterator onto the group :math:`\text{N}(S) / S`, where :math:`S` is
the stabilizer group describing this code. Each member of the group
is specified by a coset representative drawn from the respective
elements of :math:`\text{N}(S) / S`. These representatives are
chosen to be the logical :math:`X` and :math:`Z` operators specified
as properties of this instance.
:param bool incl_identity: If ``False``, the identity coset :math:`S`
is excluded from this iterator.
:yields: A representative for each element of :math:`\text{N}(S) / S`.
"""
return p.from_generators(self.logical_xs + self.logical_zs,
incl_identity=incl_identity)
def solve_commutation_constraints(
commutation_constraints=[],
anticommutation_constraints=[],
search_in_gens=None,
search_in_set=None
):
r"""
Given commutation constraints on a Pauli operator, yields an iterator onto
all solutions of those constraints.
:param commutation_constraints: A list of operators :math:`\{A_i\}` such
that each solution :math:`P` yielded by this function must satisfy
:math:`[A_i, P] = 0` for all :math:`i`.
:param anticommutation_constraints: A list of operators :math:`\{B_i\}` such
that each solution :math:`P` yielded by this function must satisfy
:math:`\{B_i, P\} = 0` for all :math:`i`.
:param search_in_gens: A list of operators :math:`\{N_i\}` that generate
the group in which to search for solutions. If ``None``, defaults to
the elementary generators of the pc.Pauli group on :math:`n` qubits, where
:math:`n` is given by the length of the commutation and anticommutation
constraints.
:param search_in_set: An iterable of operators to which the search for
satisfying assignments is restricted. This differs from ``search_in_gens``
in that it specifies the entire set, not a generating set. When this
parameter is specified, a brute-force search is executed. Use only
when the search set is small, and cannot be expressed using its generating
set.
:returns: An iterator ``it`` such that ``list(it)`` contains all operators
within the group :math:`G = \langle N_1, \dots, N_k \rangle`
given by ``search_in_gens``, consistent with the commutation and
anticommutation constraints.
This function is based on finding the generators of the centralizer groups
of each commutation constraint, and is thus faster than a predicate-based
search over the entire group of interest. The resulting iterator can be
used in conjunction with other filters, however.
>>> import qecc as q
>>> list(q.solve_commutation_constraints(q.PauliList('XXI', 'IZZ', 'IYI'), q.PauliList('YIY')))
[i^0 XII, i^0 IIZ, i^0 YYX, i^0 ZYY]
>>> from itertools import ifilter
>>> list(ifilter(lambda P: P.wt <= 2, q.solve_commutation_constraints(q.PauliList('XXI', 'IZZ', 'IYI'), q.PauliList('YIY'))))
[i^0 XII, i^0 IIZ]
"""
# Normalize our arguments to be PauliLists, so that we can obtain
# centralizers easily.
if not isinstance(commutation_constraints, PauliList):
commutation_constraints = PauliList(commutation_constraints)
if not isinstance(anticommutation_constraints, PauliList):
# This is probably not necessary, strictly speaking, but it keeps me
# slightly more sane to have both constraints represented by the same
# sequence type.
anticommutation_constraints = PauliList(anticommutation_constraints)
# Then check that the arguments make sense.
if len(commutation_constraints) == 0 and len(anticommutation_constraints) == 0:
raise ValueError("At least one constraint must be specified.")
#We default to executing a brute-force search if the search set is
#explicitly specified:
if search_in_set is not None:
commutation_predicate = AllPredicate(*map(
lambda acc: (lambda P: pc.com(P, acc) == 0),
commutation_constraints
))
commuters = filter(commutation_predicate, search_in_set)
anticommutation_predicate = AllPredicate(*map(
lambda acc: (lambda P: pc.com(P, acc) == 1),
anticommutation_constraints
))
return filter(anticommutation_predicate, commuters)
# We finish putting arguments in the right form by defaulting to searching
# over the pc.Pauli group on $n$ qubits.
if search_in_gens is None:
nq = len(commutation_constraints[0] if len(commutation_constraints) > 0 else anticommutation_constraints[0])
Xs, Zs = pc.elem_gens(nq)
search_in_gens = Xs + Zs
# Now we update our search by restricting to the centralizer of the
# commutation constraints.
search_in_gens = commutation_constraints.centralizer_gens(group_gens=search_in_gens)
# Finally, we return a filter iterator on the elements of the given
# centralizer that selects elements which anticommute appropriately.
anticommutation_predicate = AllPredicate(*map(
lambda acc: (lambda P: pc.com(P, acc) == 1),
anticommutation_constraints
))
assert len(search_in_gens) > 0
return ifilter(anticommutation_predicate, pc.from_generators(search_in_gens))
def generated_group(self, coset_rep=None):
"""
Yields an iterator onto the group generated by this list of Pauli
operators. See also :obj:`qecc.from_generators`.
"""
return pc.from_generators(self, coset_rep)
def generated_group(self, coset_rep=None):
"""
Yields an iterator onto the group generated by this list of Pauli
operators. See also :obj:`qecc.from_generators`.
"""
return pc.from_generators(self, coset_rep)
