# Source code for qecc.constraint_solvers

```
#!/usr/bin/python
# -*- coding: utf-8 -*-
##
# constraint_solvers.py: Solvers for various pc.Pauli and Clifford constraint
# problems.
##
# © 2012 Christopher E. Granade (cgranade@gmail.pc.com) and
# Ben Criger (bcriger@gmail.pc.com).
# This file is a part of the QuaEC project.
# Licensed under the AGPL version 3.
##
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU Affero General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Affero General Public License for more details.
#
# You should have received a copy of the GNU Affero General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
##
## ALL ##
__all__ = [
'solve_commutation_constraints'
]
## IMPORTS ##
from sys import version_info
if version_info[0] == 3:
PY3 = True
from importlib import reload
elif version_info[0] == 2:
PY3 = False
else:
raise EnvironmentError("sys.version_info refers to a version of "
"Python neither 2 nor 3. This is not permitted. "
"sys.version_info = {}".format(version_info))
if PY3:
from . import PauliClass as pc
from .paulicollections import PauliList
from .pred import AllPredicate
else:
import PauliClass as pc
from paulicollections import PauliList
from pred import AllPredicate
## FUNCTIONS ##
[docs]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(*[(lambda P: pc.com(P, acc) == 0) for acc in commutation_constraints])
commuters = list(filter(commutation_predicate, search_in_set))
anticommutation_predicate = AllPredicate(*[(lambda P: pc.com(P, acc) == 1) for acc in anticommutation_constraints])
return list(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(*[(lambda P: pc.com(P, acc) == 1) for acc in anticommutation_constraints])
assert len(search_in_gens) > 0
return filter(anticommutation_predicate, pc.from_generators(search_in_gens))
```