def expression2CG(io_config, expression):
"""
Inputs:
io_config - input/output configuration
expression - matrix representing the coefficients s_ab|xy of the Bell expression.
Eg: for io_config = [[2,2],[2,2]],
| s00|00 s01|00 s00|01 s01|01 |
expression = | s10|00 s11|00 s10|01 s11|01 |
| s00|10 s01|10 s00|11 s01|11 |
| s10|10 s11|10 s10|11 s11|11 |
Returns:
CG-reduction of the expression matrix.
"""
#Adding one to each output results in all the operators being defined
A = ncp.generate_measurements([int(a + 1) for a in io_config[0]], 'A')
B = ncp.generate_measurements([int(b + 1) for b in io_config[1]], 'B')
op_mat = np.outer(np.array(ncp.flatten(A)), np.array(ncp.flatten(B)))
#Obtain the overall expression
expr = np.sum(expression * op_mat)
#Removing the terms we don't want
for i in range(len(A)):
expr = expr.subs(A[i][-1], 1 - np.sum(A[i][:-1]))
for j in range(len(B)):
expr = expr.subs(B[j][-1], 1 - np.sum(B[j][:-1]))
# Now we get our full expression
expr = expr.expand()
# Removing the additional operator
for i in range(len(A)):
A[i] = A[i][:-1]
for i in range(len(B)):
B[i] = B[i][:-1]
A = np.array([1] + ncp.flatten(A))
B = np.array([1] + ncp.flatten(B))
# Construct the operator cg-matrix and then find relevant coefficients
op_mat = np.outer(A, B)
coefs = expr.as_coefficients_dict()
for i in range(len(A)):
for j in range(len(B)):
if op_mat[i][j] in coefs:
op_mat[i][j] = coefs[op_mat[i, j]]
else:
op_mat[i][j] = 0
return op_mat
def test_ground_state(self):
length, n, h, U, t = 2, 0.8, 3.8, -6, 1
fu = generate_operators('fu', length)
fd = generate_operators('fd', length)
_b = flatten([fu, fd])
monomials = [[ci for ci in _b]]
monomials[-1].extend([Dagger(ci) for ci in _b])
monomials.append([cj*ci for ci in _b for cj in _b])
monomials.append([Dagger(cj)*ci for ci in _b for cj in _b])
monomials[-1].extend([cj*Dagger(ci)
for ci in _b for cj in _b])
monomials.append([Dagger(cj)*Dagger(ci)
for ci in _b for cj in _b])
hamiltonian = 0
for j in range(length):
hamiltonian += U * (Dagger(fu[j])*Dagger(fd[j]) * fd[j]*fu[j])
hamiltonian += -h/2*(Dagger(fu[j])*fu[j] - Dagger(fd[j])*fd[j])
for k in get_neighbors(j, len(fu), width=1):
hamiltonian += -t*Dagger(fu[j])*fu[k]-t*Dagger(fu[k])*fu[j]
hamiltonian += -t*Dagger(fd[j])*fd[k]-t*Dagger(fd[k])*fd[j]
momentequalities = [n-sum(Dagger(br)*br for br in _b)]
sdpRelaxation = SdpRelaxation(_b, verbose=0)
sdpRelaxation.get_relaxation(-1,
objective=hamiltonian,
momentequalities=momentequalities,
substitutions=fermionic_constraints(_b),
extramonomials=monomials)
sdpRelaxation.solve()
s = 0.5*(sum((Dagger(u)*u) for u in fu) -
sum((Dagger(d)*d) for d in fd))
magnetization = sdpRelaxation[s]
self.assertTrue(abs(magnetization-0.021325317328560453) < 10e-5)
def test_apply_substitutions(self):
def apply_correct_substitutions(monomial, substitutions):
if isinstance(monomial, int) or isinstance(monomial, float):
return monomial
original_monomial = monomial
changed = True
while changed:
for lhs, rhs in substitutions.items():
monomial = monomial.subs(lhs, rhs)
if original_monomial == monomial:
changed = False
original_monomial = monomial
return monomial
length, h, U, t = 2, 3.8, -6, 1
fu = generate_operators('fu', length)
fd = generate_operators('fd', length)
_b = flatten([fu, fd])
hamiltonian = 0
for j in range(length):
hamiltonian += U * (Dagger(fu[j])*Dagger(fd[j]) * fd[j]*fu[j])
hamiltonian += -h/2*(Dagger(fu[j])*fu[j] - Dagger(fd[j])*fd[j])
for k in get_neighbors(j, len(fu), width=1):
hamiltonian += -t*Dagger(fu[j])*fu[k]-t*Dagger(fu[k])*fu[j]
hamiltonian += -t*Dagger(fd[j])*fd[k]-t*Dagger(fd[k])*fd[j]
substitutions = fermionic_constraints(_b)
monomials = expand(hamiltonian).as_coeff_mul()[1][0].as_coeff_add()[1]
substituted_hamiltonian = sum([apply_substitutions(monomial,
substitutions)
for monomial in monomials])
correct_hamiltonian = sum([apply_correct_substitutions(monomial,
substitutions)
for monomial in monomials])
self.assertTrue(substituted_hamiltonian == expand(correct_hamiltonian))
def test_ground_state(self):
length, n, h, U, t = 2, 0.8, 3.8, -6, 1
fu = generate_operators('fu', length)
fd = generate_operators('fd', length)
_b = flatten([fu, fd])
monomials = [[ci for ci in _b]]
monomials[-1].extend([Dagger(ci) for ci in _b])
monomials.append([cj * ci for ci in _b for cj in _b])
monomials.append([Dagger(cj) * ci for ci in _b for cj in _b])
monomials[-1].extend([cj * Dagger(ci) for ci in _b for cj in _b])
monomials.append([Dagger(cj) * Dagger(ci) for ci in _b for cj in _b])
hamiltonian = 0
for j in range(length):
hamiltonian += U * (Dagger(fu[j]) * Dagger(fd[j]) * fd[j] * fu[j])
hamiltonian += -h / 2 * (Dagger(fu[j]) * fu[j] -
Dagger(fd[j]) * fd[j])
for k in get_neighbors(j, len(fu), width=1):
hamiltonian += -t * Dagger(fu[j]) * fu[k] - t * Dagger(
fu[k]) * fu[j]
hamiltonian += -t * Dagger(fd[j]) * fd[k] - t * Dagger(
fd[k]) * fd[j]
momentequalities = [n - sum(Dagger(br) * br for br in _b)]
sdpRelaxation = SdpRelaxation(_b, verbose=0)
sdpRelaxation.get_relaxation(-1,
objective=hamiltonian,
momentequalities=momentequalities,
substitutions=fermionic_constraints(_b),
extramonomials=monomials)
sdpRelaxation.solve()
s = 0.5 * (sum((Dagger(u) * u) for u in fu) - sum(
(Dagger(d) * d) for d in fd))
magnetization = sdpRelaxation[s]
self.assertTrue(abs(magnetization - 0.021325317328560453) < 10e-5)
def test_violation(self):
I = [[0, -1, 0], [-1, 1, 1], [0, 1, -1]]
P = Probability([2, 2], [2, 2])
objective = define_objective_with_I(I, P)
sdpRelaxation = MoroderHierarchy([flatten(P.parties[0]), flatten(P.parties[1])], verbose=0, normalized=False)
sdpRelaxation.get_relaxation(1, objective=objective, substitutions=P.substitutions)
Problem = convert_to_picos(sdpRelaxation, duplicate_moment_matrix=True)
X = Problem.get_variable("X")
Y = Problem.get_variable("Y")
Z = Problem.add_variable("Z", (sdpRelaxation.block_struct[0], sdpRelaxation.block_struct[0]))
Problem.add_constraint(Y.partial_transpose() >> 0)
Problem.add_constraint(Z.partial_transpose() >> 0)
Problem.add_constraint(X - Y + Z == 0)
Problem.add_constraint(Z[0, 0] == 1)
solution = Problem.solve(verbose=0)
self.assertTrue(abs(solution["obj"] - 0.139) < 10e-3)
def test_apply_substitutions(self):
def apply_correct_substitutions(monomial, substitutions):
if isinstance(monomial, int) or isinstance(monomial, float):
return monomial
original_monomial = monomial
changed = True
while changed:
for lhs, rhs in substitutions.items():
monomial = monomial.subs(lhs, rhs)
if original_monomial == monomial:
changed = False
original_monomial = monomial
return monomial
length, h, U, t = 2, 3.8, -6, 1
fu = generate_operators('fu', length)
fd = generate_operators('fd', length)
_b = flatten([fu, fd])
hamiltonian = 0
for j in range(length):
hamiltonian += U * (Dagger(fu[j])*Dagger(fd[j]) * fd[j]*fu[j])
hamiltonian += -h/2*(Dagger(fu[j])*fu[j] - Dagger(fd[j])*fd[j])
for k in get_neighbors(j, len(fu), width=1):
hamiltonian += -t*Dagger(fu[j])*fu[k]-t*Dagger(fu[k])*fu[j]
hamiltonian += -t*Dagger(fd[j])*fd[k]-t*Dagger(fd[k])*fd[j]
substitutions = fermionic_constraints(_b)
monomials = expand(hamiltonian).as_coeff_mul()[1][0].as_coeff_add()[1]
substituted_hamiltonian = sum([apply_substitutions(monomial,
substitutions)
for monomial in monomials])
correct_hamiltonian = sum([apply_correct_substitutions(monomial,
substitutions)
for monomial in monomials])
self.assertTrue(substituted_hamiltonian == expand(correct_hamiltonian))
def __init__(self, lattice_length, lattice_width, solver, outputDir,
periodic=0, window_length=0, removeequalities=False, parallel=True):
SecondQuantizedModel.__init__(self, lattice_length, lattice_width,
solver, outputDir, periodic,
window_length, removeequalities, parallel=parallel)
self._fu = generate_variables(
'fu', lattice_length * lattice_width, commutative=False)
self._fd = generate_variables(
'fd', lattice_length * lattice_width, commutative=False)
self._b = flatten([self._fu, self._fd])
self.mu, self.t, self.h, self.U = 0, 0, 0, 0
def test_violation(self):
I = [[0, -1, 0],
[-1, 1, 1],
[0, 1, -1]]
P = Probability([2, 2], [2, 2])
objective = define_objective_with_I(I, P)
sdpRelaxation = MoroderHierarchy([flatten(P.parties[0]),
flatten(P.parties[1])],
verbose=0, normalized=False)
sdpRelaxation.get_relaxation(1, objective=objective,
substitutions=P.substitutions)
Problem = sdpRelaxation.convert_to_picos(duplicate_moment_matrix=True)
X = Problem.get_variable('X')
Y = Problem.get_variable('Y')
Z = Problem.add_variable('Z', (sdpRelaxation.block_struct[0],
sdpRelaxation.block_struct[0]))
Problem.add_constraint(Y.partial_transpose() >> 0)
Problem.add_constraint(Z.partial_transpose() >> 0)
Problem.add_constraint(X - Y + Z == 0)
Problem.add_constraint(Z[0, 0] == 1)
solution = Problem.solve(verbose=0)
self.assertTrue(abs(solution["obj"] - 0.1236) < 10e-3)
def optimiseQubitGP(device, starting_angles, eta=1.0, vis=1.0, tol=1e-6):
"""
Iteratively optimise the angle choices by trying to minimise the extracted dual functional
"""
a_in, b_in = device.num_inputs
bounds = [[0, pi / 2]] + [[-pi, pi] for k in range(a_in + b_in)]
old_ang, new_ang = starting_angles[:], starting_angles[:]
cg_shift = device._cgshift
old_gp = 2.0
new_gp = 1.0
# Start iteratively optimising
while new_gp < old_gp - tol:
old_ang = new_ang[:]
old_gp = new_gp
# Get the current dual solution
av_curr, lv_curr, _, _, status = device.dualSolution(
angles2Score(device, old_ang, eta, vis))
# If this is not a bad solve then proceed with an optimisation.
if status == 'optimal':
# function to optimise
def f0(x):
ang = [x[0], x[1:a_in + 1], x[1 + a_in:]]
scr = angles2Score(device, ang, eta, vis)
return av_curr + np.dot(lv_curr, scr - cg_shift)
# Find minimising angles
res = minimize(f0, ncp.flatten(old_ang), bounds=bounds)
# record results
new_gp = res.fun
new_ang = [
res.x[0], res.x[1:a_in + 1].tolist(),
res.x[a_in + 1:].tolist()
]
else:
break
return old_gp, old_ang
def _relax(self):
"""
Creates the sdp relaxation object from ncpol2sdpa.
"""
if self.solver == None:
self.solver = self.DEFAULT_SOLVER_PATH
self._eq_cons = [] # equality constraints
self._proj_cons = {} # projective constraints
self._A_ops = [] # Alice's operators
self._B_ops = [] # Bob's operators
self._obj = 0 # Objective function
self._obj_const = '' # Extra objective normalisation constant
self._sdp = None # SDP object
# Creating the operator constraints
nrm = ''
# Need as many decompositions as there are generating outcomes
for k in range(np.prod(self.generation_output_size)):
self._A_ops += [
ncp.generate_measurements(self.io_config[0],
'A' + str(k) + '_')
]
self._B_ops += [
ncp.generate_measurements(self.io_config[1],
'B' + str(k) + '_')
]
self._proj_cons.update(
ncp.projective_measurement_constraints(self._A_ops[k],
self._B_ops[k]))
#Also building a normalisation string for next step
nrm += '+' + str(k) + '[0,0]'
# Adding the constraints
# Normalisation constraint
self._eq_cons.append(nrm + '-1')
self._base_constraint_expressions = []
# Create the game expressions
for game in self.games:
tmp_expr = 0
for k in range(np.prod(self.generation_output_size)):
tmp_expr += -ncp.define_objective_with_I(
game._cgmatrix, self._A_ops[k], self._B_ops[k])
self._base_constraint_expressions.append(tmp_expr)
# Specify the scores for these expressions including any shifts
for i, game in enumerate(self.games):
#We must account for overshifting in the score coming from the decomposition
self._eq_cons.append(self._base_constraint_expressions[i] -
game.score - game._cgshift *
(np.prod(self.generation_output_size) - 1))
self._obj, self._obj_const = guessingProbabilityObjectiveFunction(
self.io_config, self.generation_inputs, self._A_ops, self._B_ops)
# Initialising SDP
ops = [
ncp.flatten([self._A_ops[0], self._B_ops[0]]),
ncp.flatten([self._A_ops[1], self._B_ops[1]]),
ncp.flatten([self._A_ops[2], self._B_ops[2]]),
ncp.flatten([self._A_ops[3], self._B_ops[3]])
]
self._sdp = ncp.SdpRelaxation(ops,
verbose=self.verbose,
normalized=False)
self._sdp.get_relaxation(level=self._relaxation_level,
momentequalities=self._eq_cons,
objective=self._obj,
substitutions=self._proj_cons,
extraobjexpr=self._obj_const)
# We include some extra monomials in the relaxation to boost rates
extra_monos = []
for v in V1 + V2:
for Ax in A:
for a in Ax:
for By in B:
for b in By:
extra_monos += [a * b * v]
extra_monos += [a * b * Dagger(v)]
# Objective function
obj = A[0][0] * (V1[0] + Dagger(V1[0])) / 2.0 + A[0][1] * (V1[1] +
Dagger(V1[1])) / 2.0
ops = ncp.flatten([A, B, V1, V2])
sdp = ncp.SdpRelaxation(ops, verbose=1, normalized=True, parallel=0)
sdp.get_relaxation(level=LEVEL,
equalities=operator_equalities,
inequalities=operator_inequalities,
momentequalities=moment_equalities,
momentinequalities=moment_inequalities,
objective=-obj,
substitutions=substitutions,
extramonomials=extra_monos)
sdp.solve('mosek')
print(
f"For detection efficiency {test_eta} the system {test_sys} achieves a DI-QKD rate of {rate(sdp,test_sys,test_eta)}"
)
substitutions.update({v * b: b * v})
substitutions.update({Dagger(v) * b: b * Dagger(v)})
# We can also impose the relations coming from the dilation theorem
for i in range(len(V)):
substitutions.update({V[i] * Dagger(V[i]): 1})
for j in range(len(V)):
if i != j:
substitutions.update({V[i] * Dagger(V[j]): 0})
# V* V <= 1
operator_ineqs += [1 - (Dagger(V[0])*V[0] + Dagger(V[1])*V[1] + \
Dagger(V[2])*V[2] + Dagger(V[3])*V[3])]
# We build a localizing set of monomials for the constraint V* V <= 1
# This set includes the monomials AB
localizing_set = ncp.nc_utils.get_all_monomials(ncp.flatten([A, B, V]),
extramonomials=None,
substitutions=substitutions,
degree=1)
localizing_set += AB
# We add this set to the localizing_monomials
localizing_monos += [localizing_set]
# We must also specify localizing mmonomials for the other constraints of the
# problem but by specifying None ncpol2sdpa uses a default set
localizing_monos += [None, None, None]
moment_equalities = moment_eqs[:]
moment_inequalities = moment_ineqs[:] + score_con[:]
operator_equalities = operator_eqs[:]
operator_inequalities = operator_ineqs[:]
This example calculates the maximum quantum violation of the CHSH inequality in
the probability picture with a mixed-level relaxation of 1+AB.
Created on Mon Dec 1 14:19:08 2014
@author: Peter Wittek
"""
from ncpol2sdpa import generate_measurements, \
projective_measurement_constraints, flatten, \
SdpRelaxation, define_objective_with_I, solve_sdp
level = 1
A_configuration = [2, 2]
B_configuration = [2, 2]
I = [[ 0, -1, 0 ],
[-1, 1, 1 ],
[ 0, 1, -1 ]]
A = generate_measurements(A_configuration, 'A')
B = generate_measurements(B_configuration, 'B')
monomial_substitutions = projective_measurement_constraints(
A, B)
objective = define_objective_with_I(I, A, B)
AB = [Ai*Bj for Ai in flatten(A) for Bj in flatten(B)]
sdpRelaxation = SdpRelaxation(flatten([A, B]))
sdpRelaxation.get_relaxation(level, objective=objective,
substitutions=monomial_substitutions,
extramonomials=AB)
print(solve_sdp(sdpRelaxation))
def get_factorization_constraints(parties,
moments,
substitutions,
level,
all_parties=False,
return_column_names=False):
'''Creates all the constraints in moments related to n-locality, namely
the factorizations <P_i P_j...> = <P_i><P_j>..., where P_i, P_j, ... are
collections of operators of spacelike-separated parties i, j...
:param parties: list of measurements for each of the parties spacelike
separated. The structure is [party1, party2...], where
party_i are the measurements of party i, arranged in
the structure [meas1, meas2...], where meas_i is a
list containing the symbols of the operators of
measurement i.
:type parties: list of lists of lists of sympy.core.Symbol
:param moments: known moments obtained with get_moment_constraints.
:type moments: dict
:param substitutions: Measurement constraints (commuting, projective...)
for the operators involved. Required by
get_all_monomials.
:type substitutions: dict
:param level: level of the moment matrix
:type level: int
:param all_parties: Optional flag for specifying whether the columns for all
parties should be created instead of a minimal set.
:type all_parties: bool
:param return_column_names: Optional flag to return the moments to which
the extra columns correspond
:type return_column_names: bool
:returns momentsbilocal: dict with <P_i P_j...> = <P_i><P_j>... constraints.
:returns extracolumns: list of sympy.core.Symbol that correspond to the
extra monomials created.
:returns columnsnames: list of original monomials corresponding to the
names of the extra monomials created.
'''
# Factorizations will at most be of size 1|2*level-1
parties_moments = [
get_all_monomials(flatten(party), None, substitutions,
2 * level - 1)[1:] for party in parties
]
# Strictly speaking, we need to build extra columns of all except one party
# to sucessfully implement all factorization constraints.
if not all_parties:
parties_moments = parties_moments[:-1]
# Generate commuting symbols for all the additional columns
parties_columns = [
generate_variables(
str(party_moments[0])[0].split('_')[0].lower() + '_',
len(party_moments)) for party_moments in parties_moments
]
# Substitute symbols whose values are known
parties_columns = [[
moments.get(parties_moments[i][parties_columns[i].index(element)],
element) for element in party
] for i, party in enumerate(parties_columns)]
# Obtain actual additional columns
extracolumns = [
extra for extra in flatten(parties_columns)
if isinstance(extra, Symbol)
]
columnsnames = [
symbol for i, symbol in enumerate(flatten(parties_moments))
if flatten(parties_columns)[i] in extracolumns
]
# Add identities for later computations
partiesplus1 = [
party_moments + [S.One] for party_moments in parties_moments
]
extramomentsplus1 = [extracols + [1] for extracols in parties_columns]
momentsbilocal = {}
for monomial in get_all_monomials(flatten(parties), None, substitutions,
2 * level)[1:]:
monomial = monomial.as_ordered_factors()
party_monomials = [[
element for element in monomial if element in flatten(party)
] for party in parties]
factors = [prod(party) for party in party_monomials]
if len(monomial) <= level:
for z in flatten([extracolumns, 1]):
key = prod([factor for factor in [z] + factors if factor != 1])
if all_parties:
item = prod([z] + [
extramomentsplus1[i][partiesplus1[i].index(factors[i])]
for i in range(len(factors))
])
else:
item = prod([z] + [
extramomentsplus1[i][partiesplus1[i].index(factors[i])]
for i in range(len(factors) - 1)
] + [moments.get(factors[-1], factors[-1])])
if key != item:
momentsbilocal[key] = item
else:
if prod([
len(party_monomial) < 2 * level
for party_monomial in party_monomials
]):
key = prod([factor for factor in factors if factor != 1])
if all_parties:
item = prod([
extramomentsplus1[i][partiesplus1[i].index(factors[i])]
for i in range(len(factors))
])
else:
item = prod([
extramomentsplus1[i][partiesplus1[i].index(factors[i])]
for i in range(len(factors) - 1)
] + [moments.get(factors[-1], factors[-1])])
if key != item:
momentsbilocal[key] = item
if return_column_names:
return momentsbilocal, extracolumns, columnsnames
return momentsbilocal, extracolumns
