class ExampleNoncommutative(unittest.TestCase):
def setUp(self):
X = generate_operators('x', 2, hermitian=True)
self.sdpRelaxation = SdpRelaxation(X)
self.sdpRelaxation.get_relaxation(2, objective=X[0]*X[1] + X[1]*X[0],
inequalities=[-X[1]**2 + X[1] + 0.5],
substitutions={X[0]**2: X[0]})
def tearDown(self):
clear_cache()
def test_solving_with_sdpa(self):
self.sdpRelaxation.solve(solver="sdpa")
self.assertTrue(abs(self.sdpRelaxation.primal + 0.75) < 10e-5)
def test_solving_with_mosek(self):
self.sdpRelaxation.solve(solver="mosek")
self.assertTrue(abs(self.sdpRelaxation.primal + 0.75) < 10e-5)
def test_solving_with_cvxopt(self):
self.sdpRelaxation.solve(solver="cvxopt")
self.assertTrue(abs(self.sdpRelaxation.primal + 0.75) < 10e-5)
def test_solving_with_cvxpy(self):
self.sdpRelaxation.solve(solver="cvxpy")
self.assertTrue(abs(self.sdpRelaxation.primal + 0.75) < 10e-5)
def setUp(self):
X = generate_operators('x', 2, hermitian=True)
self.sdpRelaxation = SdpRelaxation(X)
self.sdpRelaxation.get_relaxation(2,
objective=X[0] * X[1] + X[1] * X[0],
inequalities=[-X[1]**2 + X[1] + 0.5],
substitutions={X[0]**2: X[0]})
class ExampleNoncommutative(unittest.TestCase):
def setUp(self):
X = generate_operators('x', 2, hermitian=True)
self.sdpRelaxation = SdpRelaxation(X)
self.sdpRelaxation.get_relaxation(2,
objective=X[0] * X[1] + X[1] * X[0],
inequalities=[-X[1]**2 + X[1] + 0.5],
substitutions={X[0]**2: X[0]})
def tearDown(self):
clear_cache()
def test_solving_with_sdpa(self):
self.sdpRelaxation.solve(solver="sdpa")
self.assertTrue(abs(self.sdpRelaxation.primal + 0.75) < 10e-5)
def test_solving_with_mosek(self):
self.sdpRelaxation.solve(solver="mosek")
self.assertTrue(abs(self.sdpRelaxation.primal + 0.75) < 10e-5)
def test_solving_with_cvxopt(self):
self.sdpRelaxation.solve(solver="cvxopt")
self.assertTrue(abs(self.sdpRelaxation.primal + 0.75) < 10e-5)
def test_solving_with_cvxpy(self):
self.sdpRelaxation.solve(solver="cvxpy")
self.assertTrue(abs(self.sdpRelaxation.primal + 0.75) < 10e-5)
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_maximum_violation(self):
def expectation_values(measurement, outcomes):
exp_values = []
for k in range(len(measurement)):
exp_value = 0
for j in range(len(measurement[k])):
exp_value += outcomes[k][j] * measurement[k][j]
exp_values.append(exp_value)
return exp_values
E = generate_operators('E', 8, hermitian=True)
M, outcomes = [], []
for i in range(4):
M.append([E[2 * i], E[2 * i + 1]])
outcomes.append([1, -1])
A = [M[0], M[1]]
B = [M[2], M[3]]
substitutions = projective_measurement_constraints(A, B)
C = expectation_values(M, outcomes)
chsh = -(C[0] * C[2] + C[0] * C[3] + C[1] * C[2] - C[1] * C[3])
sdpRelaxation = SdpRelaxation(E, verbose=0)
sdpRelaxation.get_relaxation(1, objective=chsh,
substitutions=substitutions)
sdpRelaxation.solve()
self.assertTrue(abs(sdpRelaxation.primal + 2*np.sqrt(2)) < 10e-5)
def __main__():
# chsh quantum bound : 4
# mermin quantum bound : 4
# svetlichny quantum bound 4sqrt(2) ~ 5.65
P = Probability([2, 2], [2, 2], [2, 2])
Chsh = chsh(P)
Mermin = mermin(P)
Svetlichny = svetlichny(P)
dx = 0
div = 1.0
step = 0.4
lim = 5.6
x = []
y = []
while dx <= lim :
print (dx)
ineq = []
ineq.append(Svetlichny - dx)
ineq.append(-1*Svetlichny + dx)
sdpRelaxation = SdpRelaxation(P.get_all_operators(), verbose=0)
sdpRelaxation.get_relaxation(2, substitutions = P.substitutions,inequalities = ineq)
sdpRelaxation.set_objective(-Chsh)
solve_sdp(sdpRelaxation, solver="cvxopt")
x.append(dx)
y.append(abs(sdpRelaxation.primal)/div)
dx = dx + step
plt.plot(x,y,'k')
plt.show()
def test_solving(self):
x = generate_variables(2, commutative=True)
g0 = 4 * x[0] ** 2 + x[0] * x[1] - 4 * x[1] ** 2 - 2.1 * x[0] ** 4 + 4 * x[1] ** 4 + x[0] ** 6 / 3
sdpRelaxation = SdpRelaxation(x)
sdpRelaxation.get_relaxation(3, objective=g0)
sdpRelaxation.solve()
self.assertTrue(abs(sdpRelaxation.primal + 1.0316282672706911) < 10e-5)
def test_solving_with_sdpa(self):
x = generate_variables('x', 2, commutative=True)
sdpRelaxation = SdpRelaxation(x)
sdpRelaxation.get_relaxation(2, objective=x[0]*x[1] + x[1]*x[0],
inequalities=[-x[1]**2 + x[1] + 0.5],
substitutions={x[0]**2: x[0]})
sdpRelaxation.solve(solver="sdpa")
self.assertTrue(abs(sdpRelaxation.primal + 0.7320505301965234) < 10e-5)
def test_ground_state_energy(self):
N = 3
a = generate_variables(N, name="a")
substitutions = bosonic_constraints(a)
hamiltonian = sum(Dagger(a[i]) * a[i] for i in range(N))
sdpRelaxation = SdpRelaxation(a, verbose=0)
sdpRelaxation.get_relaxation(1, objective=hamiltonian, substitutions=substitutions)
sdpRelaxation.solve()
self.assertTrue(abs(sdpRelaxation.primal) < 10e-5)
def test_chordal_extension(self):
X = generate_variables(3, commutative=True)
inequalities = [1 - X[0] ** 2 - X[1] ** 2, 1 - X[1] ** 2 - X[2] ** 2]
sdpRelaxation = SdpRelaxation(X)
sdpRelaxation.get_relaxation(
2, objective=X[1] - 2 * X[0] * X[1] + X[1] * X[2], inequalities=inequalities, chordal_extension=True
)
sdpRelaxation.solve()
self.assertTrue(abs(sdpRelaxation.primal + 2.2443690631722637) < 10e-5)
def test_maximum_violation(self):
I = [[0, -1, 0], [-1, 1, 1], [0, 1, -1]]
P = Probability([2, 2], [2, 2])
relaxation = SdpRelaxation(P.get_all_operators())
relaxation.get_relaxation(1,
objective=define_objective_with_I(I, P),
substitutions=P.substitutions,
extramonomials=P.get_extra_monomials('AB'))
relaxation.solve()
self.assertTrue(abs(relaxation.primal + (np.sqrt(2)-1)/2) < 10e-5)
def test_max_cut(self):
W = np.diag(np.ones(8), 1) + np.diag(np.ones(7), 2) + np.diag([1, 1], 7) + np.diag([1], 8)
W = W + W.T
Q = (np.diag(np.dot(np.ones(len(W)).T, W)) - W) / 4
x = generate_variables(len(W), commutative=True)
equalities = [xi ** 2 - 1 for xi in x]
objective = -np.dot(x, np.dot(Q, np.transpose(x)))
sdpRelaxation = SdpRelaxation(x)
sdpRelaxation.get_relaxation(1, objective=objective, equalities=equalities, removeequalities=True)
sdpRelaxation.solve()
self.assertTrue(abs(sdpRelaxation.primal + 4.5) < 10e-5)
def setUp(self):
X = generate_variables(2, hermitian=True)
self.sdpRelaxation = SdpRelaxation(X)
self.sdpRelaxation.get_relaxation(
2,
objective=X[0] * X[1] + X[1] * X[0],
inequalities=[-X[1] ** 2 + X[1] + 0.5],
substitutions={X[0] ** 2: X[0]},
)
def test_guessing_probability(self):
p = [0.5, 0.5, 0.5, 0.5, 0.4267766952966368, 0.4267766952966368, 0.4267766952966368, 0.07322330470336313]
P = Probability([2, 2], [2, 2])
behaviour_constraint = [
P([0], [0], "A") - p[0],
P([0], [1], "A") - p[1],
P([0], [0], "B") - p[2],
P([0], [1], "B") - p[3],
P([0, 0], [0, 0]) - p[4],
P([0, 0], [0, 1]) - p[5],
P([0, 0], [1, 0]) - p[6],
P([0, 0], [1, 1]) - p[7],
]
behaviour_constraint.append("-0[0,0]+1.0")
sdpRelaxation = SdpRelaxation(P.get_all_operators(), normalized=False, verbose=0)
sdpRelaxation.get_relaxation(
1, objective=-P([0], [0], "A"), momentequalities=behaviour_constraint, substitutions=P.substitutions
)
sdpRelaxation.solve()
self.assertTrue(abs(sdpRelaxation.primal + 0.5) < 10e-5)
def __main__():
# chsh quantum bound : 4
# mermin quantum bound : 4
# svetlichny quantum bound 4sqrt(2) ~ 5.65
P = Probability([2, 2], [2, 2], [2, 2])
Chsh = chsh(P)
Mermin = mermin(P)
Svetlichny = svetlichny(P)
dx = 0
div = 1.0
step = 0.4
lim = 5.6
x = []
y = []
while dx <= lim:
print(dx)
ineq = []
ineq.append(Svetlichny - dx)
ineq.append(-1 * Svetlichny + dx)
sdpRelaxation = SdpRelaxation(P.get_all_operators(), verbose=0)
sdpRelaxation.get_relaxation(2,
substitutions=P.substitutions,
inequalities=ineq)
sdpRelaxation.set_objective(-Chsh)
solve_sdp(sdpRelaxation, solver="cvxopt")
x.append(dx)
y.append(abs(sdpRelaxation.primal) / div)
dx = dx + step
plt.plot(x, y, 'k')
plt.show()
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_guessing_probability(self):
p = [
0.5, 0.5, 0.5, 0.5, 0.4267766952966368, 0.4267766952966368,
0.4267766952966368, 0.07322330470336313
]
P = Probability([2, 2], [2, 2])
behaviour_constraint = [
P([0], [0], 'A') - p[0],
P([0], [1], 'A') - p[1],
P([0], [0], 'B') - p[2],
P([0], [1], 'B') - p[3],
P([0, 0], [0, 0]) - p[4],
P([0, 0], [0, 1]) - p[5],
P([0, 0], [1, 0]) - p[6],
P([0, 0], [1, 1]) - p[7]
]
behaviour_constraint.append("-0[0,0]+1.0")
sdpRelaxation = SdpRelaxation(P.get_all_operators(),
normalized=False,
verbose=0)
sdpRelaxation.get_relaxation(1,
objective=-P([0], [0], 'A'),
momentequalities=behaviour_constraint,
substitutions=P.substitutions)
sdpRelaxation.solve()
self.assertTrue(abs(sdpRelaxation.primal + 0.5) < 10e-5)
def test_maximum_violation(self):
def expectation_values(measurement, outcomes):
exp_values = []
for k in range(len(measurement)):
exp_value = 0
for j in range(len(measurement[k])):
exp_value += outcomes[k][j] * measurement[k][j]
exp_values.append(exp_value)
return exp_values
E = generate_operators('E', 8, hermitian=True)
M, outcomes = [], []
for i in range(4):
M.append([E[2 * i], E[2 * i + 1]])
outcomes.append([1, -1])
A = [M[0], M[1]]
B = [M[2], M[3]]
substitutions = projective_measurement_constraints(A, B)
C = expectation_values(M, outcomes)
chsh = -(C[0] * C[2] + C[0] * C[3] + C[1] * C[2] - C[1] * C[3])
sdpRelaxation = SdpRelaxation(E, verbose=0)
sdpRelaxation.get_relaxation(1,
objective=chsh,
substitutions=substitutions)
sdpRelaxation.solve()
self.assertTrue(abs(sdpRelaxation.primal + 2 * np.sqrt(2)) < 10e-5)
def test_solving_with_sdpa(self):
x = generate_variables('x', 2, commutative=True)
sdpRelaxation = SdpRelaxation(x)
sdpRelaxation.get_relaxation(2, objective=x[0]*x[1] + x[1]*x[0],
inequalities=[-x[1]**2 + x[1] + 0.5],
substitutions={x[0]**2: x[0]})
sdpRelaxation.solve(solver="sdpa")
self.assertTrue(abs(sdpRelaxation.primal + 0.7320505301965234) < 10e-5)
def test_solving(self):
x = generate_variables('x', 2, commutative=True)
g0 = 4 * x[0] ** 2 + x[0] * x[1] - 4 * x[1] ** 2 - \
2.1 * x[0] ** 4 + 4 * x[1] ** 4 + x[0] ** 6 / 3
sdpRelaxation = SdpRelaxation(x)
sdpRelaxation.get_relaxation(3, objective=g0)
sdpRelaxation.solve()
self.assertTrue(abs(sdpRelaxation.primal + 1.0316282672706911) < 10e-5)
def test_chordal_extension(self):
X = generate_variables('x', 3, commutative=True)
inequalities = [1-X[0]**2-X[1]**2, 1-X[1]**2-X[2]**2]
sdpRelaxation = SdpRelaxation(X)
sdpRelaxation.get_relaxation(2,
objective=X[1] - 2*X[0]*X[1] + X[1]*X[2],
inequalities=inequalities,
chordal_extension=True)
sdpRelaxation.solve()
self.assertTrue(abs(sdpRelaxation.primal + 2.2443690631722637) < 10e-5)
def test_ground_state_energy(self):
N = 3
a = generate_operators('a', N)
substitutions = bosonic_constraints(a)
hamiltonian = sum(Dagger(a[i]) * a[i] for i in range(N))
sdpRelaxation = SdpRelaxation(a, verbose=0)
sdpRelaxation.get_relaxation(1, objective=hamiltonian,
substitutions=substitutions)
sdpRelaxation.solve()
self.assertTrue(abs(sdpRelaxation.primal) < 10e-5)
def test_maximum_violation(self):
I = [[0, -1, 0], [-1, 1, 1], [0, 1, -1]]
P = Probability([2, 2], [2, 2])
relaxation = SdpRelaxation(P.get_all_operators())
relaxation.get_relaxation(1,
objective=define_objective_with_I(I, P),
substitutions=P.substitutions,
extramonomials=P.get_extra_monomials('AB'))
relaxation.solve()
self.assertTrue(abs(relaxation.primal + (np.sqrt(2) - 1) / 2) < 10e-5)
def test_max_cut(self):
W = np.diag(np.ones(8), 1) + np.diag(np.ones(7), 2) + \
np.diag([1, 1], 7) + np.diag([1], 8)
W = W + W.T
Q = (np.diag(np.dot(np.ones(len(W)).T, W)) - W) / 4
x = generate_variables('x', len(W), commutative=True)
equalities = [xi ** 2 - 1 for xi in x]
objective = -np.dot(x, np.dot(Q, np.transpose(x)))
sdpRelaxation = SdpRelaxation(x)
sdpRelaxation.get_relaxation(1, objective=objective,
equalities=equalities,
removeequalities=True)
sdpRelaxation.solve()
self.assertTrue(abs(sdpRelaxation.primal + 13.5) < 10e-5)
Kim, S. & Kojima, M. (2012). Exploiting Sparsity in SDP Relaxation of Polynomial Optimization Problems. In Handbook on Semidefinite, Conic and Polynomial Optimization. Springer, 2012, 499--531. Created on Sun Nov 30 19:18:04 2014 @author: Peter Wittek """ from ncpol2sdpa import generate_variables, SdpRelaxation, solve_sdp # Number of variables n_vars = 3 # Level of relaxation level = 2 # Get commutative variables X = generate_variables(n_vars, commutative=True) # Define the objective function obj = X[1] - 2 * X[0] * X[1] + X[1] * X[2] # Inequality constraints inequalities = [1 - X[0]**2 - X[1]**2, 1 - X[1]**2 - X[2]**2] # Obtain SDP relaxation sdpRelaxation = SdpRelaxation(X, hierarchy="npa_chordal") sdpRelaxation.get_relaxation(level, objective=obj, inequalities=inequalities) print(solve_sdp(sdpRelaxation))
"""
from ncpol2sdpa import generate_variables, SdpRelaxation, convert_to_picos
# Number of Hermitian variables
n_vars = 2
# Level of relaxation
level = 2
# Get Hermitian variables
X = generate_variables(n_vars, hermitian=True)
# Define the objective function
obj = X[0] * X[1] + X[1] * X[0]
# Inequality constraints
inequalities = [-X[1]**2 + X[1] + 0.5]
# Equality constraints
equalities = []
equalities.append(X[0]**2 - X[0])
# Obtain SDP relaxation
sdpRelaxation = SdpRelaxation(X)
sdpRelaxation.get_relaxation(level,
objective=obj,
inequalities=inequalities,
equalities=equalities)
P = convert_to_picos(sdpRelaxation)
P.solve()
Section 5.12 of the following paper:
Henrion, D.; Lasserre, J. & Löfberg, J. GloptiPoly 3: moments, optimization and
semidefinite programming. Optimization Methods & Software, 2009, 24, 761-779
Created on Thu May 15 12:12:40 2014
@author: wittek
"""
import numpy as np
from ncpol2sdpa import SdpRelaxation, generate_variables
W = np.diag(np.ones(8), 1) + np.diag(np.ones(7), 2) + np.diag([1, 1], 7) + \
np.diag([1], 8)
W = W + W.T
n = len(W)
e = np.ones(n)
Q = (np.diag(np.dot(e.T, W)) - W) / 4
x = generate_variables(n, commutative=True)
equalities = [xi ** 2 - 1 for xi in x]
objective = -np.dot(x, np.dot(Q, np.transpose(x)))
level = 1
sdpRelaxation = SdpRelaxation(x)
sdpRelaxation.get_relaxation(level, objective=objective, equalities=equalities,
removeequalities=True)
sdpRelaxation.write_to_file("max_cut.dat-s")
"""
import time
from sympy.physics.quantum.dagger import Dagger
from ncpol2sdpa import generate_variables, SdpRelaxation,\
bosonic_constraints
# Level of relaxation
level = 1
# Number of variables
N = 3
# Parameters for the Hamiltonian
hbar, omega = 1, 1
# Define ladder operators
a = generate_variables(N, name='a')
substitutions = bosonic_constraints(a)
hamiltonian = sum(hbar * omega * (Dagger(a[i]) * a[i]) for i in range(N))
time0 = time.time()
# Obtain SDP relaxation
sdpRelaxation = SdpRelaxation(a, verbose=1)
sdpRelaxation.get_relaxation(level, objective=hamiltonian,
substitutions=substitutions)
# Export relaxation to SDPA format
sdpRelaxation.write_to_file("harmonic_oscillator.dat-s")
print('%0.2f s' % ((time.time() - time0)))
return exp_values
# Number of Hermitian variables
n_vars = 8
# Level of relaxation
level = 1
# Get Hermitian variables
E = generate_variables(n_vars, name='E', hermitian=True)
# Define measurements and outcomes
M, outcomes = [], []
for i in range(int(n_vars / 2)):
M.append([E[2 * i], E[2 * i + 1]])
outcomes.append([1, -1])
# Define which measurements Alice and Bob have
A = [M[0], M[1]]
B = [M[2], M[3]]
substitutions = projective_measurement_constraints(A, B)
C = expectation_values(M, outcomes)
chsh = -(C[0] * C[2] + C[0] * C[3] + C[1] * C[2] - C[1] * C[3])
sdpRelaxation = SdpRelaxation(E, verbose=2)
sdpRelaxation.get_relaxation(level, objective=chsh,
substitutions=substitutions)
print(solve_sdp(sdpRelaxation))
# -*- coding: utf-8 -*-
"""
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 Probability, SdpRelaxation, define_objective_with_I
level = 1
I = [[ 0, -1, 0 ],
[-1, 1, 1 ],
[ 0, 1, -1 ]]
P = Probability([2, 2], [2, 2])
objective = define_objective_with_I(I, P)
sdpRelaxation = SdpRelaxation(P.get_all_operators())
sdpRelaxation.get_relaxation(level, objective=objective,
substitutions=P.substitutions,
extramonomials=P.get_extra_monomials('AB'))
sdpRelaxation.solve()
print(sdpRelaxation.primal, sdpRelaxation.status)
@author: Peter Wittek
"""
from ncpol2sdpa import generate_variables, SdpRelaxation, convert_to_picos
# Number of Hermitian variables
n_vars = 2
# Level of relaxation
level = 2
# Get Hermitian variables
X = generate_variables(n_vars, hermitian=True)
# Define the objective function
obj = X[0] * X[1] + X[1] * X[0]
# Inequality constraints
inequalities = [-X[1] ** 2 + X[1] + 0.5]
# Equality constraints
equalities = []
equalities.append(X[0] ** 2 - X[0])
# Obtain SDP relaxation
sdpRelaxation = SdpRelaxation(X)
sdpRelaxation.get_relaxation(level, objective=obj,
inequalities=inequalities,
equalities=equalities)
P = convert_to_picos(sdpRelaxation)
P.solve()
write_to_sdpa, bosonic_constraints
# Level of relaxation
level = 1
# Number of variables
N = 3
# Parameters for the Hamiltonian
hbar, omega = 1, 1
# Define ladder operators
a = generate_variables(N, name='a')
substitutions = bosonic_constraints(a)
hamiltonian = 0
for i in range(N):
hamiltonian += hbar * omega * (Dagger(a[i]) * a[i])
time0 = time.time()
# Obtain SDP relaxation
print("Obtaining SDP relaxation...")
sdpRelaxation = SdpRelaxation(a, verbose=1)
sdpRelaxation.get_relaxation(level, objective=hamiltonian,
substitutions=substitutions)
# Export relaxation to SDPA format
write_to_sdpa(sdpRelaxation, 'harmonic_oscillator.dat-s')
print('%0.2f s' % ((time.time() - time0)))
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))
# Get Hermitian variables
X = generate_variables(n_vars, hermitian=True)
# Define the objective function
obj = 0
for i in range(n_vars):
for j in range(n_vars):
obj += X[i] * X[j]
# Equality constraints
equalities = []
for i in range(n_vars):
equalities.append(X[i] * X[i] - 1.0)
# Simple monomial substitutions
substitutions = {}
for i in range(n_vars):
for j in range(i + 1, n_vars):
# [X_i, X_j] = 0
substitutions[X[i] * X[j]] = X[j] * X[i]
# Obtain SDP relaxation
time0 = time.time()
sdpRelaxation = SdpRelaxation(X)
sdpRelaxation.get_relaxation(level,
objective=obj,
equalities=equalities,
substitutions=substitutions)
write_to_sdpa(sdpRelaxation, 'benchmark.dat-s')
print('%0.2f s' % ((time.time() - time0)))
# -*- coding: utf-8 -*-
"""
This script replicates the results of gloptipolydemo.m, which is packaged with
Gloptipoly3.
Created on Thu May 15 11:16:58 2014
@author: wittek
"""
from ncpol2sdpa import SdpRelaxation, generate_variables
# Get commutative variables
x = generate_variables(2, commutative=True)
g0 = 4 * x[0] ** 2 + x[0] * x[1] - 4 * x[1] ** 2 - \
2.1 * x[0] ** 4 + 4 * x[1] ** 4 + x[0] ** 6 / 3
# Obtain SDP relaxation
sdpRelaxation = SdpRelaxation(x)
sdpRelaxation.get_relaxation(3, objective=g0)
sdpRelaxation.write_to_file("gloptipoly_demo.dat-s")
Created on Fri May 10 09:45:11 2013
@author: Peter Wittek
"""
from ncpol2sdpa import generate_variables, SdpRelaxation
# Number of variables
n_vars = 2
# Level of relaxation
level = 2
# Get commutative variables
X = generate_variables(n_vars, commutative=True)
# Define the objective function
obj = X[0] * X[1] + X[1] * X[0]
# Inequality constraints
inequalities = [-X[1] ** 2 + X[1] + 0.5]
# Simple monomial substitutions
monomial_substitution = {}
monomial_substitution[X[0] ** 2] = X[0]
# Obtain SDP relaxation
sdpRelaxation = SdpRelaxation(X)
sdpRelaxation.get_relaxation(level, objective=obj, inequalities=inequalities,
substitutions=monomial_substitution)
sdpRelaxation.write_to_file('example_commutative.dat-s')
# -*- coding: utf-8 -*-
"""
This script replicates the results of gloptipolydemo.m, which is packaged with
Gloptipoly3.
Created on Thu May 15 11:16:58 2014
@author: wittek
"""
from sympy.physics.quantum.operator import HermitianOperator
from ncpol2sdpa import SdpRelaxation, write_to_sdpa
# Get commutative variables
x1 = HermitianOperator("x1")
x1.is_commutative = True
x2 = HermitianOperator("x2")
x2.is_commutative = True
g0 = 4 * x1 ** 2 + x1 * x2 - 4 * x2 ** 2 - \
2.1 * x1 ** 4 + 4 * x2 ** 4 + x1 ** 6 / 3
# Obtain SDP relaxation
sdpRelaxation = SdpRelaxation([x1, x2])
sdpRelaxation.get_relaxation(3, objective=g0)
write_to_sdpa(sdpRelaxation, 'gloptipoly_demo.dat-s')
level = 1
# Get Hermitian variables
X = generate_variables(n_vars, hermitian=True)
# Define the objective function
obj = 0
for i in range(n_vars):
for j in range(n_vars):
obj += X[i] * X[j]
# Equality constraints
equalities = []
for i in range(n_vars):
equalities.append(X[i] * X[i] - 1.0)
# Simple monomial substitutions
substitutions = {}
for i in range(n_vars):
for j in range(i + 1, n_vars):
# [X_i, X_j] = 0
substitutions[X[i] * X[j]] = X[j] * X[i]
# Obtain SDP relaxation
time0 = time.time()
sdpRelaxation = SdpRelaxation(X)
sdpRelaxation.get_relaxation(level, objective=obj, equalities=equalities,
substitutions=substitutions)
sdpRelaxation.write_to_file('benchmark.dat-s')
print('%0.2f s' % ((time.time() - time0)))
Section 5.12 of the following paper:
Henrion, D.; Lasserre, J. & Löfberg, J. GloptiPoly 3: moments, optimization and
semidefinite programming. Optimization Methods & Software, 2009, 24, 761-779
Created on Thu May 15 12:12:40 2014
@author: wittek
"""
import numpy as np
from ncpol2sdpa import SdpRelaxation, write_to_sdpa, generate_variables
W = np.diag(np.ones(8), 1) + np.diag(np.ones(7), 2) + np.diag([1, 1], 7) + \
np.diag([1], 8)
W = W + W.T
n = len(W)
e = np.ones(n)
Q = (np.diag(np.dot(e.T, W)) - W) / 4
x = generate_variables(n, commutative=True)
equalities = [xi ** 2 - 1 for xi in x]
objective = -np.dot(x, np.dot(Q, np.transpose(x)))
level = 1
sdpRelaxation = SdpRelaxation(x)
sdpRelaxation.get_relaxation(level, objective=objective, equalities=equalities,
removeequalities=True)
write_to_sdpa(sdpRelaxation, 'max_cut.dat-s')
@author: Peter Wittek
"""
from ncpol2sdpa import generate_variables, SdpRelaxation, write_to_sdpa
# Number of variables
n_vars = 2
# Level of relaxation
level = 2
# Get commutative variables
X = generate_variables(n_vars, commutative=True)
# Define the objective function
obj = X[0] * X[1] + X[1] * X[0]
# Inequality constraints
inequalities = [-X[1]**2 + X[1] + 0.5]
# Simple monomial substitutions
monomial_substitution = {}
monomial_substitution[X[0]**2] = X[0]
# Obtain SDP relaxation
sdpRelaxation = SdpRelaxation(X)
sdpRelaxation.get_relaxation(level,
objective=obj,
inequalities=inequalities,
substitutions=monomial_substitution)
write_to_sdpa(sdpRelaxation, 'example_commutative.dat-s')
# Number of variables
N = 3
# Parameters for the Hamiltonian
hbar, omega = 1, 1
# Define ladder operators
a = generate_variables(N, name='a')
hamiltonian = 0
for i in range(N):
hamiltonian += hbar * omega * (Dagger(a[i]) * a[i] + 0.5)
substitutions, equalities = bosonic_constraints(a)
inequalities = []
time0 = time.time()
# Obtain SDP relaxation
print("Obtaining SDP relaxation...")
sdpRelaxation = SdpRelaxation(a, verbose=1)
sdpRelaxation.get_relaxation(level, objective=hamiltonian,
equalities=equalities,
substitutions=substitutions,
removeequalities=True)
# Export relaxation to SDPA format
print("Writing to disk...")
write_to_sdpa(sdpRelaxation, 'harmonic_oscillator.dat-s')
print('%0.2f s' % ((time.time() - time0)))
Created on Fri May 10 09:45:11 2013
@author: Peter Wittek
"""
from ncpol2sdpa import generate_variables, SdpRelaxation, write_to_sdpa
# Number of variables
n_vars = 2
# Level of relaxation
level = 2
# Get commutative variables
X = generate_variables(n_vars, commutative=True)
# Define the objective function
obj = X[0] * X[1] + X[1] * X[0]
# Inequality constraints
inequalities = [-X[1] ** 2 + X[1] + 0.5]
# Simple monomial substitutions
monomial_substitution = {}
monomial_substitution[X[0] ** 2] = X[0]
# Obtain SDP relaxation
sdpRelaxation = SdpRelaxation(X)
sdpRelaxation.get_relaxation(level, objective=obj, inequalities=inequalities,
substitutions=monomial_substitution)
write_to_sdpa(sdpRelaxation, 'example_commutative.dat-s')
# -*- coding: utf-8 -*-
"""
This example 18.1 from the following paper:
Kim, S. & Kojima, M. (2012). Exploiting Sparsity in SDP Relaxation of
Polynomial Optimization Problems. In Handbook on Semidefinite, Conic and
Polynomial Optimization. Springer, 2012, 499--531.
Created on Sun Nov 30 19:18:04 2014
@author: Peter Wittek
"""
from ncpol2sdpa import generate_variables, SdpRelaxation
level = 2
X = generate_variables(3, commutative=True)
inequalities = [1-X[0]**2-X[1]**2 >= 0,
1-X[1]**2-X[2]**2 >= 0]
sdpRelaxation = SdpRelaxation(X, verbose=2)
sdpRelaxation.get_relaxation(level, objective=X[1] - 2*X[0]*X[1] + X[1]*X[2],
inequalities=inequalities, chordal_extension=True)
sdpRelaxation.solve()
sdpRelaxation.write_to_file("/home/wittek/sparse2.csv")
print(sdpRelaxation.primal, sdpRelaxation.dual, sdpRelaxation.status)
