def test_fast_substitute(self):
f = generate_variables(2, name="f")
substitutions = {}
substitutions[Dagger(f[0]) * f[0]] = -f[0] * Dagger(f[0])
monomial = Dagger(f[0]) * f[0]
lhs = Dagger(f[0]) * f[0]
rhs = -f[0] * Dagger(f[0])
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial.subs(lhs, rhs))
monomial = Dagger(f[0]) * f[0] ** 2
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial.subs(lhs, rhs))
monomial = Dagger(f[0]) ** 2 * f[0]
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial.subs(lhs, rhs))
monomial = Dagger(f[0]) ** 2 * f[0]
lhs = Dagger(f[0]) ** 2
rhs = -f[0] * Dagger(f[0])
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial.subs(lhs, rhs))
g = generate_variables(2, name="g")
monomial = 2 * g[0] ** 3 * g[1] * Dagger(f[0]) ** 2 * f[0]
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial.subs(lhs, rhs))
monomial = S.One
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial.subs(lhs, rhs))
monomial = 5
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial)
monomial = 2 * g[0] ** 3 * g[1] * Dagger(f[0]) ** 2 * f[0] + f[1]
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial.subs(lhs, rhs))
monomial = f[1] * Dagger(f[0]) ** 2 * f[0]
lhs = f[1]
rhs = 1.0 + f[0]
self.assertTrue(fast_substitute(monomial, lhs, rhs) == expand(monomial.subs(lhs, rhs)))
monomial = f[1] ** 2 * Dagger(f[0]) ** 2 * f[0]
result = fast_substitute(fast_substitute(monomial, lhs, rhs), lhs, rhs)
self.assertTrue(result == expand(monomial.subs(lhs, rhs)))
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_variables(length, name="fu")
fd = generate_variables(length, name="fd")
_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_variables(length, name="fu")
fd = generate_variables(length, name="fd")
_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 = get_xmat_value(s, sdpRelaxation)
self.assertTrue(abs(magnetization - 0.021325317328560453) < 10e-5)
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_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_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_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 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_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_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_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 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)
def __init__(self, lattice_length, lattice_width, solver, outputDir,
periodic, window_length, removeequalities, parallel=True):
LatticeModel.__init__(self, lattice_length, lattice_width, solver,
outputDir, removeequalities)
self._b = generate_variables('b', lattice_length * lattice_width,
commutative=False)
self._level = -1
self._periodic = periodic
self.nmax = None
self.localNmax = None
self.n = None
self.nmin = None
if window_length == 0:
self.window_length = lattice_length * lattice_width
else:
if self._lattice_width != 1:
raise NotImplementedError(
"Windowed models in more than 1D not implemented!")
self.window_length = window_length
self._parallel = parallel
self._debug_mode = False
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_variables(8, name="E", 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)
on Optimization, SIAM, 2010, 20, 2157-2180.
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)
on Optimization, SIAM, 2010, 20, 2157-2180.
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,
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
# 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)
on Optimization, SIAM, 2010, 20, 2157-2180.
Created on Fri May 10 09:45:11 2013
@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,
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')
on Optimization, SIAM, 2010, 20, 2157-2180.
Created on Fri May 10 09:45:11 2013
@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,
import time
from sympy.physics.quantum.dagger import Dagger
from ncpol2sdpa import generate_variables, bosonic_constraints, SdpRelaxation,\
write_to_sdpa
# 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')
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,
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
"""
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)))