def test_ComplexSuperApply(self):
"""
Superoperator: Efficient numerics and reference return same result,
acting on non-composite system
"""
rho_list = list(map(rand_dm, [2, 3, 2, 3, 2]))
rho_input = tensor(rho_list)
superop = kraus_to_super(rand_kraus_map(3))
analytic_result = rho_list
analytic_result[1] = Qobj(vec2mat(superop.data.todense() *
mat2vec(analytic_result[1].data.todense())))
analytic_result[3] = Qobj(vec2mat(superop.data.todense() *
mat2vec(analytic_result[3].data.todense())))
analytic_result = tensor(analytic_result)
naive_result = subsystem_apply(rho_input, superop,
[False, True, False, True, False],
reference=True)
naive_diff = (analytic_result - naive_result).data.todense()
assert_(norm(naive_diff) < 1e-12)
efficient_result = subsystem_apply(rho_input, superop,
[False, True, False, True, False])
efficient_diff = (efficient_result - analytic_result).data.todense()
assert_(norm(efficient_diff) < 1e-12)
def _steadystate_svd_dense(L, ss_args):
"""
Find the steady state(s) of an open quantum system by solving for the
nullspace of the Liouvillian.
"""
atol = 1e-12
rtol = 1e-12
if settings.debug:
print('Starting SVD solver...')
u, s, vh = svd(L.full(), full_matrices=False)
tol = max(atol, rtol * s[0])
nnz = (s >= tol).sum()
ns = vh[nnz:].conj().T
if ss_args['all_states']:
rhoss_list = []
for n in range(ns.shape[1]):
rhoss = Qobj(vec2mat(ns[:, n]), dims=L.dims[0])
rhoss_list.append(rhoss / rhoss.tr())
return rhoss_list
else:
rhoss = Qobj(vec2mat(ns[:, 0]), dims=L.dims[0])
return rhoss / rhoss.tr()
def test_ComplexSuperApply(self):
"""
Superoperator: Efficient numerics and reference return same result,
acting on non-composite system
"""
tol = 1e-10
rho_list = list(map(rand_dm, [2, 3, 2, 3, 2]))
rho_input = tensor(rho_list)
superop = kraus_to_super(rand_kraus_map(3))
analytic_result = rho_list
analytic_result[1] = Qobj(
vec2mat(superop.full() @ mat2vec(analytic_result[1].full())))
analytic_result[3] = Qobj(
vec2mat(superop.full() @ mat2vec(analytic_result[3].full())))
analytic_result = tensor(analytic_result)
naive_result = subsystem_apply(rho_input,
superop,
[False, True, False, True, False],
reference=True)
naive_diff = (analytic_result - naive_result).full()
naive_diff_norm = norm(naive_diff)
assert_(naive_diff_norm < tol,
msg="ComplexSuper: naive_diff_norm {} "
"is beyond tolerance {}".format(naive_diff_norm, tol))
efficient_result = subsystem_apply(rho_input, superop,
[False, True, False, True, False])
efficient_diff = (efficient_result - analytic_result).full()
efficient_diff_norm = norm(efficient_diff)
assert_(efficient_diff_norm < tol,
msg="ComplexSuper: efficient_diff_norm {} "
"is beyond tolerance {}".format(efficient_diff_norm, tol))
def _steadystate_svd_dense(L, ss_args):
"""
Find the steady state(s) of an open quantum system by solving for the
nullspace of the Liouvillian.
"""
ss_args['info'].pop('weight', None)
atol = 1e-12
rtol = 1e-12
if settings.debug:
logger.debug('Starting SVD solver.')
_svd_start = time.time()
u, s, vh = svd(L.full(), full_matrices=False)
tol = max(atol, rtol * s[0])
nnz = (s >= tol).sum()
ns = vh[nnz:].conj().T
_svd_end = time.time()
ss_args['info']['solution_time'] = _svd_end - _svd_start
if ss_args['all_states']:
rhoss_list = []
for n in range(ns.shape[1]):
rhoss = Qobj(vec2mat(ns[:, n]), dims=L.dims[0])
rhoss_list.append(rhoss / rhoss.tr())
if ss_args['return_info']:
return rhoss_list, ss_args['info']
else:
if ss_args['return_info']:
return rhoss_list, ss_args['info']
else:
return rhoss_list
else:
rhoss = Qobj(vec2mat(ns[:, 0]), dims=L.dims[0])
return rhoss / rhoss.tr()
def test_ComplexSuperApply(self):
"""
Superoperator: Efficient numerics and reference return same result,
acting on non-composite system
"""
rho_list = list(map(rand_dm, [2, 3, 2, 3, 2]))
rho_input = tensor(rho_list)
superop = kraus_to_super(rand_kraus_map(3))
analytic_result = rho_list
analytic_result[1] = Qobj(
vec2mat(superop.data.todense() *
mat2vec(analytic_result[1].data.todense())))
analytic_result[3] = Qobj(
vec2mat(superop.data.todense() *
mat2vec(analytic_result[3].data.todense())))
analytic_result = tensor(analytic_result)
naive_result = subsystem_apply(rho_input,
superop,
[False, True, False, True, False],
reference=True)
naive_diff = (analytic_result - naive_result).data.todense()
assert_(norm(naive_diff) < 1e-12)
efficient_result = subsystem_apply(rho_input, superop,
[False, True, False, True, False])
efficient_diff = (efficient_result - analytic_result).data.todense()
assert_(norm(efficient_diff) < 1e-12)
def _steadystate_svd_dense(L, ss_args):
"""
Find the steady state(s) of an open quantum system by solving for the
nullspace of the Liouvillian.
"""
ss_args['info'].pop('weight', None)
atol = 1e-12
rtol = 1e-12
if settings.debug:
logger.debug('Starting SVD solver.')
_svd_start = time.time()
u, s, vh = svd(L.full(), full_matrices=False)
tol = max(atol, rtol * s[0])
nnz = (s >= tol).sum()
ns = vh[nnz:].conj().T
_svd_end = time.time()
ss_args['info']['solution_time'] = _svd_end-_svd_start
if ss_args['all_states']:
rhoss_list = []
for n in range(ns.shape[1]):
rhoss = Qobj(vec2mat(ns[:, n]), dims=L.dims[0])
rhoss_list.append(rhoss / rhoss.tr())
if ss_args['return_info']:
return rhoss_list, ss_args['info']
else:
if ss_args['return_info']:
return rhoss_list, ss_args['info']
else:
return rhoss_list
else:
rhoss = Qobj(vec2mat(ns[:, 0]), dims=L.dims[0])
return rhoss / rhoss.tr()
def gradient_CFI(self, M):
'''
Input:
1) M:
TYPE: list of Qobj (matrix)
DESCRIPTION: merasurement. It takes the form [M1,M2,...], where M1, M2 ...
are matrices and satisfies M1+M2+... = identity matrix.
Output:
1) updated values of self.control_coefficients.
Notice: To run this function, the function 'propagation_single()' has to run first.
'''
num = len(self.times)
dH0 = 1.j * Qobj(liouvillian(self.Hamiltonian_derivative[0]).full())
dim = self.freeHamiltonian.dims[0][0]
dt = self.times[1] - self.times[0]
Hc_coeff = self.control_coefficients
D = self.propagator
rhoT_vec = self.rho[num-1]
drhoT_vec = self.rho_derivative[num-1]
rhoT = Qobj(vec2mat(rhoT_vec.full()))
drhoT = Qobj(vec2mat(drhoT_vec.full()))
L1 = Qobj(np.array([[0. for i in range(0,dim)] for k in range(0,dim)]))
L2 = Qobj(np.array([[0. for i in range(0,dim)] for k in range(0,dim)]))
for mi in range(0, len(M)):
ptemp = (rhoT * M[mi]).tr()
dptemp = (drhoT * M[mi]).tr()
if ptemp != 0:
L1 += (dptemp / ptemp) * M[mi]
L2 += ((dptemp / ptemp)**2) * M[mi]
for ki in range(0, len(self.control_Hamiltonian)):
Hk = 1.j * Qobj(liouvillian(self.control_Hamiltonian[ki]).full())
Hc_ki = Hc_coeff[ki]
for ti in range(0,num):
Mj1 = 1.j * D[ti+1][num-1] * Hk * self.rho[ti]
Mj2 = Qobj(np.zeros((dim*dim, 1)))
for ri in range(0, ti+1):
Mj2 += D[ti+1][num-1] * Hk * D[ri+1][ti] * dH0 * self.rho[ri]
Mj3 = Qobj(np.zeros((dim*dim, 1)))
for ri in range(ti+1, num):
Mj3 += D[ri+1][num-1] * dH0 * D[ti+1][ri] * Hk * self.rho[ti]
Mj1mat = Qobj(vec2mat(Mj1.full()))
Mj2mat = Qobj(vec2mat(Mj2.full()))
Mj3mat = Qobj(vec2mat(Mj3.full()))
term1 = dt * (L2 * Mj1mat).tr()
term2 = -2 * (dt * dt) * (L1 * (Mj2mat+Mj3mat)).tr()
delta = np.real(term1 + term2)
Hc_ki[ti] += Hc_ki[ti] + self.epsilon * delta
Hc_coeff[ki] = Hc_ki
self.control_coefficients = Hc_coeff
def _steadystate_eigen(L, ss_args):
"""
Internal function for solving the steady state problem by
finding the eigenvector corresponding to the zero eigenvalue
of the Liouvillian using ARPACK.
"""
if settings.debug:
print('Starting Eigen solver...')
dims = L.dims[0]
shape = prod(dims[0])
L = L.data.tocsc()
if ss_args['use_rcm']:
if settings.debug:
old_band = sp_bandwidth(L)[0]
print('Original bandwidth:', old_band)
perm = reverse_cuthill_mckee(L)
rev_perm = np.argsort(perm)
L = sp_permute(L, perm, perm, 'csc')
if settings.debug:
rcm_band = sp_bandwidth(L)[0]
print('RCM bandwidth:', rcm_band)
print('Bandwidth reduction factor:', round(old_band/rcm_band, 1))
eigval, eigvec = eigs(L, k=1, sigma=1e-15, tol=ss_args['tol'],
which='LM', maxiter=ss_args['maxiter'])
if ss_args['use_rcm']:
eigvec = eigvec[np.ix_(rev_perm,)]
data = vec2mat(eigvec)
data = 0.5 * (data + data.conj().T)
out = Qobj(data, dims=dims, isherm=True)
return out/out.tr()
def qpt(U, op_basis_list):
"""
Calculate the quantum process tomography chi matrix for a given
(possibly nonunitary) transformation matrix U, which transforms a
density matrix in vector form according to:
vec(rho) = U * vec(rho0)
or
rho = vec2mat(U * mat2vec(rho0))
U can be calculated for an open quantum system using the QuTiP propagator
function.
"""
E_ops = []
# loop over all index permutations
for inds in index_permutations([len(op_list) for op_list in op_basis_list]):
# loop over all composite systems
E_op_list = [op_basis_list[k][inds[k]] for k in range(len(op_basis_list))]
E_ops.append(tensor(E_op_list))
EE_ops = [spre(E1) * spost(E2.dag()) for E1 in E_ops for E2 in E_ops]
M = hstack([mat2vec(EE.full()) for EE in EE_ops])
Uvec = mat2vec(U.full())
chi_vec = la.solve(M, Uvec)
return vec2mat(chi_vec)
def _smepdpsolve_single_trajectory(L, dt, tlist, N_store, N_substeps, rho_t,
c_ops, e_ops, data):
"""
Internal function.
"""
states_list = []
rho_t = np.copy(rho_t)
prng = RandomState() # todo: seed it
r_jump, r_op = prng.rand(2)
jump_times = []
jump_op_idx = []
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
data.expect[e_idx, t_idx] += expect_rho_vec(e, rho_t)
else:
states_list.append(Qobj(vec2mat(rho_t)))
for j in range(N_substeps):
if expect_rho_vec(d_op, sigma_t) < r_jump:
# jump occurs
p = np.array([rho_expect(c.dag() * c, rho_t) for c in c_ops])
p = np.cumsum(p / np.sum(p))
n = np.where(p >= r_op)[0][0]
# apply jump
rho_t = c_ops[n] * psi_t * c_ops[n].dag()
rho_t /= rho_expect(c.dag() * c, rho_t)
rho_t = np.copy(rho_t)
# store info about jump
jump_times.append(tlist[t_idx] + dt * j)
jump_op_idx.append(n)
# get new random numbers for next jump
r_jump, r_op = prng.rand(2)
# deterministic evolution wihtout correction for norm decay
dsigma_t = spmv(L.data.data,
L.data.indices,
L.data.indptr, sigma_t) * dt
# deterministic evolution with correction for norm decay
drho_t = spmv(L.data.data,
L.data.indices,
L.data.indptr, rho_t) * dt
rho_t += drho_t
# increment density matrices
sigma_t += dsigma_t
rho_t += drho_t
return states_list, jump_times, jump_op_idx
def _steadystate_direct_dense(L, ss_args):
"""
Direct solver that use numpy dense matrices. Suitable for
small system, with a few states.
"""
if settings.debug:
logger.debug('Starting direct dense solver.')
dims = L.dims[0]
n = int(np.sqrt(L.shape[0]))
b = np.zeros(n**2)
b[0] = ss_args['weight']
L = L.data.todense()
L[0, :] = np.diag(ss_args['weight'] * np.ones(n)).reshape((1, n**2))
_dense_start = time.time()
v = np.linalg.solve(L, b)
_dense_end = time.time()
ss_args['info']['solution_time'] = _dense_end - _dense_start
if ss_args['return_info']:
ss_args['info']['residual_norm'] = la.norm(b - L * v, np.inf)
data = vec2mat(v)
data = 0.5 * (data + data.conj().T)
return Qobj(data, dims=dims, isherm=True)
def propagator_steadystate(U):
"""Find the steady state for successive applications of the propagator
:math:`U`.
Parameters
----------
U : qobj
Operator representing the propagator.
Returns
-------
a : qobj
Instance representing the steady-state density matrix.
"""
evals, evecs = la.eig(U.full())
ev_min, ev_idx = _get_min_and_index(abs(evals - 1.0))
evecs = evecs.T
rho = Qobj(vec2mat(evecs[ev_idx]), dims=U.dims[0])
rho = rho * (1.0 / rho.tr())
rho = 0.5 * (rho + rho.dag()) # make sure rho is herm
return rho
def _subsystem_apply_reference(state, channel, mask):
if isket(state):
state = ket2dm(state)
if isoper(channel):
full_oper = tensor([channel if mask[j]
else qeye(state.dims[0][j])
for j in range(len(state.dims[0]))])
return full_oper * state * full_oper.dag()
else:
# Go to Choi, then Kraus
# chan_mat = array(channel.data.todense())
choi_matrix = super_to_choi(channel)
vals, vecs = eig(choi_matrix.full())
vecs = list(map(array, zip(*vecs)))
kraus_list = [sqrt(vals[j]) * vec2mat(vecs[j])
for j in range(len(vals))]
# Kraus operators to be padded with identities:
k_qubit_kraus_list = product(kraus_list, repeat=sum(mask))
rho_out = Qobj(inpt=zeros(state.shape), dims=state.dims)
for operator_iter in k_qubit_kraus_list:
operator_iter = iter(operator_iter)
op_iter_list = [next(operator_iter).conj().T if mask[j]
else qeye(state.dims[0][j])
for j in range(len(state.dims[0]))]
full_oper = tensor(list(map(Qobj, op_iter_list)))
rho_out = rho_out + full_oper * state * full_oper.dag()
return Qobj(rho_out)
def _steadystate_direct_dense(L, ss_args):
"""
Direct solver that use numpy dense matrices. Suitable for
small system, with a few states.
"""
if settings.debug:
logger.debug('Starting direct dense solver.')
dims = L.dims[0]
n = int(np.sqrt(L.shape[0]))
b = np.zeros(n ** 2)
b[0] = ss_args['weight']
L = L.data.todense()
L[0, :] = np.diag(ss_args['weight']*np.ones(n)).reshape((1, n ** 2))
_dense_start = time.time()
v = np.linalg.solve(L, b)
_dense_end = time.time()
ss_args['info']['solution_time'] = _dense_end-_dense_start
if ss_args['return_info']:
ss_args['info']['residual_norm'] = la.norm(b - L*v, np.inf)
data = vec2mat(v)
data = 0.5 * (data + data.conj().T)
return Qobj(data, dims=dims, isherm=True)
def _smepdpsolve_single_trajectory(L, dt, tlist, N_store, N_substeps, rho_t,
c_ops, e_ops, data):
"""
Internal function.
"""
states_list = []
rho_t = np.copy(rho_t)
prng = RandomState() # todo: seed it
r_jump, r_op = prng.rand(2)
jump_times = []
jump_op_idx = []
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
data.expect[e_idx, t_idx] += expect_rho_vec(e, rho_t)
else:
states_list.append(Qobj(vec2mat(rho_t)))
for j in range(N_substeps):
if expect_rho_vec(d_op, sigma_t) < r_jump:
# jump occurs
p = np.array([rho_expect(c.dag() * c, rho_t) for c in c_ops])
p = np.cumsum(p / np.sum(p))
n = np.where(p >= r_op)[0][0]
# apply jump
rho_t = c_ops[n] * psi_t * c_ops[n].dag()
rho_t /= rho_expect(c.dag() * c, rho_t)
rho_t = np.copy(rho_t)
# store info about jump
jump_times.append(tlist[t_idx] + dt * j)
jump_op_idx.append(n)
# get new random numbers for next jump
r_jump, r_op = prng.rand(2)
# deterministic evolution wihtout correction for norm decay
dsigma_t = spmv(L.data.data,
L.data.indices,
L.data.indptr, sigma_t) * dt
# deterministic evolution with correction for norm decay
drho_t = spmv(L.data.data,
L.data.indices,
L.data.indptr, rho_t) * dt
rho_t += drho_t
# increment density matrices
sigma_t += dsigma_t
rho_t += drho_t
return states_list, jump_times, jump_op_idx
def _subsystem_apply_reference(state, channel, mask):
if isket(state):
state = ket2dm(state)
if isoper(channel):
full_oper = tensor([
channel if mask[j] else qeye(state.dims[0][j])
for j in range(len(state.dims[0]))
])
return full_oper * state * full_oper.dag()
else:
# Go to Choi, then Kraus
# chan_mat = array(channel.data.todense())
choi_matrix = super_to_choi(channel)
vals, vecs = eig(choi_matrix.full())
vecs = list(map(array, zip(*vecs)))
kraus_list = [
sqrt(vals[j]) * vec2mat(vecs[j]) for j in range(len(vals))
]
# Kraus operators to be padded with identities:
k_qubit_kraus_list = product(kraus_list, repeat=sum(mask))
rho_out = Qobj(inpt=zeros(state.shape), dims=state.dims)
for operator_iter in k_qubit_kraus_list:
operator_iter = iter(operator_iter)
op_iter_list = [
next(operator_iter) if mask[j] else qeye(state.dims[0][j])
for j in range(len(state.dims[0]))
]
full_oper = tensor(list(map(Qobj, op_iter_list)))
rho_out = rho_out + full_oper * state * full_oper.dag()
return Qobj(rho_out)
def propagator_steadystate(U):
"""Find the steady state for successive applications of the propagator
:math:`U`.
Parameters
----------
U : qobj
Operator representing the propagator.
Returns
-------
a : qobj
Instance representing the steady-state density matrix.
"""
evals, evecs = la.eig(U.full())
ev_min, ev_idx = _get_min_and_index(abs(evals - 1.0))
evecs = evecs.T
rho = Qobj(vec2mat(evecs[ev_idx]), dims=U.dims[0])
rho = rho * (1.0 / rho.tr())
rho = 0.5 * (rho + rho.dag()) # make sure rho is herm
return rho
def _smesolve_single_trajectory(L, dt, tlist, N_store, N_substeps, rho_t,
A_ops, e_ops, data, rhs, d1, d2):
"""
Internal function. See smesolve.
"""
dW = np.sqrt(dt) * scipy.randn(len(A_ops), N_store, N_substeps)
states_list = []
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
# XXX: need to keep hilbert space structure
data.expect[e_idx, t_idx] += expect(e, Qobj(vec2mat(rho_t)))
else:
states_list.append(Qobj(rho_t)) # dito
for j in range(N_substeps):
drho_t = spmv(
L.data.data, L.data.indices, L.data.indptr, rho_t) * dt
for a_idx, A in enumerate(A_ops):
drho_t += rhs(
L.data, rho_t, A, dt, dW[a_idx, t_idx, j], d1, d2)
rho_t += drho_t
return states_list
def np_to_kraus(U):
from scipy.linalg import eig
from qutip.superoperator import vec2mat
vals, vecs = eig(U)
vecs = [np.array(_) for _ in zip(*vecs)]
return [np.sqrt(val) * vec2mat(vec) for val, vec in zip(vals, vecs)]
def _smesolve_single_trajectory(L, dt, tlist, N_store, N_substeps, rho_t, A_ops, e_ops, data, rhs, d1, d2):
"""
Internal function. See smesolve.
"""
dW = np.sqrt(dt) * scipy.randn(len(A_ops), N_store, N_substeps)
states_list = []
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
# XXX: need to keep hilbert space structure
data.expect[e_idx, t_idx] += expect(e, Qobj(vec2mat(rho_t)))
else:
states_list.append(Qobj(rho_t)) # dito
for j in range(N_substeps):
drho_t = spmv(L.data.data, L.data.indices, L.data.indptr, rho_t) * dt
for a_idx, A in enumerate(A_ops):
drho_t += rhs(L.data, rho_t, A, dt, dW[a_idx, t_idx, j], d1, d2)
rho_t += drho_t
return states_list
def qpt(U, op_basis_list):
"""
Calculate the quantum process tomography chi matrix for a given
(possibly nonunitary) transformation matrix U, which transforms a
density matrix in vector form according to:
vec(rho) = U * vec(rho0)
or
rho = vec2mat(U * mat2vec(rho0))
U can be calculated for an open quantum system using the QuTiP propagator
function.
"""
E_ops = []
# loop over all index permutations
for inds in index_permutations([len(op_list) for op_list in op_basis_list]):
# loop over all composite systems
E_op_list = [op_basis_list[k][inds[k]] for k in range(len(op_basis_list))]
E_ops.append(tensor(E_op_list))
EE_ops = [spre(E1) * spost(E2.dag()) for E1 in E_ops for E2 in E_ops]
M = hstack([mat2vec(EE.full()) for EE in EE_ops])
Uvec = mat2vec(U.full())
chi_vec = la.solve(M, Uvec)
return vec2mat(chi_vec)
def _steadystate_direct_sparse(L, ss_args):
"""
Direct solver that uses scipy sparse matrices
"""
if settings.debug:
print('Starting direct solver...')
dims = L.dims[0]
weight = np.abs(L.data.data.max())
n = prod(L.dims[0][0])
b = np.zeros((n ** 2, 1), dtype=complex)
b[0, 0] = weight
L = L.data + sp.csr_matrix(
(weight*np.ones(n), (np.zeros(n), [nn * (n + 1) for nn in range(n)])),
shape=(n ** 2, n ** 2))
L.sort_indices()
use_solver(assumeSortedIndices=True, useUmfpack=ss_args['use_umfpack'])
if ss_args['use_rcm']:
perm = symrcm(L)
L = sp_permute(L, perm, perm)
b = b[np.ix_(perm,)]
v = spsolve(L, b)
if ss_args['use_rcm']:
rev_perm = np.argsort(perm)
v = v[np.ix_(rev_perm,)]
data = vec2mat(v)
data = 0.5 * (data + data.conj().T)
return Qobj(data, dims=dims, isherm=True)
def gradient_QFI(self):
'''
Output:
1) updated values of self.control_coefficients
Notice: To run this function, the function 'propagation_single()' has to run first.
'''
num = len(self.times)
dH0 = 1.j * Qobj(liouvillian(self.Hamiltonian_derivative[0]).full())
rhomat_final = Qobj(vec2mat(self.rho[num - 1].full()))
drhomat_final = Qobj(vec2mat(self.rho_derivative[num - 1].full()))
SLD_final = CR.SLD(rhomat_final, drhomat_final)
dim = self.freeHamiltonian.dims[0][0]
dt = self.times[1] - self.times[0]
Hc_coeff = self.control_coefficients
D = self.propagator
for ki in range(0, len(self.control_Hamiltonian)):
Hk = 1.j * Qobj(liouvillian(self.control_Hamiltonian[ki]).full())
Hc_ki = Hc_coeff[ki]
for ti in range(0, num):
Mj1 = 1.j * D[ti + 1][num - 1] * Hk * self.rho[ti]
Mj2 = Qobj(np.zeros((dim * dim, 1)))
for ri in range(0, ti + 1):
Mj2 += D[ti + 1][num -
1] * Hk * D[ri +
1][ti] * dH0 * self.rho[ri]
Mj3 = Qobj(np.zeros((dim * dim, 1)))
for ri in range(ti + 1, num):
Mj3 += D[ri + 1][num -
1] * dH0 * D[ti +
1][ri] * Hk * self.rho[ti]
Mj1mat = Qobj(vec2mat(Mj1.full()))
Mj2mat = Qobj(vec2mat(Mj2.full()))
Mj3mat = Qobj(vec2mat(Mj3.full()))
term1 = dt * (SLD_final * SLD_final * Mj1mat).tr()
term2 = -2 * (dt * dt) * (SLD_final * (Mj2mat + Mj3mat)).tr()
delta = np.real(term1 + term2)
Hc_ki[ti] += self.epsilon * delta
Hc_coeff[ki] = Hc_ki
self.control_coefficients = Hc_coeff
def choi_to_kraus(q_oper):
"""
Takes a Choi matrix and returns a list of Kraus operators.
TODO: Create a new class structure for quantum channels, perhaps as a
strict sub-class of Qobj.
"""
vals, vecs = eig(q_oper.data.todense())
vecs = [array(_) for _ in zip(*vecs)]
return [Qobj(inpt=sqrt(val)*vec2mat(vec)) for val, vec in zip(vals, vecs)]
def __call__(self,
H,
state,
dt,
c_ops=None,
backwards=False,
initialize=False):
"""Evaluation of a single propagation step
Args:
H (list): A Liouvillian superoperator in qutip's nested-list
format, with a scalar value in the place of a time-dependency.
For example, ``[L0, [L1, u]]`` for a drift Liouvillian ``L0``,
a control Liouvillian ``H1``, and a scalar value ``u`` that is
a time-dependent control evaluated for a particular point in
time. If `initialize` is False, only the control values are
taken into account; any operators are assumed to be identical
to the internally cached values of `H` during initialization.
state (qutip.Qobj): The density matrix to propagate. The passed
value is ignored unless `initialize` is given as True.
Otherwise, it is assumed that `state` matches the (internally
stored) state that was the result from the previous propagation
step.
dt (float): The time step over which to propagate
c_ops (list or None): An empty list, or None. Since this propagator
assumes a full Liouvillian, it cannot be combined with Lindblad
operators.
backwards (bool): Whether the propagation is forward in time or
backward in time. Since the equation of motion for a
Liouvillian and conjugate Liouvillian is the same, this
parameter has no effect. Instead, for the backward propagation,
the conjugate Liouvillian must be passed for `L`.
initialize (bool): Whether to (re-)initialize for a new
propagation. This caches `H` (except for the control values)
and `state` internally.
"""
# H is really an L, but it's a very bad idea for a subclass not to
# follow the interface of the parent (including argument names).
# Internally, however, we'll use L instead of H
if initialize or self.reentrant:
self._initialize(H, state, dt, c_ops, backwards)
else:
if self.reentrant:
self._initialize_integrator(self._y)
# only update the control values
for i in self._control_indices:
self._L_list[i][1] = H[i][1]
self._t += dt
self._r.integrate(self._t)
self._y = self._r.y
return qutip.Qobj(
dense2D_to_fastcsr_fmode(vec2mat(self._y), state.shape[0],
state.shape[1]),
dims=state.dims,
isherm=True,
)
def _smesolve_single_trajectory(L, dt, tlist, N_store, N_substeps, rho_t,
A_ops, e_ops, m_ops, data, rhs, d1, d2, d2_len,
homogeneous, distribution, args,
store_measurement=False,
store_states=False, noise=None):
"""
Internal function. See smesolve.
"""
if noise is None:
if homogeneous:
if distribution == 'normal':
dW = np.sqrt(dt) * scipy.randn(len(A_ops), N_store, N_substeps, d2_len)
else:
raise TypeError('Unsupported increment distribution for homogeneous process.')
else:
if distribution != 'poisson':
raise TypeError('Unsupported increment distribution for inhomogeneous process.')
dW = np.zeros((len(A_ops), N_store, N_substeps, d2_len))
else:
dW = noise
states_list = []
measurements = np.zeros((len(tlist), len(m_ops)), dtype=complex)
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
s = expect_rho_vec(e.data, rho_t)
data.expect[e_idx, t_idx] += s
data.ss[e_idx, t_idx] += s ** 2
if store_states or not e_ops:
# XXX: need to keep hilbert space structure
states_list.append(Qobj(vec2mat(rho_t)))
rho_prev = np.copy(rho_t)
for j in range(N_substeps):
if noise is None and not homogeneous:
for a_idx, A in enumerate(A_ops):
dw_expect = np.real(cy_expect_rho_vec(A[4], rho_t)) * dt
dW[a_idx, t_idx, j, :] = np.random.poisson(dw_expect, d2_len)
rho_t = rhs(L.data, rho_t, t + dt * j,
A_ops, dt, dW[:, t_idx, j, :], d1, d2, args)
if store_measurement:
for m_idx, m in enumerate(m_ops):
# TODO: allow using more than one increment
measurements[t_idx, m_idx] = cy_expect_rho_vec(m.data, rho_prev) * dt * N_substeps + dW[m_idx, t_idx, :, 0].sum()
return states_list, dW, measurements
def _smesolve_single_trajectory(L, dt, tlist, N_store, N_substeps, rho_t,
A_ops, e_ops, data, rhs, d1, d2, d2_len,
homogeneous, distribution, args,
store_measurement=False,
store_states=False, noise=None):
"""
Internal function. See smesolve.
"""
if noise is None:
if homogeneous:
if distribution == 'normal':
dW = np.sqrt(dt) * scipy.randn(len(A_ops), N_store, N_substeps, d2_len)
else:
raise TypeError('Unsupported increment distribution for homogeneous process.')
else:
if distribution != 'poisson':
raise TypeError('Unsupported increment distribution for inhomogeneous process.')
dW = np.zeros((len(A_ops), N_store, N_substeps, d2_len))
else:
dW = noise
states_list = []
measurements = np.zeros((len(tlist), len(A_ops)), dtype=complex)
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
s = expect_rho_vec(e.data, rho_t)
data.expect[e_idx, t_idx] += s
data.ss[e_idx, t_idx] += s ** 2
if store_states or not e_ops:
# XXX: need to keep hilbert space structure
states_list.append(Qobj(vec2mat(rho_t)))
rho_prev = np.copy(rho_t)
for j in range(N_substeps):
if noise is None and not homogeneous:
for a_idx, A in enumerate(A_ops):
dw_expect = np.real(cy_expect_rho_vec(A[4], rho_t)) * dt
dW[a_idx, t_idx, j, :] = np.random.poisson(dw_expect, d2_len)
rho_t = rhs(L.data, rho_t, t + dt * j,
A_ops, dt, dW[:, t_idx, j, :], d1, d2, args)
if store_measurement:
for a_idx, A in enumerate(A_ops):
measurements[t_idx, a_idx] = cy_expect_rho_vec(A[0], rho_prev) * dt * N_substeps + dW[a_idx, t_idx, :, 0].sum()
return states_list, dW, measurements
def choi_to_kraus(q_oper):
"""
Takes a Choi matrix and returns a list of Kraus operators.
TODO: Create a new class structure for quantum channels, perhaps as a
strict sub-class of Qobj.
"""
vals, vecs = eig(q_oper.data.todense())
vecs = [array(_) for _ in zip(*vecs)]
return [
Qobj(inpt=sqrt(val) * vec2mat(vec)) for val, vec in zip(vals, vecs)
]
def choi_to_kraus(q_oper):
"""
Takes a Choi matrix and returns a list of Kraus operators.
# TODO Create a new class structure for quantum channels, perhaps as a
strict sub-class of Qobj.
"""
vals, vecs = eig(q_oper.data.todense())
vecs = list(map(array, zip(*vecs)))
return list(
map(lambda x: Qobj(inpt=x),
[sqrt(vals[j]) * vec2mat(vecs[j]) for j in range(len(vals))]))
def choi_to_kraus(q_oper):
"""
Takes a Choi matrix and returns a list of Kraus operators.
TODO: Create a new class structure for quantum channels, perhaps as a
strict sub-class of Qobj.
"""
vals, vecs = eig(q_oper.data.todense())
vecs = list(map(array, zip(*vecs)))
return list(map(lambda x: Qobj(inpt=x),
[sqrt(vals[j]) * vec2mat(vecs[j])
for j in range(len(vals))]))
def choi_to_kraus(q_oper, tol=1e-9):
"""
Takes a Choi matrix and returns a list of Kraus operators.
TODO: Create a new class structure for quantum channels, perhaps as a
strict sub-class of Qobj.
"""
vals, vecs = eig(q_oper.data.todense())
vecs = [array(_) for _ in zip(*vecs)]
shape = [np.prod(q_oper.dims[0][i]) for i in range(2)][::-1]
return [Qobj(inpt=sqrt(val)*vec2mat(vec, shape=shape),
dims=q_oper.dims[0][::-1])
for val, vec in zip(vals, vecs) if abs(val) >= tol]
def _steadystate_iterative(L, ss_args):
"""
Iterative steady state solver using the LGMRES algorithm
and a sparse incomplete LU preconditioner.
"""
if settings.debug:
print('Starting GMRES solver...')
dims = L.dims[0]
n = prod(L.dims[0][0])
b = np.zeros(n ** 2)
b[0] = 1.0
L = L.data.tocsc() + sp.csc_matrix(
(1e-1 * np.ones(n), (np.zeros(n), [nn * (n + 1) for nn in range(n)])),
shape=(n ** 2, n ** 2))
if ss_args['use_rcm']:
if settings.debug:
print('Original bandwidth ', sp_bandwidth(L))
perm = symrcm(L)
rev_perm = np.argsort(perm)
L = sp_permute(L, perm, perm, 'csc')
b = b[np.ix_(perm,)]
if settings.debug:
print('RCM bandwidth ', sp_bandwidth(L))
L.sort_indices()
if ss_args['M'] is None and ss_args['use_precond']:
M = _iterative_precondition(L, n, ss_args['drop_tol'],
ss_args['diag_pivot_thresh'],
ss_args['fill_factor'])
v, check = gmres(L, b, tol=ss_args['tol'], M=ss_args['M'], maxiter=ss_args['maxiter'])
if check > 0:
raise Exception("Steadystate solver did not reach tolerance after " +
str(check) + " steps.")
elif check < 0:
raise Exception(
"Steadystate solver failed with fatal error: " + str(check) + ".")
if ss_args['use_rcm']:
v = v[np.ix_(rev_perm,)]
data = vec2mat(v)
data = 0.5 * (data + data.conj().T)
return Qobj(data, dims=dims, isherm=True)
def _steadystate_eigen(L, ss_args):
"""
Internal function for solving the steady state problem by
finding the eigenvector corresponding to the zero eigenvalue
of the Liouvillian using ARPACK.
"""
ss_args['info'].pop('weight', None)
if settings.debug:
logger.debug('Starting Eigen solver.')
dims = L.dims[0]
L = L.data.tocsc()
if ss_args['use_rcm']:
ss_args['info']['perm'].append('rcm')
if settings.debug:
old_band = sp_bandwidth(L)[0]
logger.debug('Original bandwidth: %i' % old_band)
perm = sp.csgraph.reverse_cuthill_mckee(L)
rev_perm = np.argsort(perm)
L = sp_permute(L, perm, perm, 'csc')
if settings.debug:
rcm_band = sp_bandwidth(L)[0]
logger.debug('RCM bandwidth: %i' % rcm_band)
logger.debug('Bandwidth reduction factor: %f' %
(old_band / rcm_band))
_eigen_start = time.time()
eigval, eigvec = eigs(L,
k=1,
sigma=1e-15,
tol=ss_args['tol'],
which='LM',
maxiter=ss_args['maxiter'])
_eigen_end = time.time()
ss_args['info']['solution_time'] = _eigen_end - _eigen_start
if ss_args['return_info']:
ss_args['info']['residual_norm'] = la.norm(L * eigvec, np.inf)
if ss_args['use_rcm']:
eigvec = eigvec[np.ix_(rev_perm, )]
_temp = vec2mat(eigvec)
data = dense2D_to_fastcsr_fmode(_temp, _temp.shape[0], _temp.shape[1])
data = 0.5 * (data + data.H)
out = Qobj(data, dims=dims, isherm=True)
if ss_args['return_info']:
return out / out.tr(), ss_args['info']
else:
return out / out.tr()
def test_SimpleSuperApply(self):
"""
Non-composite system, operator on Liouville space.
"""
rho_3 = rand_dm(3)
superop = kraus_to_super(rand_kraus_map(3))
analytic_result = vec2mat(superop.data.todense() *
mat2vec(rho_3.data.todense()))
naive_result = subsystem_apply(rho_3, superop, [True], reference=True)
naive_diff = (analytic_result - naive_result).data.todense()
assert_(norm(naive_diff) < 1e-12)
efficient_result = subsystem_apply(rho_3, superop, [True])
efficient_diff = (efficient_result - analytic_result).data.todense()
assert_(norm(efficient_diff) < 1e-12)
def _steadystate_eigen(L, ss_args):
"""
Internal function for solving the steady state problem by
finding the eigenvector corresponding to the zero eigenvalue
of the Liouvillian using ARPACK.
"""
ss_args['info'].pop('weight', None)
if settings.debug:
print('Starting Eigen solver...')
dims = L.dims[0]
shape = prod(dims[0])
L = L.data.tocsc()
if ss_args['use_rcm']:
ss_args['info']['perm'].append('rcm')
if settings.debug:
old_band = sp_bandwidth(L)[0]
print('Original bandwidth:', old_band)
perm = reverse_cuthill_mckee(L)
rev_perm = np.argsort(perm)
L = sp_permute(L, perm, perm, 'csc')
if settings.debug:
rcm_band = sp_bandwidth(L)[0]
print('RCM bandwidth:', rcm_band)
print('Bandwidth reduction factor:', round(old_band / rcm_band, 1))
_eigen_start = time.time()
eigval, eigvec = eigs(L,
k=1,
sigma=1e-15,
tol=ss_args['tol'],
which='LM',
maxiter=ss_args['maxiter'])
_eigen_end = time.time()
ss_args['info']['solution_time'] = _eigen_end - _eigen_start
if ss_args['use_rcm']:
eigvec = eigvec[np.ix_(rev_perm, )]
data = vec2mat(eigvec)
data = 0.5 * (data + data.conj().T)
out = Qobj(data, dims=dims, isherm=True)
if ss_args['return_info']:
return out / out.tr(), ss_args['info']
else:
return out / out.tr()
def test_SimpleSuperApply(self):
"""
Non-composite system, operator on Liouville space.
"""
rho_3 = rand_dm(3)
superop = kraus_to_super(rand_kraus_map(3))
analytic_result = vec2mat(superop.data.todense() *
mat2vec(rho_3.data.todense()))
naive_result = subsystem_apply(rho_3, superop, [True],
reference=True)
naive_diff = (analytic_result - naive_result).data.todense()
assert_(norm(naive_diff) < 1e-12)
efficient_result = subsystem_apply(rho_3, superop, [True])
efficient_diff = (efficient_result - analytic_result).data.todense()
assert_(norm(efficient_diff) < 1e-12)
def _steadystate_eigen(L, ss_args):
"""
Internal function for solving the steady state problem by
finding the eigenvector corresponding to the zero eigenvalue
of the Liouvillian using ARPACK.
"""
ss_args['info'].pop('weight', None)
if settings.debug:
logger.debug('Starting Eigen solver.')
dims = L.dims[0]
L = L.data.tocsc()
if ss_args['use_rcm']:
ss_args['info']['perm'].append('rcm')
if settings.debug:
old_band = sp_bandwidth(L)[0]
logger.debug('Original bandwidth: %i' % old_band)
perm = reverse_cuthill_mckee(L)
rev_perm = np.argsort(perm)
L = sp_permute(L, perm, perm, 'csc')
if settings.debug:
rcm_band = sp_bandwidth(L)[0]
logger.debug('RCM bandwidth: %i' % rcm_band)
logger.debug('Bandwidth reduction factor: %f' %
(old_band/rcm_band))
_eigen_start = time.time()
eigval, eigvec = eigs(L, k=1, sigma=1e-15, tol=ss_args['tol'],
which='LM', maxiter=ss_args['maxiter'])
_eigen_end = time.time()
ss_args['info']['solution_time'] = _eigen_end - _eigen_start
if ss_args['return_info']:
ss_args['info']['residual_norm'] = la.norm(L*eigvec, np.inf)
if ss_args['use_rcm']:
eigvec = eigvec[np.ix_(rev_perm,)]
_temp = vec2mat(eigvec)
data = dense2D_to_fastcsr_fmode(_temp, _temp.shape[0], _temp.shape[1])
data = 0.5 * (data + data.H)
out = Qobj(data, dims=dims, isherm=True)
if ss_args['return_info']:
return out/out.tr(), ss_args['info']
else:
return out/out.tr()
def _steadystate_power(L, ss_args):
"""
Inverse power method for steady state solving.
"""
if settings.debug:
print('Starting iterative power method Solver...')
tol=ss_args['tol']
maxiter=ss_args['maxiter']
use_solver(assumeSortedIndices=True)
rhoss = Qobj()
sflag = issuper(L)
if sflag:
rhoss.dims = L.dims[0]
else:
rhoss.dims = [L.dims[0], 1]
n = prod(rhoss.shape)
L = L.data.tocsc() - (tol ** 2) * sp.eye(n, n, format='csc')
L.sort_indices()
v = mat2vec(rand_dm(rhoss.shape[0], 0.5 / rhoss.shape[0] + 0.5).full())
it = 0
while (la.norm(L * v, np.inf) > tol) and (it < maxiter):
v = spsolve(L, v)
v = v / la.norm(v, np.inf)
it += 1
if it >= maxiter:
raise Exception('Failed to find steady state after ' +
str(maxiter) + ' iterations')
# normalise according to type of problem
if sflag:
trow = sp.eye(rhoss.shape[0], rhoss.shape[0], format='coo')
trow = sp_reshape(trow, (1, n))
data = v / sum(trow.dot(v))
else:
data = data / la.norm(v)
data = sp.csr_matrix(vec2mat(data))
rhoss.data = 0.5 * (data + data.conj().T)
rhoss.isherm = True
return rhoss
def _steadystate_direct_dense(L):
"""
Direct solver that use numpy dense matrices. Suitable for
small system, with a few states.
"""
if settings.debug:
print('Starting direct dense solver...')
dims = L.dims[0]
n = prod(L.dims[0][0])
b = np.zeros(n ** 2)
b[0] = ss_args['weight']
L = L.data.todense()
L[0, :] = np.diag(ss_args['weight']*np.ones(n)).reshape((1, n ** 2))
v = np.linalg.solve(L, b)
data = vec2mat(v)
data = 0.5 * (data + data.conj().T)
return Qobj(data, dims=dims, isherm=True)
def test_SimpleSuperApply(self):
"""
Non-composite system, operator on Liouville space.
"""
tol = 1e-12
rho_3 = rand_dm(3)
superop = kraus_to_super(rand_kraus_map(3))
analytic_result = vec2mat(superop.full() @ mat2vec(rho_3.full()))
naive_result = subsystem_apply(rho_3, superop, [True], reference=True)
naive_diff = (analytic_result - naive_result).full()
naive_diff_norm = norm(naive_diff)
assert_(naive_diff_norm < tol,
msg="SimpleSuper: naive_diff_norm {} "
"is beyond tolerance {}".format(naive_diff_norm, tol))
efficient_result = subsystem_apply(rho_3, superop, [True])
efficient_diff = (efficient_result - analytic_result).full()
efficient_diff_norm = norm(efficient_diff)
assert_(efficient_diff_norm < tol,
msg="SimpleSuper: efficient_diff_norm {} "
"is beyond tolerance {}".format(efficient_diff_norm, tol))
def _steadystate_direct_dense(L):
"""
Direct solver that use numpy dense matrices. Suitable for
small system, with a few states.
"""
if settings.debug:
print('Starting direct dense solver...')
dims = L.dims[0]
n = prod(L.dims[0][0])
b = np.zeros(n**2)
b[0] = ss_args['weight']
L = L.data.todense()
L[0, :] = np.diag(ss_args['weight'] * np.ones(n)).reshape((1, n**2))
_dense_start = time.time()
v = np.linalg.solve(L, b)
_dense_end = time.time()
ss_args['info']['solution_time'] = _dense_end - _dense_start
data = vec2mat(v)
data = 0.5 * (data + data.conj().T)
return Qobj(data, dims=dims, isherm=True)
def integrate(L,E0,ti,tf,opt=qt.Options()):
"""
Basic ode integrator
"""
def _rhs(t,y,L):
ym = y.reshape(L.shape)
return (L*ym).flatten()
from qutip.superoperator import vec2mat
r = sp.integrate.ode(_rhs)
r.set_f_params(L.data)
initial_vector = E0.data.toarray().flatten()
r.set_integrator('zvode', method=opt.method, order=opt.order,
atol=opt.atol, rtol=opt.rtol, nsteps=opt.nsteps,
first_step=opt.first_step, min_step=opt.min_step,
max_step=opt.max_step)
r.set_initial_value(initial_vector, ti)
r.integrate(tf)
if not r.successful():
raise Exception("ODE integration error: Try to increase "
"the allowed number of substeps by increasing "
"the nsteps parameter in the Options class.")
return qt.Qobj(vec2mat(r.y)).trans()
def test_SimpleSuperApply(self):
"""
Non-composite system, operator on Liouville space.
"""
tol = 1e-12
rho_3 = rand_dm(3)
superop = kraus_to_super(rand_kraus_map(3))
analytic_result = vec2mat(superop.data.todense() * mat2vec(rho_3.data.todense()))
naive_result = subsystem_apply(rho_3, superop, [True], reference=True)
naive_diff = (analytic_result - naive_result).data.todense()
naive_diff_norm = norm(naive_diff)
assert_(
naive_diff_norm < tol,
msg="SimpleSuper: naive_diff_norm {} " "is beyond tolerance {}".format(naive_diff_norm, tol),
)
efficient_result = subsystem_apply(rho_3, superop, [True])
efficient_diff = (efficient_result - analytic_result).data.todense()
efficient_diff_norm = norm(efficient_diff)
assert_(
efficient_diff_norm < tol,
msg="SimpleSuper: efficient_diff_norm {} " "is beyond tolerance {}".format(efficient_diff_norm, tol),
)
def ode2es(L, rho0):
"""Creates an exponential series that describes the time evolution for the
initial density matrix (or state vector) `rho0`, given the Liouvillian
(or Hamiltonian) `L`.
.. deprecated:: 4.6.0
:obj:`~ode2es` will be removed in QuTiP 5. Please use
:obj:`Qobj.eigenstates` to get the eigenstates and -values, and use
:obj:`~QobjEvo` for general time-dependence.
Parameters
----------
L : qobj
Liouvillian of the system.
rho0 : qobj
Initial state vector or density matrix.
Returns
-------
eseries : :class:`qutip.eseries`
``eseries`` represention of the system dynamics.
"""
if issuper(L):
# check initial state
if isket(rho0):
# Got a wave function as initial state: convert to density matrix.
rho0 = rho0 * rho0.dag()
# check if state is below error threshold
if abs(rho0.full()).sum() < 1e-10 + 1e-24:
# enforce zero operator
return eseries(qzero(rho0.dims[0]))
w, v = L.eigenstates()
v = np.hstack([ket.full() for ket in v])
# w[i] = eigenvalue i
# v[:,i] = eigenvector i
rlen = np.prod(rho0.shape)
r0 = mat2vec(rho0.full())
v0 = la.solve(v, r0)
vv = v * sp.spdiags(v0.T, 0, rlen, rlen)
out = None
for i in range(rlen):
qo = Qobj(vec2mat(vv[:, i]), dims=rho0.dims, shape=rho0.shape)
if out:
out += eseries(qo, w[i])
else:
out = eseries(qo, w[i])
elif isoper(L):
if not isket(rho0):
raise TypeError('Second argument must be a ket if first' +
'is a Hamiltonian.')
# check if state is below error threshold
if abs(rho0.full()).sum() < 1e-5 + 1e-20:
# enforce zero operator
dims = rho0.dims
return eseries(
Qobj(sp.csr_matrix((dims[0][0], dims[1][0]), dtype=complex)))
w, v = L.eigenstates()
v = np.hstack([ket.full() for ket in v])
# w[i] = eigenvalue i
# v[:,i] = eigenvector i
rlen = np.prod(rho0.shape)
r0 = rho0.full()
v0 = la.solve(v, r0)
vv = v * sp.spdiags(v0.T, 0, rlen, rlen)
out = None
for i in range(rlen):
qo = Qobj(np.array(vv[:, i]).T, dims=rho0.dims, shape=rho0.shape)
if out:
out += eseries(qo, -1.0j * w[i])
else:
out = eseries(qo, -1.0j * w[i])
else:
raise TypeError('First argument must be a Hamiltonian or Liouvillian.')
return estidy(out)
def _td_brmesolve(H, psi0, tlist, a_ops=[], e_ops=[], c_ops=[], args={},
use_secular=True, sec_cutoff=0.1,
tol=qset.atol, options=None,
progress_bar=None,_safe_mode=True,
verbose=False,
_prep_time=0):
if isket(psi0):
rho0 = ket2dm(psi0)
else:
rho0 = psi0
nrows = rho0.shape[0]
H_terms = []
H_td_terms = []
H_obj = []
A_terms = []
A_td_terms = []
C_terms = []
C_td_terms = []
CA_obj = []
spline_count = [0,0]
coupled_ops = []
coupled_lengths = []
coupled_spectra = []
if isinstance(H, Qobj):
H_terms.append(H.full('f'))
H_td_terms.append('1')
else:
for kk, h in enumerate(H):
if isinstance(h, Qobj):
H_terms.append(h.full('f'))
H_td_terms.append('1')
elif isinstance(h, list):
H_terms.append(h[0].full('f'))
if isinstance(h[1], Cubic_Spline):
H_obj.append(h[1].coeffs)
spline_count[0] += 1
H_td_terms.append(h[1])
else:
raise Exception('Invalid Hamiltonian specification.')
for kk, c in enumerate(c_ops):
if isinstance(c, Qobj):
C_terms.append(c.full('f'))
C_td_terms.append('1')
elif isinstance(c, list):
C_terms.append(c[0].full('f'))
if isinstance(c[1], Cubic_Spline):
CA_obj.append(c[1].coeffs)
spline_count[0] += 1
C_td_terms.append(c[1])
else:
raise Exception('Invalid collapse operator specification.')
coupled_offset = 0
for kk, a in enumerate(a_ops):
if isinstance(a, list):
if isinstance(a[0], Qobj):
A_terms.append(a[0].full('f'))
A_td_terms.append(a[1])
if isinstance(a[1], tuple):
if not len(a[1])==2:
raise Exception('Tuple must be len=2.')
if isinstance(a[1][0],Cubic_Spline):
spline_count[1] += 1
if isinstance(a[1][1],Cubic_Spline):
spline_count[1] += 1
elif isinstance(a[0], tuple):
if not isinstance(a[1], tuple):
raise Exception('Invalid bath-coupling specification.')
if (len(a[0])+1) != len(a[1]):
raise Exception('BR a_ops tuple lengths not compatible.')
coupled_ops.append(kk+coupled_offset)
coupled_lengths.append(len(a[0]))
coupled_spectra.append(a[1][0])
coupled_offset += len(a[0])-1
if isinstance(a[1][0],Cubic_Spline):
spline_count[1] += 1
for nn, _a in enumerate(a[0]):
A_terms.append(_a.full('f'))
A_td_terms.append(a[1][nn+1])
if isinstance(a[1][nn+1],Cubic_Spline):
CA_obj.append(a[1][nn+1].coeffs)
spline_count[1] += 1
else:
raise Exception('Invalid bath-coupling specification.')
string_list = []
for kk,_ in enumerate(H_td_terms):
string_list.append("H_terms[{0}]".format(kk))
for kk,_ in enumerate(H_obj):
string_list.append("H_obj[{0}]".format(kk))
for kk,_ in enumerate(C_td_terms):
string_list.append("C_terms[{0}]".format(kk))
for kk,_ in enumerate(CA_obj):
string_list.append("CA_obj[{0}]".format(kk))
for kk,_ in enumerate(A_td_terms):
string_list.append("A_terms[{0}]".format(kk))
#Add nrows to parameters
string_list.append('nrows')
for name, value in args.items():
if isinstance(value, np.ndarray):
raise TypeError('NumPy arrays not valid args for BR solver.')
else:
string_list.append(str(value))
parameter_string = ",".join(string_list)
if verbose:
print('BR prep time:', time.time()-_prep_time)
#
# generate and compile new cython code if necessary
#
if not options.rhs_reuse or config.tdfunc is None:
if options.rhs_filename is None:
config.tdname = "rhs" + str(os.getpid()) + str(config.cgen_num)
else:
config.tdname = opt.rhs_filename
if verbose:
_st = time.time()
cgen = BR_Codegen(h_terms=len(H_terms),
h_td_terms=H_td_terms, h_obj=H_obj,
c_terms=len(C_terms),
c_td_terms=C_td_terms, c_obj=CA_obj,
a_terms=len(A_terms), a_td_terms=A_td_terms,
spline_count=spline_count,
coupled_ops = coupled_ops,
coupled_lengths = coupled_lengths,
coupled_spectra = coupled_spectra,
config=config, sparse=False,
use_secular = use_secular,
sec_cutoff = sec_cutoff,
args=args,
use_openmp=options.use_openmp,
omp_thresh=qset.openmp_thresh if qset.has_openmp else None,
omp_threads=options.num_cpus,
atol=tol)
cgen.generate(config.tdname + ".pyx")
code = compile('from ' + config.tdname + ' import cy_td_ode_rhs',
'<string>', 'exec')
exec(code, globals())
config.tdfunc = cy_td_ode_rhs
if verbose:
print('BR compile time:', time.time()-_st)
initial_vector = mat2vec(rho0.full()).ravel()
_ode = scipy.integrate.ode(config.tdfunc)
code = compile('_ode.set_f_params(' + parameter_string + ')',
'<string>', 'exec')
_ode.set_integrator('zvode', method=options.method,
order=options.order, atol=options.atol,
rtol=options.rtol, nsteps=options.nsteps,
first_step=options.first_step,
min_step=options.min_step,
max_step=options.max_step)
_ode.set_initial_value(initial_vector, tlist[0])
exec(code, locals())
#
# prepare output array
#
n_tsteps = len(tlist)
e_sops_data = []
output = Result()
output.solver = "brmesolve"
output.times = tlist
if options.store_states:
output.states = []
if isinstance(e_ops, types.FunctionType):
n_expt_op = 0
expt_callback = True
elif isinstance(e_ops, list):
n_expt_op = len(e_ops)
expt_callback = False
if n_expt_op == 0:
# fall back on storing states
output.states = []
options.store_states = True
else:
output.expect = []
output.num_expect = n_expt_op
for op in e_ops:
e_sops_data.append(spre(op).data)
if op.isherm:
output.expect.append(np.zeros(n_tsteps))
else:
output.expect.append(np.zeros(n_tsteps, dtype=complex))
else:
raise TypeError("Expectation parameter must be a list or a function")
#
# start evolution
#
if type(progress_bar)==BaseProgressBar and verbose:
_run_time = time.time()
progress_bar.start(n_tsteps)
rho = Qobj(rho0)
dt = np.diff(tlist)
for t_idx, t in enumerate(tlist):
progress_bar.update(t_idx)
if not _ode.successful():
raise Exception("ODE integration error: Try to increase "
"the allowed number of substeps by increasing "
"the nsteps parameter in the Options class.")
if options.store_states or expt_callback:
rho.data = dense2D_to_fastcsr_fmode(vec2mat(_ode.y), rho.shape[0], rho.shape[1])
if options.store_states:
output.states.append(Qobj(rho, isherm=True))
if expt_callback:
# use callback method
e_ops(t, rho)
for m in range(n_expt_op):
if output.expect[m].dtype == complex:
output.expect[m][t_idx] = expect_rho_vec(e_sops_data[m],
_ode.y, 0)
else:
output.expect[m][t_idx] = expect_rho_vec(e_sops_data[m],
_ode.y, 1)
if t_idx < n_tsteps - 1:
_ode.integrate(_ode.t + dt[t_idx])
progress_bar.finished()
if type(progress_bar)==BaseProgressBar and verbose:
print('BR runtime:', time.time()-_run_time)
if (not options.rhs_reuse) and (config.tdname is not None):
_cython_build_cleanup(config.tdname)
if options.store_final_state:
rho.data = dense2D_to_fastcsr_fmode(vec2mat(_ode.y), rho.shape[0], rho.shape[1])
output.final_state = Qobj(rho, dims=rho0.dims, isherm=True)
return output
def _steadystate_power(L, ss_args):
"""
Inverse power method for steady state solving.
"""
ss_args['info'].pop('weight', None)
if settings.debug:
logger.debug('Starting iterative inverse-power method solver.')
tol = ss_args['tol']
mtol = ss_args['mtol']
if mtol is None:
mtol = max(0.1 * tol, 1e-15)
maxiter = ss_args['maxiter']
use_solver(assumeSortedIndices=True)
rhoss = Qobj()
sflag = issuper(L)
if sflag:
rhoss.dims = L.dims[0]
else:
rhoss.dims = [L.dims[0], 1]
n = L.shape[0]
# Build Liouvillian
if ss_args['solver'] == 'mkl' and ss_args['method'] == 'power':
has_mkl = 1
else:
has_mkl = 0
L, perm, perm2, rev_perm, ss_args = _steadystate_power_liouvillian(
L, ss_args, has_mkl)
orig_nnz = L.nnz
# start with all ones as RHS
v = np.ones(n, dtype=complex)
if ss_args['use_rcm']:
v = v[np.ix_(perm2, )]
# Do preconditioning
if ss_args['solver'] == 'scipy':
if ss_args['M'] is None and ss_args['use_precond'] and \
ss_args['method'] in ['power-gmres',
'power-lgmres',
'power-bicgstab']:
ss_args['M'], ss_args = _iterative_precondition(
L, int(np.sqrt(n)), ss_args)
if ss_args['M'] is None:
warnings.warn("Preconditioning failed. Continuing without.",
UserWarning)
ss_iters = {'iter': 0}
def _iter_count(r):
ss_iters['iter'] += 1
return
_power_start = time.time()
# Get LU factors
if ss_args['method'] == 'power':
if ss_args['solver'] == 'mkl':
lu = mkl_splu(L,
max_iter_refine=ss_args['max_iter_refine'],
scaling_vectors=ss_args['scaling_vectors'],
weighted_matching=ss_args['weighted_matching'])
else:
lu = splu(L,
permc_spec=ss_args['permc_spec'],
diag_pivot_thresh=ss_args['diag_pivot_thresh'],
options=dict(ILU_MILU=ss_args['ILU_MILU']))
if settings.debug and _scipy_check:
L_nnz = lu.L.nnz
U_nnz = lu.U.nnz
logger.debug('L NNZ: %i ; U NNZ: %i' % (L_nnz, U_nnz))
logger.debug('Fill factor: %f' % ((L_nnz + U_nnz) / orig_nnz))
it = 0
# FIXME: These atol keyword except checks can be removed once scipy 1.1
# is a minimum requirement
while (la.norm(L * v, np.inf) > tol) and (it < maxiter):
check = 0
if ss_args['method'] == 'power':
v = lu.solve(v)
elif ss_args['method'] == 'power-gmres':
try:
v, check = gmres(L,
v,
tol=mtol,
atol=ss_args['matol'],
M=ss_args['M'],
x0=ss_args['x0'],
restart=ss_args['restart'],
maxiter=ss_args['maxiter'],
callback=_iter_count)
except TypeError as e:
if "unexpected keyword argument 'atol'" in str(e):
v, check = gmres(L,
v,
tol=mtol,
M=ss_args['M'],
x0=ss_args['x0'],
restart=ss_args['restart'],
maxiter=ss_args['maxiter'],
callback=_iter_count)
elif ss_args['method'] == 'power-lgmres':
try:
v, check = lgmres(L,
v,
tol=mtol,
atol=ss_args['matol'],
M=ss_args['M'],
x0=ss_args['x0'],
maxiter=ss_args['maxiter'],
callback=_iter_count)
except TypeError as e:
if "unexpected keyword argument 'atol'" in str(e):
v, check = lgmres(L,
v,
tol=mtol,
M=ss_args['M'],
x0=ss_args['x0'],
maxiter=ss_args['maxiter'],
callback=_iter_count)
elif ss_args['method'] == 'power-bicgstab':
try:
v, check = bicgstab(L,
v,
tol=mtol,
atol=ss_args['matol'],
M=ss_args['M'],
x0=ss_args['x0'],
maxiter=ss_args['maxiter'],
callback=_iter_count)
except TypeError as e:
if "unexpected keyword argument 'atol'" in str(e):
v, check = bicgstab(L,
v,
tol=mtol,
M=ss_args['M'],
x0=ss_args['x0'],
maxiter=ss_args['maxiter'],
callback=_iter_count)
else:
raise Exception("Invalid iterative solver method.")
if check > 0:
raise Exception("{} failed to find solution in "
"{} iterations.".format(ss_args['method'], check))
if check < 0:
raise Exception("Breakdown in {}".format(ss_args['method']))
v = v / la.norm(v, np.inf)
it += 1
if ss_args['method'] == 'power' and ss_args['solver'] == 'mkl':
lu.delete()
if ss_args['return_info']:
ss_args['info']['max_iter_refine'] = ss_args['max_iter_refine']
ss_args['info']['scaling_vectors'] = ss_args['scaling_vectors']
ss_args['info']['weighted_matching'] = ss_args['weighted_matching']
if it >= maxiter:
raise Exception('Failed to find steady state after ' + str(maxiter) +
' iterations')
_power_end = time.time()
ss_args['info']['solution_time'] = _power_end - _power_start
ss_args['info']['iterations'] = it
if ss_args['return_info']:
ss_args['info']['residual_norm'] = la.norm(L * v, np.inf)
if settings.debug:
logger.debug('Number of iterations: %i' % it)
if ss_args['use_rcm']:
v = v[np.ix_(rev_perm, )]
# normalise according to type of problem
if sflag:
trow = v[::rhoss.shape[0] + 1]
data = v / np.sum(trow)
else:
data = data / la.norm(v)
data = dense2D_to_fastcsr_fmode(vec2mat(data), rhoss.shape[0],
rhoss.shape[0])
rhoss.data = 0.5 * (data + data.H)
rhoss.isherm = True
if ss_args['return_info']:
return rhoss, ss_args['info']
else:
return rhoss
def floquet_markov_mesolve(R,
ekets,
rho0,
tlist,
e_ops,
f_modes_table=None,
options=None,
floquet_basis=True):
"""
Solve the dynamics for the system using the Floquet-Markov master equation.
"""
if options is None:
opt = Options()
else:
opt = options
if opt.tidy:
R.tidyup()
#
# check initial state
#
if isket(rho0):
# Got a wave function as initial state: convert to density matrix.
rho0 = ket2dm(rho0)
#
# prepare output array
#
n_tsteps = len(tlist)
dt = tlist[1] - tlist[0]
output = Result()
output.solver = "fmmesolve"
output.times = tlist
if isinstance(e_ops, FunctionType):
n_expt_op = 0
expt_callback = True
elif isinstance(e_ops, list):
n_expt_op = len(e_ops)
expt_callback = False
if n_expt_op == 0:
output.states = []
else:
if not f_modes_table:
raise TypeError("The Floquet mode table has to be provided " +
"when requesting expectation values.")
output.expect = []
output.num_expect = n_expt_op
for op in e_ops:
if op.isherm:
output.expect.append(np.zeros(n_tsteps))
else:
output.expect.append(np.zeros(n_tsteps, dtype=complex))
else:
raise TypeError("Expectation parameter must be a list or a function")
#
# transform the initial density matrix to the eigenbasis: from
# computational basis to the floquet basis
#
if ekets is not None:
rho0 = rho0.transform(ekets)
#
# setup integrator
#
initial_vector = mat2vec(rho0.full())
r = scipy.integrate.ode(cy_ode_rhs)
r.set_f_params(R.data.data, R.data.indices, R.data.indptr)
r.set_integrator('zvode',
method=opt.method,
order=opt.order,
atol=opt.atol,
rtol=opt.rtol,
max_step=opt.max_step)
r.set_initial_value(initial_vector, tlist[0])
#
# start evolution
#
rho = Qobj(rho0)
t_idx = 0
for t in tlist:
if not r.successful():
break
rho = Qobj(vec2mat(r.y), rho0.dims, rho0.shape)
if expt_callback:
# use callback method
if floquet_basis:
e_ops(t, Qobj(rho))
else:
f_modes_table_t, T = f_modes_table
f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T)
e_ops(t, Qobj(rho).transform(f_modes_t, True))
else:
# calculate all the expectation values, or output rho if
# no operators
if n_expt_op == 0:
if floquet_basis:
output.states.append(Qobj(rho))
else:
f_modes_table_t, T = f_modes_table
f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T)
output.states.append(Qobj(rho).transform(f_modes_t, True))
else:
f_modes_table_t, T = f_modes_table
f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T)
for m in range(0, n_expt_op):
output.expect[m][t_idx] = \
expect(e_ops[m], rho.transform(f_modes_t, False))
r.integrate(r.t + dt)
t_idx += 1
return output
def _steadystate_iterative(L, ss_args):
"""
Iterative steady state solver using the GMRES, LGMRES, or BICGSTAB
algorithm and a sparse incomplete LU preconditioner.
"""
ss_iters = {'iter': 0}
def _iter_count(r):
ss_iters['iter'] += 1
return
if settings.debug:
logger.debug('Starting %s solver.' % ss_args['method'])
dims = L.dims[0]
n = int(np.sqrt(L.shape[0]))
b = np.zeros(n**2)
b[0] = ss_args['weight']
L, perm, perm2, rev_perm, ss_args = _steadystate_LU_liouvillian(L, ss_args)
if np.any(perm):
b = b[np.ix_(perm, )]
if np.any(perm2):
b = b[np.ix_(perm2, )]
use_solver(assumeSortedIndices=True)
if ss_args['M'] is None and ss_args['use_precond']:
ss_args['M'], ss_args = _iterative_precondition(L, n, ss_args)
if ss_args['M'] is None:
warnings.warn("Preconditioning failed. Continuing without.",
UserWarning)
# Select iterative solver type
_iter_start = time.time()
# FIXME: These atol keyword except checks can be removed once scipy 1.1
# is a minimum requirement
extra = {"callback_type": 'legacy'} if scipy.__version__ >= "1.4" else {}
if ss_args['method'] == 'iterative-gmres':
try:
v, check = gmres(L,
b,
tol=ss_args['tol'],
atol=ss_args['matol'],
M=ss_args['M'],
x0=ss_args['x0'],
restart=ss_args['restart'],
maxiter=ss_args['maxiter'],
callback=_iter_count,
**extra)
except TypeError as e:
if "unexpected keyword argument 'atol'" in str(e):
v, check = gmres(L,
b,
tol=ss_args['tol'],
M=ss_args['M'],
x0=ss_args['x0'],
restart=ss_args['restart'],
maxiter=ss_args['maxiter'],
callback=_iter_count)
elif ss_args['method'] == 'iterative-lgmres':
try:
v, check = lgmres(L,
b,
tol=ss_args['tol'],
atol=ss_args['matol'],
M=ss_args['M'],
x0=ss_args['x0'],
maxiter=ss_args['maxiter'],
callback=_iter_count)
except TypeError as e:
if "unexpected keyword argument 'atol'" in str(e):
v, check = lgmres(L,
b,
tol=ss_args['tol'],
M=ss_args['M'],
x0=ss_args['x0'],
maxiter=ss_args['maxiter'],
callback=_iter_count)
elif ss_args['method'] == 'iterative-bicgstab':
try:
v, check = bicgstab(L,
b,
tol=ss_args['tol'],
atol=ss_args['matol'],
M=ss_args['M'],
x0=ss_args['x0'],
maxiter=ss_args['maxiter'],
callback=_iter_count)
except TypeError as e:
if "unexpected keyword argument 'atol'" in str(e):
v, check = bicgstab(L,
b,
tol=ss_args['tol'],
M=ss_args['M'],
x0=ss_args['x0'],
maxiter=ss_args['maxiter'],
callback=_iter_count)
else:
raise Exception("Invalid iterative solver method.")
_iter_end = time.time()
ss_args['info']['iter_time'] = _iter_end - _iter_start
if 'precond_time' in ss_args['info'].keys():
ss_args['info']['solution_time'] = (ss_args['info']['iter_time'] +
ss_args['info']['precond_time'])
else:
ss_args['info']['solution_time'] = ss_args['info']['iter_time']
ss_args['info']['iterations'] = ss_iters['iter']
if ss_args['return_info']:
ss_args['info']['residual_norm'] = la.norm(b - L * v, np.inf)
if settings.debug:
logger.debug('Number of Iterations: %i' % ss_iters['iter'])
logger.debug('Iteration. time: %f' % (_iter_end - _iter_start))
if check > 0:
raise Exception("Steadystate error: Did not reach tolerance after " +
str(ss_args['maxiter']) + " steps." +
"\nResidual norm: " +
str(ss_args['info']['residual_norm']))
elif check < 0:
raise Exception("Steadystate error: Failed with fatal error: " +
str(check) + ".")
if ss_args['use_rcm']:
v = v[np.ix_(rev_perm, )]
data = vec2mat(v)
data = 0.5 * (data + data.conj().T)
if ss_args['return_info']:
return Qobj(data, dims=dims, isherm=True), ss_args['info']
else:
return Qobj(data, dims=dims, isherm=True)
def propagator(H, t, c_op_list, H_args=None, opt=None):
"""
Calculate the propagator U(t) for the density matrix or wave function such
that :math:`\psi(t) = U(t)\psi(0)` or
:math:`\\rho_{\mathrm vec}(t) = U(t) \\rho_{\mathrm vec}(0)`
where :math:`\\rho_{\mathrm vec}` is the vector representation of the
density matrix.
Parameters
----------
H : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
H_args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
if opt is None:
opt = Odeoptions()
opt.rhs_reuse = True
tlist = [0, t] if isinstance(t, (int, float, np.int64, np.float64)) else t
if len(c_op_list) == 0:
# calculate propagator for the wave function
if isinstance(H, types.FunctionType):
H0 = H(0.0, H_args)
N = H0.shape[0]
dims = H0.dims
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
N = H0.shape[0]
dims = H0.dims
else:
N = H.shape[0]
dims = H.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
for n in range(0, N):
psi0 = basis(N, n)
output = mesolve(H, psi0, tlist, [], [], H_args, opt)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
# todo: evolving a batch of wave functions:
#psi_0_list = [basis(N, n) for n in range(N)]
#psi_t_list = mesolve(H, psi_0_list, [0, t], [], [], H_args, opt)
#for n in range(0, N):
# u[:,n] = psi_t_list[n][1].full().T
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
if isinstance(H, types.FunctionType):
H0 = H(0.0, H_args)
N = H0.shape[0]
dims = [H0.dims, H0.dims]
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
N = H0.shape[0]
dims = [H0.dims, H0.dims]
else:
N = H.shape[0]
dims = [H.dims, H.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
for n in range(0, N * N):
psi0 = basis(N * N, n)
rho0 = Qobj(vec2mat(psi0.full()))
output = mesolve(H, rho0, tlist, c_op_list, [], H_args, opt)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
if len(tlist) == 2:
return Qobj(u[:, :, 1], dims=dims)
else:
return [Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))]
def _generic_ode_solve(r, rho0, tlist, e_ops, opt, progress_bar):
"""
Internal function for solving ME. Solve an ODE which solver parameters
already setup (r). Calculate the required expectation values or invoke
callback function at each time step.
"""
#
# prepare output array
#
n_tsteps = len(tlist)
e_sops_data = []
output = Result()
output.solver = "mesolve"
output.times = tlist
if opt.store_states:
output.states = []
if isinstance(e_ops, types.FunctionType):
n_expt_op = 0
expt_callback = True
elif isinstance(e_ops, list):
n_expt_op = len(e_ops)
expt_callback = False
if n_expt_op == 0:
# fall back on storing states
output.states = []
opt.store_states = True
else:
output.expect = []
output.num_expect = n_expt_op
for op in e_ops:
e_sops_data.append(spre(op).data)
if op.isherm and rho0.isherm:
output.expect.append(np.zeros(n_tsteps))
else:
output.expect.append(np.zeros(n_tsteps, dtype=complex))
else:
raise TypeError("Expectation parameter must be a list or a function")
#
# start evolution
#
progress_bar.start(n_tsteps)
rho = Qobj(rho0)
dt = np.diff(tlist)
for t_idx, t in enumerate(tlist):
progress_bar.update(t_idx)
if not r.successful():
raise Exception("ODE integration error: Try to increase "
"the allowed number of substeps by increasing "
"the nsteps parameter in the Options class.")
if opt.store_states or expt_callback:
rho.data = dense2D_to_fastcsr_fmode(vec2mat(r.y), rho.shape[0], rho.shape[1])
if opt.store_states:
output.states.append(Qobj(rho, isherm=True))
if expt_callback:
# use callback method
e_ops(t, rho)
for m in range(n_expt_op):
if output.expect[m].dtype == complex:
output.expect[m][t_idx] = expect_rho_vec(e_sops_data[m],
r.y, 0)
else:
output.expect[m][t_idx] = expect_rho_vec(e_sops_data[m],
r.y, 1)
if t_idx < n_tsteps - 1:
r.integrate(r.t + dt[t_idx])
progress_bar.finished()
if (not opt.rhs_reuse) and (config.tdname is not None):
_cython_build_cleanup(config.tdname)
if opt.store_final_state:
rho.data = dense2D_to_fastcsr_fmode(vec2mat(r.y), rho.shape[0], rho.shape[1])
output.final_state = Qobj(rho, dims=rho0.dims, isherm=True)
return output
def get_curr_state_data(r):
return vec2mat(r.y)
def propagator(H, t, c_op_list, args=None, options=None, sparse=False):
"""
Calculate the propagator U(t) for the density matrix or wave function such
that :math:`\psi(t) = U(t)\psi(0)` or
:math:`\\rho_{\mathrm vec}(t) = U(t) \\rho_{\mathrm vec}(0)`
where :math:`\\rho_{\mathrm vec}` is the vector representation of the
density matrix.
Parameters
----------
H : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
if options is None:
options = Options()
options.rhs_reuse = True
rhs_clear()
if isinstance(t, (int, float, np.integer, np.floating)):
tlist = [0, t]
else:
tlist = t
if isinstance(H, (types.FunctionType, types.BuiltinFunctionType,
functools.partial)):
H0 = H(0.0, args)
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
else:
H0 = H
if len(c_op_list) == 0 and H0.isoper:
# calculate propagator for the wave function
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
for n in range(0, N):
psi0 = basis(N, n)
output = sesolve(H, psi0, tlist, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
# todo: evolving a batch of wave functions:
# psi_0_list = [basis(N, n) for n in range(N)]
# psi_t_list = mesolve(H, psi_0_list, [0, t], [], [], args, options)
# for n in range(0, N):
# u[:,n] = psi_t_list[n][1].full().T
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
for n in range(0, N):
psi0 = basis(N, n)
rho0 = Qobj(vec2mat(psi0.full()))
output = mesolve(H, rho0, tlist, [], [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if sparse:
for n in range(N * N):
psi0 = basis(N * N, n)
psi0.dims = [dims[0], 1]
rho0 = vector_to_operator(psi0)
output = mesolve(H, rho0, tlist, c_op_list, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = operator_to_vector(
output.states[k]).full(squeeze=True)
else:
for n in range(N * N):
psi0 = basis(N * N, n)
rho0 = Qobj(vec2mat(psi0.full()))
output = mesolve(H, rho0, tlist, c_op_list, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
if len(tlist) == 2:
return Qobj(u[:, :, 1], dims=dims)
else:
return [Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))]
def _steadystate_direct_sparse(L, ss_args):
"""
Direct solver that uses scipy sparse matrices
"""
if settings.debug:
logger.debug('Starting direct LU solver.')
dims = L.dims[0]
n = int(np.sqrt(L.shape[0]))
b = np.zeros(n**2, dtype=complex)
b[0] = ss_args['weight']
if ss_args['solver'] == 'mkl':
has_mkl = 1
else:
has_mkl = 0
ss_lu_liouv_list = _steadystate_LU_liouvillian(L, ss_args, has_mkl)
L, perm, perm2, rev_perm, ss_args = ss_lu_liouv_list
if np.any(perm):
b = b[np.ix_(perm, )]
if np.any(perm2):
b = b[np.ix_(perm2, )]
if ss_args['solver'] == 'scipy':
ss_args['info']['permc_spec'] = ss_args['permc_spec']
ss_args['info']['drop_tol'] = ss_args['drop_tol']
ss_args['info']['diag_pivot_thresh'] = ss_args['diag_pivot_thresh']
ss_args['info']['fill_factor'] = ss_args['fill_factor']
ss_args['info']['ILU_MILU'] = ss_args['ILU_MILU']
if not ss_args['solver'] == 'mkl':
# Use superLU solver
orig_nnz = L.nnz
_direct_start = time.time()
lu = splu(L,
permc_spec=ss_args['permc_spec'],
diag_pivot_thresh=ss_args['diag_pivot_thresh'],
options=dict(ILU_MILU=ss_args['ILU_MILU']))
v = lu.solve(b)
_direct_end = time.time()
ss_args['info']['solution_time'] = _direct_end - _direct_start
if (settings.debug or ss_args['return_info']) and _scipy_check:
L_nnz = lu.L.nnz
U_nnz = lu.U.nnz
ss_args['info']['l_nnz'] = L_nnz
ss_args['info']['u_nnz'] = U_nnz
ss_args['info']['lu_fill_factor'] = (L_nnz + U_nnz) / L.nnz
if settings.debug:
logger.debug('L NNZ: %i ; U NNZ: %i' % (L_nnz, U_nnz))
logger.debug('Fill factor: %f' % ((L_nnz + U_nnz) / orig_nnz))
else: # Use MKL solver
if len(ss_args['info']['perm']) != 0:
in_perm = np.arange(n**2, dtype=np.int32)
else:
in_perm = None
_direct_start = time.time()
v = mkl_spsolve(L,
b,
perm=in_perm,
verbose=ss_args['verbose'],
max_iter_refine=ss_args['max_iter_refine'],
scaling_vectors=ss_args['scaling_vectors'],
weighted_matching=ss_args['weighted_matching'])
_direct_end = time.time()
ss_args['info']['solution_time'] = _direct_end - _direct_start
if ss_args['return_info']:
ss_args['info']['residual_norm'] = la.norm(b - L * v, np.inf)
ss_args['info']['max_iter_refine'] = ss_args['max_iter_refine']
ss_args['info']['scaling_vectors'] = ss_args['scaling_vectors']
ss_args['info']['weighted_matching'] = ss_args['weighted_matching']
if ss_args['use_rcm']:
v = v[np.ix_(rev_perm, )]
data = dense2D_to_fastcsr_fmode(vec2mat(v), n, n)
data = 0.5 * (data + data.H)
if ss_args['return_info']:
return Qobj(data, dims=dims, isherm=True), ss_args['info']
else:
return Qobj(data, dims=dims, isherm=True)
def bloch_redfield_solve(R, ekets, rho0, tlist, e_ops=[], options=None):
"""
Evolve the ODEs defined by Bloch-Redfield master equation. The
Bloch-Redfield tensor can be calculated by the function
:func:`bloch_redfield_tensor`.
Parameters
----------
R : :class:`qutip.qobj`
Bloch-Redfield tensor.
ekets : array of :class:`qutip.qobj`
Array of kets that make up a basis tranformation for the eigenbasis.
rho0 : :class:`qutip.qobj`
Initial density matrix.
tlist : *list* / *array*
List of times for :math:`t`.
e_ops : list of :class:`qutip.qobj` / callback function
List of operators for which to evaluate expectation values.
options : :class:`qutip.Qdeoptions`
Options for the ODE solver.
Returns
-------
output: :class:`qutip.solver`
An instance of the class :class:`qutip.solver`, which contains either
an *array* of expectation values for the times specified by `tlist`.
"""
if options is None:
options = Options()
if options.tidy:
R.tidyup()
#
# check initial state
#
if isket(rho0):
# Got a wave function as initial state: convert to density matrix.
rho0 = rho0 * rho0.dag()
#
# prepare output array
#
n_tsteps = len(tlist)
dt = tlist[1] - tlist[0]
result_list = []
#
# transform the initial density matrix and the e_ops opterators to the
# eigenbasis
#
rho_eb = rho0.transform(ekets)
e_eb_ops = [e.transform(ekets) for e in e_ops]
for e_eb in e_eb_ops:
result_list.append(np.zeros(n_tsteps, dtype=complex))
#
# setup integrator
#
initial_vector = mat2vec(rho_eb.full())
r = scipy.integrate.ode(cy_ode_rhs)
r.set_f_params(R.data.data, R.data.indices, R.data.indptr)
r.set_integrator('zvode', method=options.method, order=options.order,
atol=options.atol, rtol=options.rtol,
nsteps=options.nsteps, first_step=options.first_step,
min_step=options.min_step, max_step=options.max_step)
r.set_initial_value(initial_vector, tlist[0])
#
# start evolution
#
dt = np.diff(tlist)
for t_idx, _ in enumerate(tlist):
if not r.successful():
break
rho_eb.data = vec2mat(r.y)
# calculate all the expectation values, or output rho_eb if no
# expectation value operators are given
if e_ops:
rho_eb_tmp = Qobj(rho_eb)
for m, e in enumerate(e_eb_ops):
result_list[m][t_idx] = expect(e, rho_eb_tmp)
else:
result_list.append(rho_eb.transform(ekets, True))
if t_idx < n_tsteps - 1:
r.integrate(r.t + dt[t_idx])
return result_list
def propagator(H,
t,
c_op_list,
args=None,
options=None,
sparse=False,
progress_bar=None):
"""
Calculate the propagator U(t) for the density matrix or wave function such
that :math:`\psi(t) = U(t)\psi(0)` or
:math:`\\rho_{\mathrm vec}(t) = U(t) \\rho_{\mathrm vec}(0)`
where :math:`\\rho_{\mathrm vec}` is the vector representation of the
density matrix.
Parameters
----------
H : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation. By default no progress bar
is used, and if set to True a TextProgressBar will be used.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
if options is None:
options = Options()
options.rhs_reuse = True
rhs_clear()
if isinstance(t, (int, float, np.integer, np.floating)):
tlist = [0, t]
else:
tlist = t
if isinstance(
H,
(types.FunctionType, types.BuiltinFunctionType, functools.partial)):
H0 = H(0.0, args)
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
else:
H0 = H
if len(c_op_list) == 0 and H0.isoper:
# calculate propagator for the wave function
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
psi0 = basis(N, n)
output = sesolve(H, psi0, tlist, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
progress_bar.finished()
# todo: evolving a batch of wave functions:
# psi_0_list = [basis(N, n) for n in range(N)]
# psi_t_list = mesolve(H, psi_0_list, [0, t], [], [], args, options)
# for n in range(0, N):
# u[:,n] = psi_t_list[n][1].full().T
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
psi0 = basis(N, n)
rho0 = Qobj(vec2mat(psi0.full()))
output = mesolve(H, rho0, tlist, [], [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if sparse:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
psi0 = basis(N * N, n)
psi0.dims = [dims[0], 1]
rho0 = vector_to_operator(psi0)
output = mesolve(H, rho0, tlist, c_op_list, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = operator_to_vector(
output.states[k]).full(squeeze=True)
progress_bar.finished()
else:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
psi0 = basis(N * N, n)
rho0 = Qobj(vec2mat(psi0.full()))
output = mesolve(H, rho0, tlist, c_op_list, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
if len(tlist) == 2:
return Qobj(u[:, :, 1], dims=dims)
else:
return [Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))]
def _generic_ode_solve(r, rho0, tlist, e_ops, opt, progress_bar):
"""
Internal function for solving ME. Solve an ODE which solver parameters
already setup (r). Calculate the required expectation values or invoke
callback function at each time step.
"""
#
# prepare output array
#
n_tsteps = len(tlist)
e_sops_data = []
output = Result()
output.solver = "mesolve"
output.times = tlist
if opt.store_states:
output.states = []
if isinstance(e_ops, types.FunctionType):
n_expt_op = 0
expt_callback = True
elif isinstance(e_ops, list):
n_expt_op = len(e_ops)
expt_callback = False
if n_expt_op == 0:
# fall back on storing states
output.states = []
opt.store_states = True
else:
output.expect = []
output.num_expect = n_expt_op
for op in e_ops:
e_sops_data.append(spre(op).data)
if op.isherm and rho0.isherm:
output.expect.append(np.zeros(n_tsteps))
else:
output.expect.append(np.zeros(n_tsteps, dtype=complex))
else:
raise TypeError("Expectation parameter must be a list or a function")
#
# start evolution
#
progress_bar.start(n_tsteps)
rho = Qobj(rho0)
dt = np.diff(tlist)
for t_idx, t in enumerate(tlist):
progress_bar.update(t_idx)
if not r.successful():
break
if opt.store_states or expt_callback:
rho.data = vec2mat(r.y)
if opt.store_states:
output.states.append(Qobj(rho))
if expt_callback:
# use callback method
e_ops(t, rho)
for m in range(n_expt_op):
if output.expect[m].dtype == complex:
output.expect[m][t_idx] = expect_rho_vec(e_sops_data[m], r.y, 0)
else:
output.expect[m][t_idx] = expect_rho_vec(e_sops_data[m], r.y, 1)
if t_idx < n_tsteps - 1:
r.integrate(r.t + dt[t_idx])
progress_bar.finished()
if not opt.rhs_reuse and config.tdname is not None:
try:
os.remove(config.tdname + ".pyx")
except:
pass
if opt.store_final_state:
rho.data = vec2mat(r.y)
output.final_state = Qobj(rho)
return output
def ode2es(L, rho0):
"""Creates an exponential series that describes the time evolution for the
initial density matrix (or state vector) `rho0`, given the Liouvillian
(or Hamiltonian) `L`.
Parameters
----------
L : qobj
Liouvillian of the system.
rho0 : qobj
Initial state vector or density matrix.
Returns
-------
eseries : :class:`qutip.eseries`
``eseries`` represention of the system dynamics.
"""
if issuper(L):
# check initial state
if isket(rho0):
# Got a wave function as initial state: convert to density matrix.
rho0 = rho0 * rho0.dag()
w, v = L.eigenstates()
v = np.hstack([ket.full() for ket in v])
# w[i] = eigenvalue i
# v[:,i] = eigenvector i
rlen = np.prod(rho0.shape)
r0 = mat2vec(rho0.full())
v0 = la.solve(v, r0)
vv = v * sp.spdiags(v0.T, 0, rlen, rlen)
out = None
for i in range(rlen):
qo = Qobj(vec2mat(vv[:, i]), dims=rho0.dims, shape=rho0.shape)
if out:
out += eseries(qo, w[i])
else:
out = eseries(qo, w[i])
elif isoper(L):
if not isket(rho0):
raise TypeError("Second argument must be a ket if first" + "is a Hamiltonian.")
w, v = L.eigenstates()
v = np.hstack([ket.full() for ket in v])
# w[i] = eigenvalue i
# v[:,i] = eigenvector i
rlen = np.prod(rho0.shape)
r0 = rho0.full()
v0 = la.solve(v, r0)
vv = v * sp.spdiags(v0.T, 0, rlen, rlen)
out = None
for i in range(rlen):
qo = Qobj(np.matrix(vv[:, i]).T, dims=rho0.dims, shape=rho0.shape)
if out:
out += eseries(qo, -1.0j * w[i])
else:
out = eseries(qo, -1.0j * w[i])
else:
raise TypeError("First argument must be a Hamiltonian or Liouvillian.")
return estidy(out)
def _ode_super_func(t, y, data):
ym = vec2mat(y)
return (data*ym).ravel('F')
def bloch_redfield_solve(R, ekets, rho0, tlist, e_ops=[], options=None):
"""
Evolve the ODEs defined by Bloch-Redfield master equation. The
Bloch-Redfield tensor can be calculated by the function
:func:`bloch_redfield_tensor`.
Parameters
----------
R : :class:`qutip.qobj`
Bloch-Redfield tensor.
ekets : array of :class:`qutip.qobj`
Array of kets that make up a basis tranformation for the eigenbasis.
rho0 : :class:`qutip.qobj`
Initial density matrix.
tlist : *list* / *array*
List of times for :math:`t`.
e_ops : list of :class:`qutip.qobj` / callback function
List of operators for which to evaluate expectation values.
options : :class:`qutip.Qdeoptions`
Options for the ODE solver.
Returns
-------
output: :class:`qutip.solver`
An instance of the class :class:`qutip.solver`, which contains either
an *array* of expectation values for the times specified by `tlist`.
"""
if options is None:
options = Options()
if options.tidy:
R.tidyup()
#
# check initial state
#
if isket(rho0):
# Got a wave function as initial state: convert to density matrix.
rho0 = rho0 * rho0.dag()
#
# prepare output array
#
n_tsteps = len(tlist)
dt = tlist[1] - tlist[0]
result_list = []
#
# transform the initial density matrix and the e_ops opterators to the
# eigenbasis
#
rho_eb = rho0.transform(ekets)
e_eb_ops = [e.transform(ekets) for e in e_ops]
for e_eb in e_eb_ops:
result_list.append(np.zeros(n_tsteps, dtype=complex))
#
# setup integrator
#
initial_vector = mat2vec(rho_eb.full())
r = scipy.integrate.ode(cy_ode_rhs)
r.set_f_params(R.data.data, R.data.indices, R.data.indptr)
r.set_integrator('zvode',
method=options.method,
order=options.order,
atol=options.atol,
rtol=options.rtol,
nsteps=options.nsteps,
first_step=options.first_step,
min_step=options.min_step,
max_step=options.max_step)
r.set_initial_value(initial_vector, tlist[0])
#
# start evolution
#
dt = np.diff(tlist)
for t_idx, _ in enumerate(tlist):
if not r.successful():
break
rho_eb.data = vec2mat(r.y)
# calculate all the expectation values, or output rho_eb if no
# expectation value operators are given
if e_ops:
rho_eb_tmp = Qobj(rho_eb)
for m, e in enumerate(e_eb_ops):
result_list[m][t_idx] = expect(e, rho_eb_tmp)
else:
result_list.append(rho_eb.transform(ekets, True))
if t_idx < n_tsteps - 1:
r.integrate(r.t + dt[t_idx])
return result_list
def floquet_markov_mesolve(R, ekets, rho0, tlist, e_ops, f_modes_table=None,
options=None, floquet_basis=True):
"""
Solve the dynamics for the system using the Floquet-Markov master equation.
"""
if options is None:
opt = Options()
else:
opt = options
if opt.tidy:
R.tidyup()
#
# check initial state
#
if isket(rho0):
# Got a wave function as initial state: convert to density matrix.
rho0 = ket2dm(rho0)
#
# prepare output array
#
n_tsteps = len(tlist)
dt = tlist[1] - tlist[0]
output = Result()
output.solver = "fmmesolve"
output.times = tlist
if isinstance(e_ops, FunctionType):
n_expt_op = 0
expt_callback = True
elif isinstance(e_ops, list):
n_expt_op = len(e_ops)
expt_callback = False
if n_expt_op == 0:
output.states = []
else:
if not f_modes_table:
raise TypeError("The Floquet mode table has to be provided " +
"when requesting expectation values.")
output.expect = []
output.num_expect = n_expt_op
for op in e_ops:
if op.isherm:
output.expect.append(np.zeros(n_tsteps))
else:
output.expect.append(np.zeros(n_tsteps, dtype=complex))
else:
raise TypeError("Expectation parameter must be a list or a function")
#
# transform the initial density matrix to the eigenbasis: from
# computational basis to the floquet basis
#
if ekets is not None:
rho0 = rho0.transform(ekets)
#
# setup integrator
#
initial_vector = mat2vec(rho0.full())
r = scipy.integrate.ode(cy_ode_rhs)
r.set_f_params(R.data.data, R.data.indices, R.data.indptr)
r.set_integrator('zvode', method=opt.method, order=opt.order,
atol=opt.atol, rtol=opt.rtol, max_step=opt.max_step)
r.set_initial_value(initial_vector, tlist[0])
#
# start evolution
#
rho = Qobj(rho0)
t_idx = 0
for t in tlist:
if not r.successful():
break
rho.data = vec2mat(r.y)
if expt_callback:
# use callback method
if floquet_basis:
e_ops(t, Qobj(rho))
else:
f_modes_table_t, T = f_modes_table
f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T)
e_ops(t, Qobj(rho).transform(f_modes_t, True))
else:
# calculate all the expectation values, or output rho if
# no operators
if n_expt_op == 0:
if floquet_basis:
output.states.append(Qobj(rho))
else:
f_modes_table_t, T = f_modes_table
f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T)
output.states.append(Qobj(rho).transform(f_modes_t, True))
else:
f_modes_table_t, T = f_modes_table
f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T)
for m in range(0, n_expt_op):
output.expect[m][t_idx] = \
expect(e_ops[m], rho.transform(f_modes_t, False))
r.integrate(r.t + dt)
t_idx += 1
return output