def _ptrace(rho, sel):
"""
Private function calculating the partial trace.
"""
if isinstance(sel, int):
sel = np.array([sel])
else:
sel = np.asarray(sel)
if (sel < 0).any() or (sel >= len(rho.dims[0])).any():
raise TypeError("Invalid selection index in ptrace.")
drho = rho.dims[0]
N = np.prod(drho)
M = np.prod(np.asarray(drho).take(sel))
if np.prod(rho.dims[1]) == 1:
rho = rho * rho.dag()
perm = sp.lil_matrix((M * M, N * N))
# all elements in range(len(drho)) not in sel set
rest = np.setdiff1d(np.arange(len(drho)), sel)
ilistsel = _select(sel, drho)
indsel = _list2ind(ilistsel, drho)
ilistrest = _select(rest, drho)
indrest = _list2ind(ilistrest, drho)
irest = (indrest - 1) * N + indrest - 2
# Possibly use parfor here if M > some value ?
# We have to initialise like this to get a numpy array of lists rather than
# a standard numpy 2d array. Scipy >= 1.5 requires that we do this, rather
# than just pass a 2d array, and scipy < 1.5 will accept it (because it's
# actually the correct format).
perm.rows = np.empty((M * M,), dtype=object)
perm.data = np.empty((M * M,), dtype=object)
for m in range(M * M):
perm.rows[m] = list((irest
+ (indsel[int(np.floor(m / M))] - 1)*N
+ indsel[int(np.mod(m, M))]).T[0])
perm.data[m] = [1.0] * len(perm.rows[m])
perm = perm.tocsr()
rhdata = perm * sp_reshape(rho.data, (np.prod(rho.shape), 1))
rho1_data = sp_reshape(rhdata, (M, M))
dims_kept0 = np.asarray(rho.dims[0]).take(sel)
dims_kept1 = np.asarray(rho.dims[0]).take(sel)
rho1_dims = [dims_kept0.tolist(), dims_kept1.tolist()]
rho1_shape = [np.prod(dims_kept0), np.prod(dims_kept1)]
return rho1_data, rho1_dims, rho1_shape
def operator_to_vector(op):
"""
Create a vector representation given a quantum operator in matrix form.
The passed object should have a ``Qobj.type`` of 'oper' or 'super'; this
function is not designed for general-purpose matrix reshaping.
Parameters
----------
op : Qobj or QobjEvo
Quantum operator in matrix form. This must have a type of 'oper' or
'super'.
Returns
-------
Qobj or QobjEvo
The same object, but re-cast into a column-stacked-vector form of type
'operator-ket'. The output is the same type as the passed object.
"""
if isinstance(op, QobjEvo):
return op.apply(operator_to_vector)
if not (op.isoper or op.issuper):
raise ValueError("only valid for operator matrices")
size = op.shape[0] * op.shape[1]
return Qobj(
sp_reshape(op.data.T, (size, 1)),
dims=[op.dims, [1]],
shape=(size, 1),
type='operator-ket',
copy=False,
)
def vector_to_operator(op):
"""
Create a matrix representation given a quantum operator in vector form.
The passed object should have a ``Qobj.type`` of 'operator-ket'; this
function is not designed for general-purpose matrix reshaping.
Parameters
----------
op : Qobj or QobjEvo
Quantum operator in column-stacked-vector form. This must have a type
of 'operator-ket'.
Returns
-------
Qobj or QobjEvo
The same object, but re-cast into "standard" operator form. The output
is the same type as the passed object.
"""
if isinstance(op, QobjEvo):
return op.apply(vector_to_operator)
if not op.isoperket:
raise ValueError(
"only valid for operators in column-stacked 'operator-ket' format")
# e.g. op.dims = [ [[rows], [cols]], [1]]
dims = op.dims[0]
shape = (np.prod(dims[0]), np.prod(dims[1]))
return Qobj(
sp_reshape(op.data.T, shape[::-1]).T,
dims=dims,
shape=shape,
copy=False,
)
def operator_to_vector(op):
"""
Create a vector representation of a quantum operator given
the matrix representation.
"""
q = Qobj()
q.dims = [op.dims, [1]]
q.data = sp_reshape(op.data.T, (np.prod(op.shape), 1))
return q
def vector_to_operator(op):
"""
Create a matrix representation given a quantum operator in
vector form.
"""
q = Qobj()
q.dims = op.dims[0]
q.data = sp_reshape(op.data.H, q.shape)
return q
def _ptrace(rho, sel):
"""
Private function calculating the partial trace.
"""
if isinstance(sel, int):
sel = np.array([sel])
else:
sel = np.asarray(sel)
if (sel < 0).any() or (sel >= len(rho.dims[0])).any():
raise TypeError("Invalid selection index in ptrace.")
drho = rho.dims[0]
N = np.prod(drho)
M = np.prod(np.asarray(drho).take(sel))
if np.prod(rho.dims[1]) == 1:
rho = rho * rho.dag()
perm = sp.lil_matrix((M * M, N * N))
# all elements in range(len(drho)) not in sel set
rest = np.setdiff1d(np.arange(len(drho)), sel)
ilistsel = _select(sel, drho)
indsel = _list2ind(ilistsel, drho)
ilistrest = _select(rest, drho)
indrest = _list2ind(ilistrest, drho)
irest = (indrest - 1) * N + indrest - 2
# Possibly use parfor here if M > some value ?
perm.rows = np.array(
[(irest + (indsel[int(np.floor(m / M))] - 1) * N +
indsel[int(np.mod(m, M))]).T[0]
for m in range(M ** 2)])
# perm.data = np.ones_like(perm.rows,dtype=int)
perm.data = np.ones_like(perm.rows)
perm.tocsr()
rhdata = perm * sp_reshape(rho.data, (np.prod(rho.shape), 1))
rho1_data = sp_reshape(rhdata, (M, M))
dims_kept0 = np.asarray(rho.dims[0]).take(sel)
dims_kept1 = np.asarray(rho.dims[0]).take(sel)
rho1_dims = [dims_kept0.tolist(), dims_kept1.tolist()]
rho1_shape = [np.prod(dims_kept0), np.prod(dims_kept1)]
return rho1_data, rho1_dims, rho1_shape
def vector_to_operator(op):
"""
Create a matrix representation given a quantum operator in
vector form.
"""
q = Qobj()
q.dims = op.dims[0]
n = int(np.sqrt(op.shape[0]))
q.data = sp_reshape(op.data.T, (n, n)).T
return q
def _ptrace(rho, sel):
"""
Private function calculating the partial trace.
"""
if isinstance(sel, int):
sel = np.array([sel])
else:
sel = np.asarray(sel)
if (sel < 0).any() or (sel >= len(rho.dims[0])).any():
raise TypeError("Invalid selection index in ptrace.")
drho = rho.dims[0]
N = np.prod(drho)
M = np.prod(np.asarray(drho).take(sel))
if np.prod(rho.dims[1]) == 1:
rho = rho * rho.dag()
perm = sp.lil_matrix((M * M, N * N))
# all elements in range(len(drho)) not in sel set
rest = np.setdiff1d(np.arange(len(drho)), sel)
ilistsel = _select(sel, drho)
indsel = _list2ind(ilistsel, drho)
ilistrest = _select(rest, drho)
indrest = _list2ind(ilistrest, drho)
irest = (indrest - 1) * N + indrest - 2
# Possibly use parfor here if M > some value ?
perm.rows = np.array([(irest + (indsel[int(np.floor(m / M))] - 1) * N +
indsel[int(np.mod(m, M))]).T[0]
for m in range(M**2)])
# perm.data = np.ones_like(perm.rows,dtype=int)
perm.data = np.ones_like(perm.rows)
perm = perm.tocsr()
rhdata = perm * sp_reshape(rho.data, (np.prod(rho.shape), 1))
rho1_data = sp_reshape(rhdata, (M, M))
dims_kept0 = np.asarray(rho.dims[0]).take(sel)
dims_kept1 = np.asarray(rho.dims[0]).take(sel)
rho1_dims = [dims_kept0.tolist(), dims_kept1.tolist()]
rho1_shape = [np.prod(dims_kept0), np.prod(dims_kept1)]
return rho1_data, rho1_dims, rho1_shape
def operator_to_vector(op):
"""
Create a vector representation of a quantum operator given
the matrix representation.
"""
q = Qobj()
q.shape = [prod(op.shape), 1]
q.dims = [op.dims, [1]]
q.data = sp_reshape(op.data.T, tuple(q.shape))
q.type = 'operator-vector'
return q
def vector_to_operator(op):
"""
Create a matrix representation given a quantum operator in
vector form.
"""
q = Qobj()
q.shape = [op.dims[0][0][0], op.dims[0][1][0]]
q.dims = op.dims[0]
q.data = sp_reshape(op.data.H, tuple(q.shape))
q.type = 'oper'
return q
def operator_to_vector(op):
"""
Create a vector representation of a quantum operator given
the matrix representation.
"""
q = Qobj()
q.shape = [prod(op.shape), 1]
q.dims = [op.dims, [1]]
q.data = sp_reshape(op.data.T, tuple(q.shape))
q.type = 'operator-vector'
return q
def vector_to_operator(op):
"""
Create a matrix representation given a quantum operator in
vector form.
"""
q = Qobj()
q.shape = [op.dims[0][0][0], op.dims[0][1][0]]
q.dims = op.dims[0]
q.data = sp_reshape(op.data.H, tuple(q.shape))
q.type = 'oper'
return q
def operator_to_vector(op):
"""
Create a vector representation of a quantum operator given
the matrix representation.
"""
if isinstance(op, QobjEvo):
return op.apply(operator_to_vector)
q = Qobj()
q.dims = [op.dims, [1]]
q.data = sp_reshape(op.data.T, (np.prod(op.shape), 1))
return q
def _steadystate_power(L,
maxiter=10,
tol=1e-6,
itertol=1e-5,
use_umfpack=True,
verbose=False):
"""
Inverse power method for steady state solving.
"""
if verbose:
print('Starting iterative power method Solver...')
use_solver(assumeSortedIndices=True, useUmfpack=use_umfpack)
rhoss = Qobj()
sflag = issuper(L)
if sflag:
rhoss.dims = L.dims[0]
rhoss.shape = [prod(rhoss.dims[0]), prod(rhoss.dims[1])]
else:
rhoss.dims = [L.dims[0], 1]
rhoss.shape = [prod(rhoss.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())
if verbose:
start_time = time.time()
it = 0
while (la.norm(L * v, np.inf) > tol) and (it < maxiter):
v = spsolve(L, v, use_umfpack=use_umfpack)
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
if verbose:
print('Power solver time: ', time.time() - start_time)
if qset.auto_tidyup:
return rhoss.tidyup()
else:
return rhoss
def vector_to_operator(op):
"""
Create a matrix representation given a quantum operator in
vector form.
"""
if isinstance(op, QobjEvo):
return op.apply(vector_to_operator)
q = Qobj()
# e.g. op.dims = [ [[rows], [cols]], [1]]
q.dims = op.dims[0]
shape = (np.prod(q.dims[0]), np.prod(q.dims[1]))
q.data = sp_reshape(op.data.T, shape[::-1]).T
return q
def _steadystate_power(L, maxiter=10, tol=1e-6, itertol=1e-5, use_umfpack=True,
verbose=False):
"""
Inverse power method for steady state solving.
"""
if verbose:
print('Starting iterative power method Solver...')
use_solver(assumeSortedIndices=True, useUmfpack=use_umfpack)
rhoss = Qobj()
sflag = issuper(L)
if sflag:
rhoss.dims = L.dims[0]
rhoss.shape = [prod(rhoss.dims[0]), prod(rhoss.dims[1])]
else:
rhoss.dims = [L.dims[0], 1]
rhoss.shape = [prod(rhoss.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())
if verbose:
start_time = time.time()
it = 0
while (la.norm(L * v, np.inf) > tol) and (it < maxiter):
v = spsolve(L, v, use_umfpack=use_umfpack)
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
if verbose:
print('Power solver time: ', time.time() - start_time)
if qset.auto_tidyup:
return rhoss.tidyup()
else:
return rhoss
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_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']
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
L, perm, perm2, rev_perm, ss_args = _steadystate_power_liouvillian(
L, ss_args)
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['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':
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
_tol = np.max(ss_args['tol'] / 10,
1e-15) # Should make this user accessible
while (la.norm(L * v, np.inf) > tol) and (it < maxiter):
if ss_args['method'] == 'power':
v = lu.solve(v)
elif ss_args['method'] == 'power-gmres':
v, check = gmres(L,
v,
tol=_tol,
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':
v, check = lgmres(L,
v,
tol=_tol,
M=ss_args['M'],
x0=ss_args['x0'],
maxiter=ss_args['maxiter'],
callback=_iter_count)
elif ss_args['method'] == 'power-bicgstab':
v, check = bicgstab(L,
v,
tol=_tol,
M=ss_args['M'],
x0=ss_args['x0'],
maxiter=ss_args['maxiter'],
callback=_iter_count)
else:
raise Exception("Invalid iterative solver method.")
v = v / la.norm(v, np.inf)
it += 1
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)
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 = 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
if ss_args['return_info']:
return rhoss, ss_args['info']
else:
return rhoss
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']
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() - (1e-15) * sp.eye(n, n, format='csc')
L.sort_indices()
orig_nnz = L.nnz
# start with all ones as RHS
v = np.ones(n, dtype=complex)
if ss_args['use_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')
v = v[np.ix_(perm, )]
if settings.debug:
new_band = sp_bandwidth(L)[0]
logger.debug('RCM bandwidth: %i' % new_band)
logger.debug('Bandwidth reduction factor: %f' %
round(old_band / new_band, 2))
_power_start = time.time()
# Get LU factors
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
while (la.norm(L * v, np.inf) > tol) and (it < maxiter):
v = lu.solve(v)
v = v / la.norm(v, np.inf)
it += 1
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 = 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
if ss_args['return_info']:
return rhoss, ss_args['info']
else:
return rhoss
def partial_trace(rho_data, rho_dims, sel, perm=None):
"""Compute the partial trace of the matrix
Parameters
----------
rho_data : csr_matrix or ndarray
Matrix for which the partial trace will be computed
rho_dims : list
Subsystem dimensions
sel : int/list
An ``int`` or ``list`` of components to keep after partial trace.
perm : ndarray
Optionally a pre computed permutation matrix can be provided
This may be more efficient if the same partial trace (sel)
is to be computed for matices of the same sub-system dimensions.
It can be calculated using the calc_perm function
Returns
-------
rho1_data : qobj
Matrix representing partial trace with selected components remaining.
rho1_dims : list
Subsystem dimensions for output matrix
rho1_shape : tuple
Shape of output matrix
"""
#print("rho_dims {} (type: {})".format(rho_dims, type(rho_dims)))
#print("sel {} (type: {})".format(sel, type(sel)))
if isinstance(sel, int):
sel = np.array([sel])
else:
sel = np.asarray(sel)
if (sel < 0).any() or (sel >= len(rho_dims[0])).any():
raise TypeError("Invalid selection index in ptrace.")
sparse = sp.issparse(rho_data)
if np.prod(rho_dims[1]) == 1:
rho_data = rho_data.dot(rho_data.T.conj())
rho1_dims = [
np.asarray(rho_dims[0]).take(sel).tolist(),
np.asarray(rho_dims[0]).take(sel).tolist()
]
rho1_shape = (np.prod(rho1_dims[0]), np.prod(rho1_dims[1]))
if perm is None:
perm = calc_perm(rho_dims, sel, sparse=sparse)
if sparse:
rhdata = perm * sp_reshape(rho_data, [np.prod(rho_dims[0])**2, 1])
rhdata = rhdata.tolil().reshape(rho1_shape)
rho1_data = rhdata.tocsr()
else:
rho1_data = np.reshape(
np.sum(np.reshape(rho_data.flatten()[perm.flatten()], perm.shape),
1), rho1_shape)
return rho1_data, rho1_dims, rho1_shape
def propagator(H, t, c_op_list=[], args={}, options=None,
unitary_mode='batch', parallel=False,
progress_bar=None, **kwargs):
"""
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.
unitary_mode = str ('batch', 'single')
Solve all basis vectors simulaneously ('batch') or individually
('single').
parallel : bool {False, True}
Run the propagator in parallel mode. This will override the
unitary_mode settings if set to True.
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)`.
"""
kw = _default_kwargs()
if 'num_cpus' in kwargs:
num_cpus = kwargs['num_cpus']
else:
num_cpus = kw['num_cpus']
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
td_type = _td_format_check(H, c_op_list, solver='me')
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
if parallel:
unitary_mode = 'single'
u = np.zeros([N, N, len(tlist)], dtype=complex)
output = parallel_map(_parallel_sesolve,range(N),
task_args=(N,H, tlist,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N):
for k, t in enumerate(tlist):
u[:, n, k] = output[n].states[k].full().T
else:
if unitary_mode == 'single':
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, _safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
progress_bar.finished()
elif unitary_mode =='batch':
u = np.zeros(len(tlist), dtype=object)
_rows = np.array([(N+1)*m for m in range(N)])
_cols = np.zeros_like(_rows)
_data = np.ones_like(_rows,dtype=complex)
psi0 = Qobj(sp.coo_matrix((_data,(_rows,_cols))).tocsr())
if td_type[1] > 0 or td_type[2] > 0:
H2 = []
for k in range(len(H)):
if isinstance(H[k], list):
H2.append([tensor(qeye(N), H[k][0]), H[k][1]])
else:
H2.append(tensor(qeye(N), H[k]))
else:
H2 = tensor(qeye(N), H)
output = sesolve(H2, psi0, tlist, [] , args = args, _safe_mode=False,
options=Options(normalize_output=False))
for k, t in enumerate(tlist):
u[k] = sp_reshape(output.states[k].data, (N, N))
unit_row_norm(u[k].data, u[k].indptr, u[k].shape[0])
u[k] = u[k].T.tocsr()
else:
raise Exception('Invalid unitary mode.')
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
unitary_mode = 'single'
N = H0.shape[0]
sqrt_N = int(np.sqrt(N))
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,range(N * N),
task_args=(sqrt_N,H,tlist,c_op_list,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n,(sqrt_N,sqrt_N))
rho0 = Qobj(sp.csr_matrix(([1],([row_idx],[col_idx])), shape=(sqrt_N,sqrt_N), dtype=complex))
output = mesolve(H, rho0, tlist, [], [], args, options, _safe_mode=False)
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)
unitary_mode = 'single'
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,range(N * N),
task_args=(N,H,tlist,c_op_list,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n,(N,N))
rho0 = Qobj(sp.csr_matrix(([1],([row_idx],[col_idx])), shape=(N,N), dtype=complex))
output = mesolve(H, rho0, tlist, c_op_list, [], args, options, _safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
if len(tlist) == 2:
if unitary_mode == 'batch':
return Qobj(u[-1], dims=dims)
else:
return Qobj(u[:, :, 1], dims=dims)
else:
if unitary_mode == 'batch':
return np.array([Qobj(u[k], dims=dims) for k in range(len(tlist))], dtype=object)
else:
return np.array([Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))], dtype=object)
def propagator(H,
t,
c_op_list=[],
args={},
options=None,
unitary_mode='batch',
parallel=False,
progress_bar=None,
_safe_mode=True,
**kwargs):
r"""
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.
unitary_mode = str ('batch', 'single')
Solve all basis vectors simulaneously ('batch') or individually
('single').
parallel : bool {False, True}
Run the propagator in parallel mode. This will override the
unitary_mode settings if set to True.
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)`.
"""
kw = _default_kwargs()
if 'num_cpus' in kwargs:
num_cpus = kwargs['num_cpus']
else:
num_cpus = kw['num_cpus']
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 _safe_mode:
_solver_safety_check(H, None, c_ops=c_op_list, e_ops=[], args=args)
td_type = _td_format_check(H, c_op_list, solver='me')
if isinstance(
H,
(types.FunctionType, types.BuiltinFunctionType, functools.partial)):
H0 = H(0.0, args)
if unitary_mode == 'batch':
# batch don't work with function Hamiltonian
unitary_mode = 'single'
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
if parallel:
unitary_mode = 'single'
u = np.zeros([N, N, len(tlist)], dtype=complex)
output = parallel_map(_parallel_sesolve,
range(N),
task_args=(N, H, tlist, args, options),
progress_bar=progress_bar,
num_cpus=num_cpus)
for n in range(N):
for k, t in enumerate(tlist):
u[:, n, k] = output[n].states[k].full().T
else:
if unitary_mode == 'single':
output = sesolve(H,
qeye(dims[0]),
tlist, [],
args,
options,
_safe_mode=False)
if len(tlist) == 2:
return output.states[-1]
else:
return output.states
elif unitary_mode == 'batch':
u = np.zeros(len(tlist), dtype=object)
_rows = np.array([(N + 1) * m for m in range(N)])
_cols = np.zeros_like(_rows)
_data = np.ones_like(_rows, dtype=complex)
psi0 = Qobj(sp.coo_matrix((_data, (_rows, _cols))).tocsr())
if td_type[1] > 0 or td_type[2] > 0:
H2 = []
for k in range(len(H)):
if isinstance(H[k], list):
H2.append([tensor(qeye(N), H[k][0]), H[k][1]])
else:
H2.append(tensor(qeye(N), H[k]))
else:
H2 = tensor(qeye(N), H)
options.normalize_output = False
output = sesolve(H2,
psi0,
tlist, [],
args=args,
options=options,
_safe_mode=False)
for k, t in enumerate(tlist):
u[k] = sp_reshape(output.states[k].data, (N, N))
unit_row_norm(u[k].data, u[k].indptr, u[k].shape[0])
u[k] = u[k].T.tocsr()
else:
raise Exception('Invalid unitary mode.')
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
unitary_mode = 'single'
N = H0.shape[0]
sqrt_N = int(np.sqrt(N))
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,
range(N * N),
task_args=(sqrt_N, H, tlist, c_op_list, args,
options),
progress_bar=progress_bar,
num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
rho0 = qeye(N, N)
rho0.dims = [[sqrt_N, sqrt_N], [sqrt_N, sqrt_N]]
output = mesolve(H,
psi0,
tlist, [],
args,
options,
_safe_mode=False)
if len(tlist) == 2:
return output.states[-1]
else:
return output.states
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
unitary_mode = 'single'
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,
range(N * N),
task_args=(N, H, tlist, c_op_list, args,
options),
progress_bar=progress_bar,
num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n, (N, N))
rho0 = Qobj(
sp.csr_matrix(([1], ([row_idx], [col_idx])),
shape=(N, N),
dtype=complex))
output = mesolve(H,
rho0,
tlist,
c_op_list, [],
args,
options,
_safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
if len(tlist) == 2:
if unitary_mode == 'batch':
return Qobj(u[-1], dims=dims)
else:
return Qobj(u[:, :, 1], dims=dims)
else:
if unitary_mode == 'batch':
return np.array([Qobj(u[k], dims=dims) for k in range(len(tlist))],
dtype=object)
else:
return np.array(
[Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))],
dtype=object)
def _steadystate_power(L, ss_args):
"""
Inverse power method for steady state solving.
"""
ss_args['info'].pop('weight', None)
if settings.debug:
print('Starting iterative inverse-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() - (1e-15) * sp.eye(n, n, format='csc')
L.sort_indices()
orig_nnz = L.nnz
# start with all ones as RHS
v = np.ones(n,dtype=complex)
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')
v = v[np.ix_(perm,)]
if settings.debug:
new_band = sp_bandwidth(L)[0]
print('RCM bandwidth:', new_band)
print('Bandwidth reduction factor:', round(old_band/new_band, 2))
# Get LU factors
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
print('L NNZ:', L_nnz, ';', 'U NNZ:', U_nnz)
print('Fill factor:', (L_nnz+U_nnz)/orig_nnz)
_power_start = time.time()
it = 0
while (la.norm(L * v, np.inf) > tol) and (it < maxiter):
v = lu.solve(v)
v = v / la.norm(v, np.inf)
it += 1
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 settings.debug:
print('Number of iterations:', it)
if ss_args['use_rcm']:
v = v[np.ix_(rev_perm,)]
# 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
if ss_args['return_info']:
return rhoss, ss_args['info']
else:
return rhoss
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']
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 settings.has_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['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 settings.has_mkl:
lu = mkl_splu(L)
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
_tol = max(ss_args['tol']/10, 1e-15) # Should make this user accessible
while (la.norm(L * v, np.inf) > tol) and (it < maxiter):
if ss_args['method'] == 'power':
v = lu.solve(v)
elif ss_args['method'] == 'power-gmres':
v, check = gmres(L, v, tol=_tol, 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':
v, check = lgmres(L, v, tol=_tol, M=ss_args['M'],
x0=ss_args['x0'], maxiter=ss_args['maxiter'],
callback=_iter_count)
elif ss_args['method'] == 'power-bicgstab':
v, check = bicgstab(L, v, tol=_tol, M=ss_args['M'],
x0=ss_args['x0'],
maxiter=ss_args['maxiter'], callback=_iter_count)
else:
raise Exception("Invalid iterative solver method.")
v = v / la.norm(v, np.inf)
it += 1
if ss_args['method'] == 'power' and settings.has_mkl:
lu.delete()
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)
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 = 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
if ss_args['return_info']:
return rhoss, ss_args['info']
else:
return rhoss
