def steadystate_nonlinear(L_func,
rho0,
args={},
maxiter=10,
random_initial_state=False,
tol=1e-6,
itertol=1e-5,
use_umfpack=True,
verbose=False):
"""
Steady state for the evolution subject to the nonlinear Liouvillian
(which depends on the density matrix).
.. note:: Experimental. Not at all certain that the inverse power method
works for state-dependent Liouvillian operators.
"""
use_solver(assumeSortedIndices=True, useUmfpack=use_umfpack)
if random_initial_state:
rhoss = rand_dm(rho0.shape[0], 1.0, dims=rho0.dims)
elif isket(rho0):
rhoss = ket2dm(rho0)
else:
rhoss = Qobj(rho0)
v = mat2vec(rhoss.full())
n = prod(rhoss.shape)
tr_vec = sp.eye(rhoss.shape[0], rhoss.shape[0], format='coo')
tr_vec = tr_vec.reshape((1, n))
it = 0
while it < maxiter:
L = L_func(rhoss, args)
L = L.data.tocsc() - (tol**2) * sp.eye(n, n, format='csc')
L.sort_indices()
v = spsolve(L, v, use_umfpack=use_umfpack)
v = v / la.norm(v, np.inf)
data = v / sum(tr_vec.dot(v))
data = reshape(data, (rhoss.shape[0], rhoss.shape[1])).T
rhoss.data = sp.csr_matrix(data)
it += 1
if la.norm(L * v, np.inf) <= tol:
break
if it >= maxiter:
raise ValueError('Failed to find steady state after ' + str(maxiter) +
' iterations')
rhoss = 0.5 * (rhoss + rhoss.dag())
return rhoss.tidyup() if qset.auto_tidyup else rhoss
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_nonlinear(L_func, rho0, args={}, maxiter=10,
random_initial_state=False, tol=1e-6, itertol=1e-5,
use_umfpack=True, verbose=False):
"""
Steady state for the evolution subject to the nonlinear Liouvillian
(which depends on the density matrix).
.. note:: Experimental. Not at all certain that the inverse power method
works for state-dependent Liouvillian operators.
"""
use_solver(assumeSortedIndices=True, useUmfpack=use_umfpack)
if random_initial_state:
rhoss = rand_dm(rho0.shape[0], 1.0, dims=rho0.dims)
elif isket(rho0):
rhoss = ket2dm(rho0)
else:
rhoss = Qobj(rho0)
v = mat2vec(rhoss.full())
n = prod(rhoss.shape)
tr_vec = sp.eye(rhoss.shape[0], rhoss.shape[0], format='coo')
tr_vec = tr_vec.reshape((1, n))
it = 0
while it < maxiter:
L = L_func(rhoss, args)
L = L.data.tocsc() - (tol ** 2) * sp.eye(n, n, format='csc')
L.sort_indices()
v = spsolve(L, v, use_umfpack=use_umfpack)
v = v / la.norm(v, np.inf)
data = v / sum(tr_vec.dot(v))
data = reshape(data, (rhoss.shape[0], rhoss.shape[1])).T
rhoss.data = sp.csr_matrix(data)
it += 1
if la.norm(L * v, np.inf) <= tol:
break
if it >= maxiter:
raise ValueError('Failed to find steady state after ' +
str(maxiter) + ' iterations')
rhoss = 0.5 * (rhoss + rhoss.dag())
return rhoss.tidyup() if qset.auto_tidyup else rhoss
def _steadystate_power(L, maxiter=10, tol=1e-6, itertol=1e-5,
verbose=False):
"""
Inverse power method for steady state solving.
"""
if verbose:
print('Starting iterative power method Solver...')
use_solver(assumeSortedIndices=True)
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)
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 tensor(*args):
"""Calculates the tensor product of input operators.
Parameters
----------
args : array_like
``list`` or ``array`` of quantum objects for tensor product.
Returns
-------
obj : qobj
A composite quantum object.
Examples
--------
>>> tensor([sigmax(), sigmax()]) # doctest: +SKIP
Quantum object: dims = [[2, 2], [2, 2]], \
shape = [4, 4], type = oper, isHerm = True
Qobj data =
[[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j]
[ 0.+0.j 0.+0.j 1.+0.j 0.+0.j]
[ 0.+0.j 1.+0.j 0.+0.j 0.+0.j]
[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j]]
"""
if not args:
raise TypeError("Requires at least one input argument")
if len(args) == 1 and isinstance(args[0], (list, np.ndarray)):
# this is the case when tensor is called on the form:
# tensor([q1, q2, q3, ...])
qlist = args[0]
elif len(args) == 1 and isinstance(args[0], Qobj):
# tensor is called with a single Qobj as an argument, do nothing
return args[0]
else:
# this is the case when tensor is called on the form:
# tensor(q1, q2, q3, ...)
qlist = args
if not all([isinstance(q, Qobj) for q in qlist]):
# raise error if one of the inputs is not a quantum object
raise TypeError("One of inputs is not a quantum object")
out = Qobj()
if qlist[0].issuper:
out.superrep = qlist[0].superrep
if not all([q.superrep == out.superrep for q in qlist]):
raise TypeError("In tensor products of superroperators, all must" +
"have the same representation")
out.isherm = True
for n, q in enumerate(qlist):
if n == 0:
out.data = q.data
out.dims = q.dims
else:
out.data = zcsr_kron(out.data, q.data)
out.dims = [out.dims[0] + q.dims[0], out.dims[1] + q.dims[1]]
out.isherm = out.isherm and q.isherm
if not out.isherm:
out._isherm = None
return out.tidyup() if qutip.settings.auto_tidyup else out
def tensor(*args):
"""Calculates the tensor product of input operators.
Parameters
----------
args : array_like
``list`` or ``array`` of quantum objects for tensor product.
Returns
--------
obj : qobj
A composite quantum object.
Examples
--------
>>> tensor([sigmax(), sigmax()])
Quantum object: dims = [[2, 2], [2, 2]], \
shape = [4, 4], type = oper, isHerm = True
Qobj data =
[[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j]
[ 0.+0.j 0.+0.j 1.+0.j 0.+0.j]
[ 0.+0.j 1.+0.j 0.+0.j 0.+0.j]
[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j]]
"""
if not args:
raise TypeError("Requires at least one input argument")
num_args = len(args)
step = 0
isherm = True
for n in range(num_args):
if isinstance(args[n], Qobj):
qos = args[n]
if step == 0:
dat = qos.data
dim = qos.dims
shp = qos.shape
isherm = isherm and qos.isherm
step = 1
else:
dat = sp.kron(dat, qos.data, format='csr')
isherm = isherm and qos.isherm
dim = [dim[0] + qos.dims[0],
dim[1] + qos.dims[1]] # append dimensions of Qobjs
shp = [dat.shape[0], dat.shape[1]] # new shape of matrix
elif isinstance(args[n], (list, ndarray)):
qos = args[n]
items = len(qos) # number of inputs
if not all([isinstance(k, Qobj) for k in qos]):
# raise error if one of the inputs is not a quantum object
raise TypeError("One of inputs is not a quantum object")
if items == 1: # if only one Qobj, do nothing
if step == 0:
dat = qos[0].data
dim = qos[0].dims
shp = qos[0].shape
isherm = isherm and qos[0].isherm
step = 1
else:
dat = sp.kron(dat, qos[0].data, format='csr')
isherm = isherm and qos[0].isherm
dim = [dim[0] + qos[0].dims[0],
dim[1] + qos[0].dims[1]] # append dimensions of qos
shp = [dat.shape[0], dat.shape[1]] # new shape of matrix
elif items != 1:
if step == 0:
dat = qos[0].data
dim = qos[0].dims
shp = qos[0].shape
step = 1
isherm = isherm and qos[0].isherm
for k in range(items - 1): # cycle over all items
dat = sp.kron(dat, qos[k + 1].data, format='csr')
isherm = isherm and qos[k + 1].isherm
dim = [
dim[0] + qos[k + 1].dims[0],
dim[1] + qos[k + 1].dims[1]
]
shp = [dat.shape[0], dat.shape[1]] # new shape of matrix
out = Qobj()
out.data = dat
out.dims = dim
out.shape = shp
out.type = ischeck(out)
out.isherm = isherm
if qutip.settings.auto_tidyup:
return out.tidyup() # returns tidy Qobj
else:
return out
def tensor(*args):
"""Calculates the tensor product of input operators.
Parameters
----------
args : array_like
``list`` or ``array`` of quantum objects for tensor product.
Returns
-------
obj : qobj
A composite quantum object.
Examples
--------
>>> tensor([sigmax(), sigmax()])
Quantum object: dims = [[2, 2], [2, 2]], \
shape = [4, 4], type = oper, isHerm = True
Qobj data =
[[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j]
[ 0.+0.j 0.+0.j 1.+0.j 0.+0.j]
[ 0.+0.j 1.+0.j 0.+0.j 0.+0.j]
[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j]]
"""
if not args:
raise TypeError("Requires at least one input argument")
if len(args) == 1 and isinstance(args[0], (list, np.ndarray)):
# this is the case when tensor is called on the form:
# tensor([q1, q2, q3, ...])
qlist = args[0]
elif len(args) == 1 and isinstance(args[0], Qobj):
# tensor is called with a single Qobj as an argument, do nothing
return args[0]
else:
# this is the case when tensor is called on the form:
# tensor(q1, q2, q3, ...)
qlist = args
if not all([isinstance(q, Qobj) for q in qlist]):
# raise error if one of the inputs is not a quantum object
raise TypeError("One of inputs is not a quantum object")
out = Qobj()
if qlist[0].issuper:
out.superrep = qlist[0].superrep
if not all([q.superrep == out.superrep for q in qlist]):
raise TypeError("In tensor products of superroperators, all must" +
"have the same representation")
out.isherm = True
for n, q in enumerate(qlist):
if n == 0:
out.data = q.data
out.dims = q.dims
else:
out.data = sp.kron(out.data, q.data, format='csr')
out.dims = [out.dims[0] + q.dims[0], out.dims[1] + q.dims[1]]
out.isherm = out.isherm and q.isherm
if not out.isherm:
out._isherm = None
return out.tidyup() if qutip.settings.auto_tidyup else out
def test_tidyup():
small = Qobj([[1e-2, 0], [0, 1]])
small.tidyup(1e-1)
assert small.tr() == 1
def tensor(*args):
"""Calculates the tensor product of input operators.
Parameters
----------
args : array_like
``list`` or ``array`` of quantum objects for tensor product.
Returns
--------
obj : qobj
A composite quantum object.
Examples
--------
>>> tensor([sigmax(), sigmax()])
Quantum object: dims = [[2, 2], [2, 2]], \
shape = [4, 4], type = oper, isHerm = True
Qobj data =
[[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j]
[ 0.+0.j 0.+0.j 1.+0.j 0.+0.j]
[ 0.+0.j 1.+0.j 0.+0.j 0.+0.j]
[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j]]
"""
if not args:
raise TypeError("Requires at least one input argument")
num_args = len(args)
step = 0
isherm = True
for n in range(num_args):
if isinstance(args[n], Qobj):
qos = args[n]
if step == 0:
dat = qos.data
dim = qos.dims
shp = qos.shape
isherm = isherm and qos.isherm
step = 1
else:
dat = sp.kron(dat, qos.data, format='csr')
isherm = isherm and qos.isherm
dim = [dim[0] + qos.dims[0],
dim[1] + qos.dims[1]] # append dimensions of Qobjs
shp = [dat.shape[0], dat.shape[1]] # new shape of matrix
elif isinstance(args[n], (list, ndarray)):
qos = args[n]
items = len(qos) # number of inputs
if not all([isinstance(k, Qobj) for k in qos]):
# raise error if one of the inputs is not a quantum object
raise TypeError("One of inputs is not a quantum object")
if items == 1: # if only one Qobj, do nothing
if step == 0:
dat = qos[0].data
dim = qos[0].dims
shp = qos[0].shape
isherm = isherm and qos[0].isherm
step = 1
else:
dat = sp.kron(dat, qos[0].data, format='csr')
isherm = isherm and qos[0].isherm
dim = [dim[0] + qos[0].dims[0],
dim[1] + qos[0].dims[1]] # append dimensions of qos
shp = [dat.shape[0], dat.shape[1]] # new shape of matrix
elif items != 1:
if step == 0:
dat = qos[0].data
dim = qos[0].dims
shp = qos[0].shape
step = 1
isherm = isherm and qos[0].isherm
for k in range(items - 1): # cycle over all items
dat = sp.kron(dat, qos[k + 1].data, format='csr')
isherm = isherm and qos[k + 1].isherm
dim = [dim[0] + qos[k + 1].dims[0],
dim[1] + qos[k + 1].dims[1]]
shp = [dat.shape[0], dat.shape[1]] # new shape of matrix
out = Qobj()
out.data = dat
out.dims = dim
out.shape = shp
out.type = ischeck(out)
out.isherm = isherm
if qutip.settings.auto_tidyup:
return out.tidyup() # returns tidy Qobj
else:
return out
