def liouvillian_fast(H, c_op_list):
"""Assembles the Liouvillian superoperator from a Hamiltonian
and a ``list`` of collapse operators. Like liouvillian, but with an
experimental implementation which avoids creating extra Qobj instances,
which can be advantageous for large systems.
Parameters
----------
H : qobj
System Hamiltonian.
c_op_list : array_like
A ``list`` or ``array`` of collpase operators.
Returns
-------
L : qobj
Louvillian superoperator.
.. note::
Experimental.
"""
d = H.dims[1]
L = Qobj()
L.dims = [[H.dims[0][:], d[:]], [H.dims[1][:], d[:]]]
L.shape = [prod(L.dims[0][0]) * prod(L.dims[0][1]), prod(L.dims[1][0]) * prod(L.dims[1][1])]
L.data = -1j * (
sp.kron(sp.identity(prod(d)), H.data, format="csr") - sp.kron(H.data.T, sp.identity(prod(d)), format="csr")
)
n_op = len(c_op_list)
for m in range(0, n_op):
L.data = L.data + sp.kron(sp.identity(prod(d)), c_op_list[m].data, format="csr") * sp.kron(
c_op_list[m].data.T.conj().T, sp.identity(prod(d)), format="csr"
)
cdc = c_op_list[m].data.T.conj() * c_op_list[m].data
L.data = L.data - 0.5 * sp.kron(sp.identity(prod(d)), cdc, format="csr")
L.data = L.data - 0.5 * sp.kron(cdc.T, sp.identity(prod(d)), format="csr")
return L
def spre(A):
"""Superoperator formed from pre-multiplication by operator A.
Parameters
----------
A : qobj
Quantum operator for pre-multiplication.
Returns
--------
super :qobj
Superoperator formed from input quantum object.
"""
if not isoper(A):
raise TypeError("Input is not a quantum object")
d = A.dims[1]
S = Qobj()
S.dims = [[A.dims[0][:], d[:]], [A.dims[1][:], d[:]]]
S.shape = [prod(S.dims[0][0]) * prod(S.dims[0][1]), prod(S.dims[1][0]) * prod(S.dims[1][1])]
S.data = sp.kron(sp.identity(prod(d)), A.data, format="csr")
return S
def spost(A):
"""Superoperator formed from post-multiplication by operator A
Parameters
----------
A : qobj
Quantum operator for post multiplication.
Returns
-------
super : qobj
Superoperator formed from input qauntum object.
"""
if not isoper(A):
raise TypeError('Input is not a quantum object')
d=A.dims[0]
S=Qobj()
S.dims=[[d[:],A.dims[1][:]],[d[:],A.dims[0][:]]]
S.shape=[prod(S.dims[0][0])*prod(S.dims[0][1]),prod(S.dims[1][0])*prod(S.dims[1][1])]
S.data=sp.kron(A.data.T,sp.identity(prod(d)))
return Qobj(S)
def spre(A):
"""Superoperator formed from pre-multiplication by operator A.
Parameters
----------
A : qobj
Quantum operator for pre-multiplication.
Returns
--------
super :qobj
Superoperator formed from input quantum object.
"""
if not isoper(A):
raise TypeError('Input is not a quantum object')
d=A.dims[1]
S=Qobj()
S.dims=[[A.dims[0][:],d[:]],[A.dims[1][:],d[:]]]
S.shape=[prod(S.dims[0][0])*prod(S.dims[0][1]),prod(S.dims[1][0])*prod(S.dims[1][1])]
S.data=sp.kron(sp.identity(prod(d)),A.data,format='csr')
return S
def liouvillian_fast(H, c_op_list):
"""Assembles the Liouvillian superoperator from a Hamiltonian
and a ``list`` of collapse operators. Like liouvillian, but with an
experimental implementation which avoids creating extra Qobj instances,
which can be advantageous for large systems.
Parameters
----------
H : qobj
System Hamiltonian.
c_op_list : array_like
A ``list`` or ``array`` of collpase operators.
Returns
-------
L : qobj
Louvillian superoperator.
.. note::
Experimental.
"""
d=H.dims[1]
L=Qobj()
L.dims=[[H.dims[0][:],d[:]],[H.dims[1][:],d[:]]]
L.shape=[prod(L.dims[0][0])*prod(L.dims[0][1]),prod(L.dims[1][0])*prod(L.dims[1][1])]
L.data = -1j *(sp.kron(sp.identity(prod(d)), H.data,format='csr') - sp.kron(H.data.T,sp.identity(prod(d)),format='csr'))
n_op = len(c_op_list)
for m in range(0, n_op):
L.data = L.data + sp.kron(sp.identity(prod(d)), c_op_list[m].data,format='csr') * sp.kron(c_op_list[m].data.T.conj().T,sp.identity(prod(d)),format='csr')
cdc = c_op_list[m].data.T.conj()*c_op_list[m].data
L.data = L.data - 0.5 * sp.kron(sp.identity(prod(d)), cdc,format='csr')
L.data = L.data - 0.5 * sp.kron(cdc.T,sp.identity(prod(d)),format='csr')
return L
def steady(L,maxiter=100,tol=1e-6,method='solve'):
"""Steady state for the evolution subject to the
supplied Louvillian.
Parameters
----------
L : qobj
Liouvillian superoperator.
maxiter : int
Maximum number of iterations to perform, default = 100.
tol : float
Tolerance used by iterative solver, default = 1e-6.
method : str
Method for solving linear equations. Direct solver 'solve' (default) or
iterative biconjugate gradient method 'bicg'.
Returns
--------
ket : qobj
Ket vector for steady state.
Notes
-----
Uses the inverse power method.
See any Linear Algebra book with an iterative methods section.
"""
eps=finfo(float).eps
if (not isoper(L)) & (not issuper(L)):
raise TypeError('Steady states can only be found for operators or superoperators.')
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)
L1=L.data+eps*_sp_inf_norm(L)*sp.eye(n,n,format='csr')
v=randn(n,1)
it=0
while (la.norm(L.data*v,np.inf)>tol) and (it<maxiter):
if method=='bicg':
v,check=bicg(L1,v,tol=tol)
else:
v=spsolve(L1,v,use_umfpack=False)
v=v/la.norm(v,np.inf)
it+=1
if it>=maxiter:
raise ValueError('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='lil')
trow=trow.reshape((1,n)).tocsr()
data=v/sum(trow.dot(v))
else:
data=data/la.norm(v)
data=reshape(data,(rhoss.shape[0],rhoss.shape[1])).T
data=sp.csr_matrix(data)
rhoss.data=0.5*(data+data.conj().T)
#data=sp.triu(data,format='csr')#take only upper triangle
#rhoss.data=0.5*sp.eye(rhoss.shape[0],rhoss.shape[1],format='csr')*(data+data.conj().T) #output should be hermitian, but not guarenteed using iterative meth
if qset.auto_tidyup:
return Qobj(rhoss).tidyup()
else:
return Qobj(rhoss)
def steady(L, maxiter=100, tol=1e-6, method='solve'):
"""Steady state for the evolution subject to the
supplied Louvillian.
Parameters
----------
L : qobj
Liouvillian superoperator.
maxiter : int
Maximum number of iterations to perform, default = 100.
tol : float
Tolerance used by iterative solver, default = 1e-6.
method : str
Method for solving linear equations. Direct solver 'solve' (default) or
iterative biconjugate gradient method 'bicg'.
Returns
--------
ket : qobj
Ket vector for steady state.
Notes
-----
Uses the inverse power method.
See any Linear Algebra book with an iterative methods section.
"""
eps = finfo(float).eps
if (not isoper(L)) & (not issuper(L)):
raise TypeError(
'Steady states can only be found for operators or superoperators.')
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)
L1 = L.data + eps * sp_inf_norm(L) * sp.eye(n, n, format='csr')
v = randn(n, 1)
it = 0
while (la.norm(L.data * v, inf) > tol) and (it < maxiter):
if method == 'bicg':
v, check = bicg(L1, v, tol=tol)
else:
v = spsolve(L1, v, use_umfpack=False)
v = v / la.norm(v, inf)
it += 1
if it >= maxiter:
raise ValueError('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='lil')
trow = trow.reshape((1, n)).tocsr()
data = v / sum(trow.dot(v))
else:
data = data / la.norm(v)
data = reshape(data, (rhoss.shape[0], rhoss.shape[1])).T
data = sp.csr_matrix(data)
rhoss.data = 0.5 * (data + data.conj().T)
#data=sp.triu(data,format='csr')#take only upper triangle
#rhoss.data=0.5*sp.eye(rhoss.shape[0],rhoss.shape[1],format='csr')*(data+data.conj().T) #output should be hermitian, but not guarenteed using iterative meth
if qset.auto_tidyup:
return Qobj(rhoss).tidyup()
else:
return Qobj(rhoss)
