# This file is part of QuTiP: Quantum Toolbox in Python.
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__all__ = ['liouvillian', 'liouvillian_ref', 'lindblad_dissipator',
'operator_to_vector', 'vector_to_operator', 'mat2vec', 'vec2mat',
'vec2mat_index', 'mat2vec_index', 'spost', 'spre', 'sprepost']
import scipy.sparse as sp
import numpy as np
from qutip.qobj import Qobj
from qutip.fastsparse import fast_csr_matrix, fast_identity
from qutip.sparse import sp_reshape
from qutip.cy.spmath import zcsr_kron
from functools import partial
[docs]def liouvillian(H, c_ops=[], data_only=False, chi=None):
"""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 or QobjEvo
System Hamiltonian.
c_ops : array_like of Qobj or QobjEvo
A ``list`` or ``array`` of collapse operators.
Returns
-------
L : Qobj or QobjEvo
Liouvillian superoperator.
"""
if isinstance(c_ops, (Qobj, QobjEvo)):
c_ops = [c_ops]
if chi and len(chi) != len(c_ops):
raise ValueError('chi must be a list with same length as c_ops')
h = None
if H is not None:
if isinstance(H, QobjEvo):
h = H.cte
else:
h = H
if h.isoper:
op_dims = h.dims
op_shape = h.shape
elif h.issuper:
op_dims = h.dims[0]
op_shape = [np.prod(op_dims[0]), np.prod(op_dims[0])]
else:
raise TypeError("Invalid type for Hamiltonian.")
else:
# no hamiltonian given, pick system size from a collapse operator
if isinstance(c_ops, list) and len(c_ops) > 0:
if isinstance(c_ops[0], QobjEvo):
c = c_ops[0].cte
else:
c = c_ops[0]
if c.isoper:
op_dims = c.dims
op_shape = c.shape
elif c.issuper:
op_dims = c.dims[0]
op_shape = [np.prod(op_dims[0]), np.prod(op_dims[0])]
else:
raise TypeError("Invalid type for collapse operator.")
else:
raise TypeError("Either H or c_ops must be given.")
sop_dims = [[op_dims[0], op_dims[0]], [op_dims[1], op_dims[1]]]
sop_shape = [np.prod(op_dims), np.prod(op_dims)]
spI = fast_identity(op_shape[0])
td = False
L = None
if isinstance(H, QobjEvo):
td = True
def H2L(H):
if H.isoper:
return -1.0j * (spre(H) - spost(H))
else:
return H
L = H.apply(H2L)
data = L.cte.data
elif isinstance(H, Qobj):
if H.isoper:
Ht = H.data.T
data = -1j * zcsr_kron(spI, H.data)
data += 1j * zcsr_kron(Ht, spI)
else:
data = H.data
else:
data = fast_csr_matrix(shape=(sop_shape[0], sop_shape[1]))
td_c_ops = []
for idx, c_op in enumerate(c_ops):
if isinstance(c_op, QobjEvo):
td = True
if c_op.const:
c_ = c_op.cte
elif chi:
td_c_ops.append(lindblad_dissipator(c_op, chi=chi[idx]))
continue
else:
td_c_ops.append(lindblad_dissipator(c_op))
continue
else:
c_ = c_op
if c_.issuper:
data = data + c_.data
else:
cd = c_.data.H
c = c_.data
if chi:
data = data + np.exp(1j * chi[idx]) * \
zcsr_kron(c.conj(), c)
else:
data = data + zcsr_kron(c.conj(), c)
cdc = cd * c
cdct = cdc.T
data = data - 0.5 * zcsr_kron(spI, cdc)
data = data - 0.5 * zcsr_kron(cdct, spI)
if not td:
if data_only:
return data
else:
L = Qobj()
L.dims = sop_dims
L.data = data
L.superrep = 'super'
return L
else:
if not L:
l = Qobj()
l.dims = sop_dims
l.data = data
l.superrep = 'super'
L = QobjEvo(l)
else:
L.cte.data = data
for c_op in td_c_ops:
L += c_op
return L
def liouvillian_ref(H, c_ops=[]):
"""Assembles the Liouvillian superoperator from a Hamiltonian
and a ``list`` of collapse operators.
Parameters
----------
H : qobj
System Hamiltonian.
c_ops : array_like
A ``list`` or ``array`` of collapse operators.
Returns
-------
L : qobj
Liouvillian superoperator.
"""
L = -1.0j * (spre(H) - spost(H)) if H else 0
for c in c_ops:
if c.issuper:
L += c
else:
cdc = c.dag() * c
L += spre(c) * spost(c.dag()) - 0.5 * spre(cdc) - 0.5 * spost(cdc)
return L
[docs]def lindblad_dissipator(a, b=None, data_only=False, chi=None):
"""
Lindblad dissipator (generalized) for a single pair of collapse operators
(a, b), or for a single collapse operator (a) when b is not specified:
.. math::
\\mathcal{D}[a,b]\\rho = a \\rho b^\\dagger -
\\frac{1}{2}a^\\dagger b\\rho - \\frac{1}{2}\\rho a^\\dagger b
Parameters
----------
a : Qobj or QobjEvo
Left part of collapse operator.
b : Qobj or QobjEvo (optional)
Right part of collapse operator. If not specified, b defaults to a.
Returns
-------
D : qobj, QobjEvo
Lindblad dissipator superoperator.
"""
if b is None:
b = a
ad_b = a.dag() * b
if chi:
D = spre(a) * spost(b.dag()) * np.exp(1j * chi) \
- 0.5 * spre(ad_b) - 0.5 * spost(ad_b)
else:
D = spre(a) * spost(b.dag()) - 0.5 * spre(ad_b) - 0.5 * spost(ad_b)
if isinstance(a, QobjEvo) or isinstance(b, QobjEvo):
return D
else:
return D.data if data_only else D
[docs]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,
)
[docs]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 mat2vec(mat):
"""
Private function reshaping matrix to vector.
"""
return mat.T.reshape(np.prod(np.shape(mat)), 1)
def vec2mat(vec, shape=None):
"""
Private function reshaping vector to matrix.
"""
if shape is None:
n = int(np.sqrt(len(vec)))
shape = (n, n)
return vec.reshape(shape[::-1]).T
def vec2mat_index(N, I):
"""
Convert a vector index to a matrix index pair that is compatible with the
vector to matrix rearrangement done by the vec2mat function.
"""
j = int(I / N)
i = I - N * j
return i, j
def mat2vec_index(N, i, j):
"""
Convert a matrix index pair to a vector index that is compatible with the
matrix to vector rearrangement done by the mat2vec function.
"""
return i + N * j
[docs]def spost(A):
"""Superoperator formed from post-multiplication by operator A
Parameters
----------
A : Qobj or QobjEvo
Quantum operator for post multiplication.
Returns
-------
super : Qobj or QobjEvo
Superoperator formed from input qauntum object.
"""
if isinstance(A, QobjEvo):
return A.apply(spost)
if not isinstance(A, Qobj):
raise TypeError('Input is not a quantum object')
if not A.isoper:
raise TypeError('Input is not a quantum operator')
S = Qobj(isherm=A.isherm, superrep='super')
S.dims = [[A.dims[0], A.dims[1]], [A.dims[0], A.dims[1]]]
S.data = zcsr_kron(A.data.T,
fast_identity(np.prod(A.shape[0])))
return S
[docs]def spre(A):
"""Superoperator formed from pre-multiplication by operator A.
Parameters
----------
A : Qobj or QobjEvo
Quantum operator for pre-multiplication.
Returns
--------
super :Qobj or QobjEvo
Superoperator formed from input quantum object.
"""
if isinstance(A, QobjEvo):
return A.apply(spre)
if not isinstance(A, Qobj):
raise TypeError('Input is not a quantum object')
if not A.isoper:
raise TypeError('Input is not a quantum operator')
S = Qobj(isherm=A.isherm, superrep='super')
S.dims = [[A.dims[0], A.dims[1]], [A.dims[0], A.dims[1]]]
S.data = zcsr_kron(fast_identity(np.prod(A.shape[1])), A.data)
return S
def _drop_projected_dims(dims):
"""
Eliminate subsystems that has been collapsed to only one state due to
a projection.
"""
return [d for d in dims if d != 1]
[docs]def sprepost(A, B):
"""Superoperator formed from pre-multiplication by operator A and post-
multiplication of operator B.
Parameters
----------
A : Qobj or QobjEvo
Quantum operator for pre-multiplication.
B : Qobj or QobjEvo
Quantum operator for post-multiplication.
Returns
--------
super : Qobj or QobjEvo
Superoperator formed from input quantum objects.
"""
if isinstance(A, QobjEvo) or isinstance(B, QobjEvo):
return spre(A) * spost(B)
else:
dims = [[_drop_projected_dims(A.dims[0]),
_drop_projected_dims(B.dims[1])],
[_drop_projected_dims(A.dims[1]),
_drop_projected_dims(B.dims[0])]]
data = zcsr_kron(B.data.T, A.data)
return Qobj(data, dims=dims, superrep='super')
from qutip.qobjevo import QobjEvo