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"""
Module contains functions for solving for the steady state density matrix of
open quantum systems defined by a Liouvillian or Hamiltonian and a list of
collapse operators.
"""
__all__ = ['steadystate', 'steady', 'build_preconditioner',
'pseudo_inverse']
import warnings
import time
import scipy
import numpy as np
from numpy.linalg import svd
from scipy import prod
import scipy.sparse as sp
import scipy.linalg as la
from scipy.sparse.linalg import (use_solver, splu, spilu, spsolve, eigs,
LinearOperator, gmres, lgmres, bicgstab)
from qutip.qobj import Qobj, issuper, isoper
from qutip.superoperator import liouvillian, vec2mat
from qutip.sparse import sp_permute, sp_bandwidth, sp_reshape, sp_profile
from qutip.graph import reverse_cuthill_mckee, weighted_bipartite_matching
from qutip import (mat2vec, tensor, identity, operator_to_vector)
import qutip.settings as settings
from qutip.utilities import _version2int
import qutip.logging
logger = qutip.logging.get_logger()
logger.setLevel('DEBUG')
# test if scipy is recent enought to get L & U factors from superLU
_scipy_check = _version2int(scipy.__version__) >= _version2int('0.14.0')
def _default_steadystate_args():
def_args = {'method': 'direct', 'sparse': True, 'use_rcm': False,
'use_wbm': False, 'use_umfpack': False, 'weight': None,
'use_precond': False, 'all_states': False,
'M': None, 'x0': None, 'drop_tol': 1e-4, 'fill_factor': 100,
'diag_pivot_thresh': None, 'maxiter': 1000, 'tol': 1e-9,
'permc_spec': 'COLAMD', 'ILU_MILU': 'smilu_2', 'restart': 20,
'return_info': False, 'info': {'perm': []}}
return def_args
[docs]def steadystate(A, c_op_list=[], **kwargs):
"""Calculates the steady state for quantum evolution subject to the
supplied Hamiltonian or Liouvillian operator and (if given a Hamiltonian) a
list of collapse operators.
If the user passes a Hamiltonian then it, along with the list of collapse
operators, will be converted into a Liouvillian operator in Lindblad form.
Parameters
----------
A : qobj
A Hamiltonian or Liouvillian operator.
c_op_list : list
A list of collapse operators.
method : str {'direct', 'eigen', 'iterative-gmres',
'iterative-lgmres', 'iterative-bicgstab', 'svd', 'power'}
Method for solving the underlying linear equation. Direct LU solver
'direct' (default), sparse eigenvalue problem 'eigen',
iterative GMRES method 'iterative-gmres', iterative LGMRES method
'iterative-lgmres', iterative BICGSTAB method 'iterative-bicgstab',
SVD 'svd' (dense), or inverse-power method 'power'.
return_info : bool, optional, default = False
Return a dictionary of solver-specific infomation about the
solution and how it was obtained.
sparse : bool, optional, default = True
Solve for the steady state using sparse algorithms. If set to False,
the underlying Liouvillian operator will be converted into a dense
matrix. Use only for 'smaller' systems.
use_rcm : bool, optional, default = False
Use reverse Cuthill-Mckee reordering to minimize fill-in in the
LU factorization of the Liouvillian.
use_wbm : bool, optional, default = False
Use Weighted Bipartite Matching reordering to make the Liouvillian
diagonally dominant. This is useful for iterative preconditioners
only, and is set to ``True`` by default when finding a preconditioner.
weight : float, optional
Sets the size of the elements used for adding the unity trace condition
to the linear solvers. This is set to the average abs value of the
Liouvillian elements if not specified by the user.
use_umfpack : bool {False, True}
Use umfpack solver instead of SuperLU. For SciPy 0.14+, this option
requires installing scikits.umfpack.
x0 : ndarray, optional
ITERATIVE ONLY. Initial guess for solution vector.
maxiter : int, optional, default=1000
ITERATIVE ONLY. Maximum number of iterations to perform.
tol : float, optional, default=1e-9
ITERATIVE ONLY. Tolerance used for terminating solver.
permc_spec : str, optional, default='COLAMD'
ITERATIVE ONLY. Column ordering used internally by superLU for the
'direct' LU decomposition method. Options include 'COLAMD' and
'NATURAL'. If using RCM then this is set to 'NATURAL' automatically
unless explicitly specified.
use_precond : bool optional, default = False
ITERATIVE ONLY. Use an incomplete sparse LU decomposition as a
preconditioner for the 'iterative' GMRES and BICG solvers.
Speeds up convergence time by orders of magnitude in many cases.
M : {sparse matrix, dense matrix, LinearOperator}, optional
ITERATIVE ONLY. Preconditioner for A. The preconditioner should
approximate the inverse of A. Effective preconditioning can
dramatically improve the rate of convergence for iterative methods.
If no preconditioner is given and ``use_precond = True``, then one
is generated automatically.
fill_factor : float, optional, default = 100
ITERATIVE ONLY. Specifies the fill ratio upper bound (>=1) of the iLU
preconditioner. Lower values save memory at the cost of longer
execution times and a possible singular factorization.
drop_tol : float, optional, default = 1e-4
ITERATIVE ONLY. Sets the threshold for the magnitude of preconditioner
elements that should be dropped. Can be reduced for a courser
factorization at the cost of an increased number of iterations, and a
possible singular factorization.
diag_pivot_thresh : float, optional, default = None
ITERATIVE ONLY. Sets the threshold between [0,1] for which diagonal
elements are considered acceptable pivot points when using a
preconditioner. A value of zero forces the pivot to be the diagonal
element.
ILU_MILU : str, optional, default = 'smilu_2'
ITERATIVE ONLY. Selects the incomplete LU decomposition method
algoithm used in creating the preconditoner. Should only be used by
advanced users.
Returns
-------
dm : qobj
Steady state density matrix.
info : dict, optional
Dictionary containing solver-specific information about the solution.
Notes
-----
The SVD method works only for dense operators (i.e. small systems).
"""
ss_args = _default_steadystate_args()
for key in kwargs.keys():
if key in ss_args.keys():
ss_args[key] = kwargs[key]
else:
raise Exception(
"Invalid keyword argument '"+key+"' passed to steadystate.")
# Set column perm to NATURAL if using RCM and not specified by user
if ss_args['use_rcm'] and ('permc_spec' not in kwargs.keys()):
ss_args['permc_spec'] = 'NATURAL'
# Create & check Liouvillian
A = _steadystate_setup(A, c_op_list)
# Set weight parameter to avg abs val in L if not set explicitly
if 'weight' not in kwargs.keys():
ss_args['weight'] = np.mean(np.abs(A.data.data.max()))
ss_args['info']['weight'] = ss_args['weight']
if ss_args['method'] == 'direct':
if ss_args['sparse']:
return _steadystate_direct_sparse(A, ss_args)
else:
return _steadystate_direct_dense(A, ss_args)
elif ss_args['method'] == 'eigen':
return _steadystate_eigen(A, ss_args)
elif ss_args['method'] in ['iterative-gmres',
'iterative-lgmres', 'iterative-bicgstab']:
return _steadystate_iterative(A, ss_args)
elif ss_args['method'] == 'svd':
return _steadystate_svd_dense(A, ss_args)
elif ss_args['method'] == 'power':
return _steadystate_power(A, ss_args)
else:
raise ValueError('Invalid method argument for steadystate.')
def _steadystate_setup(A, c_op_list):
"""Build Liouvillian (if necessary) and check input.
"""
if isoper(A):
if len(c_op_list) > 0:
return liouvillian(A, c_op_list)
raise TypeError('Cannot calculate the steady state for a ' +
'non-dissipative system ' +
'(no collapse operators given)')
elif issuper(A):
return A
else:
raise TypeError('Solving for steady states requires ' +
'Liouvillian (super) operators')
def _steadystate_LU_liouvillian(L, ss_args):
"""Creates modified Liouvillian for LU based SS methods.
"""
perm = None
perm2 = None
rev_perm = None
n = prod(L.dims[0][0])
L = L.data.tocsc() + sp.csc_matrix(
(ss_args['weight']*np.ones(n), (np.zeros(n), [nn * (n + 1)
for nn in range(n)])),
shape=(n ** 2, n ** 2))
if settings.debug:
old_band = sp_bandwidth(L)[0]
old_pro = sp_profile(L)[0]
logger.debug('Orig. NNZ: %i' % L.nnz)
if ss_args['use_rcm']:
logger.debug('Original bandwidth: %i' % old_band)
if ss_args['use_wbm']:
if settings.debug:
logger.debug('Calculating Weighted Bipartite Matching ordering...')
_wbm_start = time.time()
perm = weighted_bipartite_matching(L)
_wbm_end = time.time()
L = sp_permute(L, perm, [], 'csc')
ss_args['info']['perm'].append('wbm')
ss_args['info']['wbm_time'] = _wbm_end-_wbm_start
if settings.debug:
wbm_band = sp_bandwidth(L)[0]
logger.debug('WBM bandwidth: %i' % wbm_band)
if ss_args['use_rcm']:
if settings.debug:
logger.debug('Calculating Reverse Cuthill-Mckee ordering...')
_rcm_start = time.time()
perm2 = reverse_cuthill_mckee(L)
_rcm_end = time.time()
rev_perm = np.argsort(perm2)
L = sp_permute(L, perm2, perm2, 'csc')
ss_args['info']['perm'].append('rcm')
ss_args['info']['rcm_time'] = _rcm_end-_rcm_start
if settings.debug:
rcm_band = sp_bandwidth(L)[0]
rcm_pro = sp_profile(L)[0]
logger.debug('RCM bandwidth: %i' % rcm_band)
logger.debug('Bandwidth reduction factor: %f' % round(
old_band/rcm_band, 1))
logger.debug('Profile reduction factor: %f' % round(
old_pro/rcm_pro, 1))
L.sort_indices()
return L, perm, perm2, rev_perm, ss_args
def steady(L, maxiter=10, tol=1e-6, itertol=1e-5, method='solve',
use_umfpack=False, use_precond=False):
"""
Deprecated. See steadystate instead.
"""
message = "steady has been deprecated, use steadystate instead"
warnings.warn(message, DeprecationWarning)
return steadystate(L, [], maxiter=maxiter, tol=tol,
use_umfpack=use_umfpack, use_precond=use_precond)
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 = prod(L.dims[0][0])
b = np.zeros(n ** 2, dtype=complex)
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, useUmfpack=ss_args['use_umfpack'])
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['use_umfpack']:
# 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 umfpack solver
_direct_start = time.time()
v = spsolve(L, b)
_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)
if (not ss_args['use_umfpack']) and 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 _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 = 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
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 _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' %
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['return_info']:
ss_args['info']['residual_norm'] = la.norm(L*eigvec, np.inf)
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 _iterative_precondition(A, n, ss_args):
"""
Internal function for preconditioning the steadystate problem for use
with iterative solvers.
"""
if settings.debug:
logger.debug('Starting preconditioner.')
_precond_start = time.time()
try:
P = spilu(A, permc_spec=ss_args['permc_spec'],
drop_tol=ss_args['drop_tol'],
diag_pivot_thresh=ss_args['diag_pivot_thresh'],
fill_factor=ss_args['fill_factor'],
options=dict(ILU_MILU=ss_args['ILU_MILU']))
P_x = lambda x: P.solve(x)
M = LinearOperator((n ** 2, n ** 2), matvec=P_x)
_precond_end = time.time()
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']
ss_args['info']['precond_time'] = _precond_end-_precond_start
if settings.debug or ss_args['return_info']:
if settings.debug:
logger.debug('Preconditioning succeeded.')
logger.debug('Precond. time: %f' %
(_precond_end - _precond_start))
if _scipy_check:
L_nnz = P.L.nnz
U_nnz = P.U.nnz
ss_args['info']['l_nnz'] = L_nnz
ss_args['info']['u_nnz'] = U_nnz
ss_args['info']['ilu_fill_factor'] = (L_nnz+U_nnz)/A.nnz
e = np.ones(n ** 2, dtype=int)
condest = la.norm(M*e, np.inf)
ss_args['info']['ilu_condest'] = condest
if settings.debug:
logger.debug('L NNZ: %i ; U NNZ: %i' % (L_nnz, U_nnz))
logger.debug('Fill factor: %f' % ((L_nnz+U_nnz)/A.nnz))
logger.debug('iLU condest: %f' % condest)
except:
raise Exception("Failed to build preconditioner. Try increasing " +
"fill_factor and/or drop_tol.")
return M, ss_args
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 = prod(L.dims[0][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, useUmfpack=ss_args['use_umfpack'])
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()
if ss_args['method'] == 'iterative-gmres':
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':
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':
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'])
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 _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']['total_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 _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
[docs]def build_preconditioner(A, c_op_list=[], **kwargs):
"""Constructs a iLU preconditioner necessary for solving for
the steady state density matrix using the iterative linear solvers
in the 'steadystate' function.
Parameters
----------
A : qobj
A Hamiltonian or Liouvillian operator.
c_op_list : list
A list of collapse operators.
return_info : bool, optional, default = False
Return a dictionary of solver-specific infomation about the
solution and how it was obtained.
use_rcm : bool, optional, default = False
Use reverse Cuthill-Mckee reordering to minimize fill-in in the
LU factorization of the Liouvillian.
use_wbm : bool, optional, default = False
Use Weighted Bipartite Matching reordering to make the Liouvillian
diagonally dominant. This is useful for iterative preconditioners
only, and is set to ``True`` by default when finding a preconditioner.
weight : float, optional
Sets the size of the elements used for adding the unity trace condition
to the linear solvers. This is set to the average abs value of the
Liouvillian elements if not specified by the user.
permc_spec : str, optional, default='COLAMD'
Column ordering used internally by superLU for the
'direct' LU decomposition method. Options include 'COLAMD' and
'NATURAL'. If using RCM then this is set to 'NATURAL' automatically
unless explicitly specified.
fill_factor : float, optional, default = 100
Specifies the fill ratio upper bound (>=1) of the iLU
preconditioner. Lower values save memory at the cost of longer
execution times and a possible singular factorization.
drop_tol : float, optional, default = 1e-4
Sets the threshold for the magnitude of preconditioner
elements that should be dropped. Can be reduced for a courser
factorization at the cost of an increased number of iterations, and a
possible singular factorization.
diag_pivot_thresh : float, optional, default = None
Sets the threshold between [0,1] for which diagonal
elements are considered acceptable pivot points when using a
preconditioner. A value of zero forces the pivot to be the diagonal
element.
ILU_MILU : str, optional, default = 'smilu_2'
Selects the incomplete LU decomposition method algoithm used in
creating the preconditoner. Should only be used by advanced users.
Returns
-------
lu : object
Returns a SuperLU object representing iLU preconditioner.
info : dict, optional
Dictionary containing solver-specific information.
"""
ss_args = _default_steadystate_args()
for key in kwargs.keys():
if key in ss_args.keys():
ss_args[key] = kwargs[key]
else:
raise Exception("Invalid keyword argument '" + key +
"' passed to steadystate.")
# Set column perm to NATURAL if using RCM and not specified by user
if ss_args['use_rcm'] and ('permc_spec' not in kwargs.keys()):
ss_args['permc_spec'] = 'NATURAL'
L = _steadystate_setup(A, c_op_list)
# Set weight parameter to avg abs val in L if not set explicitly
if 'weight' not in kwargs.keys():
ss_args['weight'] = np.mean(np.abs(L.data.data.max()))
ss_args['info']['weight'] = ss_args['weight']
n = prod(L.dims[0][0])
L, perm, perm2, rev_perm, ss_args = _steadystate_LU_liouvillian(L, ss_args)
M, ss_args = _iterative_precondition(L, n, ss_args)
if ss_args['return_info']:
return M, ss_args['info']
else:
return M
def _pseudo_inverse_dense(L, rhoss, method='direct'):
"""
Internal function for computing the pseudo inverse of an Liouvillian using
dense matrix methods. See pseudo_inverse for details.
"""
if method == 'direct':
rho_vec = np.transpose(mat2vec(rhoss.full()))
tr_mat = tensor([identity(n) for n in L.dims[0][0]])
tr_vec = np.transpose(mat2vec(tr_mat.full()))
N = np.prod(L.dims[0][0])
I = np.identity(N * N)
P = np.kron(np.transpose(rho_vec), tr_vec)
Q = I - P
LIQ = np.linalg.solve(L.full(), Q)
R = np.dot(Q, LIQ)
return Qobj(R, dims=L.dims)
elif method == 'numpy':
return Qobj(np.linalg.pinv(L.full()), dims=L.dims)
elif method == 'scipy':
return Qobj(la.pinv(L.full()), dims=L.dims)
elif method == 'scipy2':
return Qobj(la.pinv2(L.full()), dims=L.dims)
else:
raise ValueError("Unsupported method '%s'. Use 'direct' or 'numpy'" %
method)
def _pseudo_inverse_sparse(L, rhoss, method='splu', use_umfpack=False,
use_rcm=False):
"""
Internal function for computing the pseudo inverse of an Liouvillian using
sparse matrix methods. See pseudo_inverse for details.
"""
N = np.prod(L.dims[0][0])
rhoss_vec = operator_to_vector(rhoss)
tr_op = tensor([identity(n) for n in L.dims[0][0]])
tr_op_vec = operator_to_vector(tr_op)
P = sp.kron(rhoss_vec.data, tr_op_vec.data.T, format='csc')
I = sp.eye(N*N, N*N, format='csc')
Q = I - P
if use_rcm:
perm = reverse_cuthill_mckee(L.data)
A = sp_permute(L.data, perm, perm, 'csc').tocsc()
Q = sp_permute(Q, perm, perm, 'csc')
permc_spec = 'NATURAL'
else:
A = L.data.tocsc()
A.sort_indices()
permc_spec = 'COLAMD'
if method == 'spsolve':
sp.linalg.use_solver(assumeSortedIndices=True, useUmfpack=use_umfpack)
LIQ = sp.linalg.spsolve(A, Q)
elif method == 'splu':
lu = sp.linalg.splu(A, permc_spec=permc_spec)
LIQ = lu.solve(Q.toarray())
elif method == 'spilu':
lu = sp.linalg.spilu(A, permc_spec=permc_spec,
fill_factor=10, drop_tol=1e-8)
LIQ = lu.solve(Q.toarray())
else:
raise ValueError("unsupported method '%s'" % method)
R = sp.csc_matrix(Q * LIQ)
if use_rcm:
rev_perm = np.argsort(perm)
R = sp_permute(R, rev_perm, rev_perm, 'csc')
return Qobj(R, dims=L.dims)
def pseudo_inverse(L, rhoss=None, sparse=True, method='splu', **kwargs):
"""
Compute the pseudo inverse for a Liouvillian superoperator, optionally
given its steadystate density matrix (which will be computed if not given).
Returns
-------
L : Qobj
A Liouvillian superoperator for which to compute the pseudo inverse.
rhoss : Qobj
A steadystate density matrix as Qobj instance, for the Liouvillian
superoperator L.
sparse : bool
Flag that indicate whether to use sparse or dense matrix methods when
computing the pseudo inverse.
method : string
Name of method to use. For sparse=True, allowed values are 'spsolve',
'splu' and 'spilu'. For sprase=False, allowed values are 'direct' and
'numpy'.
kwargs : dictionary
Additional keyword arguments for setting paramters for solver methods.
Currently supported arguments are use_rcm (for sparse=True),
use_umfpack (for sparse=True and method='spsolve').
Returns
-------
R : Qobj
Returns a Qobj instance representing the pseudo inverse of L.
"""
if rhoss is None:
rhoss = steadystate(L)
if sparse:
return _pseudo_inverse_sparse(L, rhoss, method=method, **kwargs)
else:
method = method if method != 'splu' else 'direct'
return _pseudo_inverse_dense(L, rhoss, method=method, **kwargs)