steadystate.py

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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.
"""
import warnings
import time
import scipy
import numpy as np
from numpy.linalg import svd
from scipy import prod, randn
import scipy.sparse as sp
import scipy.linalg as la
from scipy.sparse.linalg import *
from qutip.qobj import Qobj, issuper, isoper
from qutip.superoperator import liouvillian, vec2mat, mat2vec
from qutip.operators import qeye
from qutip.random_objects import rand_dm
from qutip.sparse import sp_permute, sp_bandwidth, sp_reshape
from qutip.graph import reverse_cuthill_mckee, weighted_bipartite_matching
from qutip.states import ket2dm, maximally_mixed_dm
import qutip.settings as settings
from qutip.utilities import _version2int

if settings.debug:
    import inspect

# 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': True, 'all_states': False,
                'M': None, 'x0' : None, 'drop_tol': 1e-4, 'fill_factor': 100,
                'diag_pivot_thresh': None, 'maxiter': 10000, 'tol': 1e-9,
                'permc_spec': 'COLAMD', 'ILU_MILU': 'smilu_2'}

    return def_args


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-bicg', 'iterative-bicgstab',
                  'iterative-gmres', 'iterative-lgmres', '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', SVD 'svd' (dense), or inverse-power method 'power'.

    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=True
        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.

    maxiter : int, optional, default=10000
        Maximum number of iterations to perform if using an iterative method.

    tol : float, optional, default=1e-9
        Tolerance used for terminating solver solution when using iterative
        solvers.

    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.

    use_precond : bool optional, default = True
        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
        Preconditioner for A. The preconditioner should approximate the inverse
        of A. Effective preconditioning dramatically improves the rate of
        convergence, for iterative methods only .  If no preconditioner is
        given and ``use_precond=True``, then one is generated automatically.

    fill_factor : float, optional, default=10
        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-3
        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'
        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.

    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'

    # Set use_wbm=True if using iterative solver with preconditioner and
    # not explicitly set to False by user
    if (ss_args['method'] in ['iterative-lgmres', 'iterative-gmres']) \
            and ('use_wbm' not in kwargs.keys()):
        ss_args['use_wbm'] = True

    n_op = len(c_op_list)

    if isoper(A):
        if n_op == 0:
            raise TypeError('Cannot calculate the steady state for a ' +
                            'non-dissipative system ' +
                            '(no collapse operators given)')
        else:
            A = liouvillian(A, c_op_list)
    if not issuper(A):
        raise TypeError('Solving for steady states requires ' +
                        'Liouvillian (super) operators')

    # 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()))

    if ss_args['method'] == 'direct':
        if ss_args['sparse']:
            return _steadystate_direct_sparse(A, ss_args)
        else:
            return _steadystate_direct_dense(A)

    elif ss_args['method'] == 'eigen':
        return _steadystate_eigen(A, ss_args)

    elif ss_args['method'] in ['iterative-gmres', 'iterative-lgmres',
                               'iterative-bicg', '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 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:
        print('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 = 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))
    L.sort_indices()
    use_solver(assumeSortedIndices=True, useUmfpack=ss_args['use_umfpack'])

    if not ss_args['use_umfpack']:
        # Use superLU solver
        orig_nnz = L.nnz
        if settings.debug:
            old_band = sp_bandwidth(L)[0]
            print('Original NNZ:', orig_nnz)
            if ss_args['use_rcm']:
                print('Original bandwidth:', old_band)

        if ss_args['use_wbm']:
            perm = weighted_bipartite_matching(L)
            L = sp_permute(L, perm, [], 'csc')
            b = b[np.ix_(perm,)]

        if ss_args['use_rcm']:
            perm2 = reverse_cuthill_mckee(L)
            rev_perm = np.argsort(perm2)
            L = sp_permute(L, perm2, perm2, 'csc')
            b = b[np.ix_(perm2,)]
            if settings.debug:
                rcm_band = sp_bandwidth(L)[0]
                print('RCM bandwidth:', rcm_band)
                print('Bandwidth reduction factor:', round(
                    old_band/rcm_band, 1))

        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)

        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)

    else:
        # Use umfpack solver
        v = spsolve(L, b)

    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)
    return Qobj(data, dims=dims, isherm=True)


def _steadystate_direct_dense(L):
    """
    Direct solver that use numpy dense matrices. Suitable for
    small system, with a few states.
    """
    if settings.debug:
        print('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))
    v = np.linalg.solve(L, b)

    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.
    """
    if settings.debug:
        print('Starting Eigen solver...')

    dims = L.dims[0]
    shape = prod(dims[0])
    L = L.data.tocsc()

    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')
        if settings.debug:
            rcm_band = sp_bandwidth(L)[0]
            print('RCM bandwidth:', rcm_band)
            print('Bandwidth reduction factor:', round(old_band/rcm_band, 1))

    eigval, eigvec = eigs(L, k=1, sigma=1e-15, tol=ss_args['tol'],
                          which='LM', maxiter=ss_args['maxiter'])

    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)
    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:
        print('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)
        if settings.debug:
            print('Preconditioning succeeded.')
            print('Precond. time:',time.time()-_precond_start)
            if _scipy_check:
                L_nnz = P.L.nnz
                U_nnz = P.U.nnz
                print('L NNZ:', L_nnz, ';', 'U NNZ:', U_nnz)
                print('Fill factor:', (L_nnz+U_nnz)/A.nnz)
                e = np.ones(n ** 2,dtype=int)
                condest = la.norm(M*e,np.inf)
                print('iLU Condest:', condest)
    except:
        warnings.warn("Preconditioning failed. Continuing without.",
                      UserWarning)
        M = None

    return M


def _steadystate_iterative(L, ss_args):
    """
    Iterative steady state solver using the GMRES or LGMRES algorithm
    and a sparse incomplete LU preconditioner.
    """

    if settings.debug:
        print('Starting '+ss_args['method']+' solver...')

    dims = L.dims[0]
    n = prod(L.dims[0][0])
    b = np.zeros(n ** 2)
    b[0] = ss_args['weight']
    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 ss_args['use_wbm']:
        perm = weighted_bipartite_matching(L)
        L = sp_permute(L, perm, [], 'csc')
        b = b[np.ix_(perm,)]

    if ss_args['use_rcm']:
        if settings.debug:
            old_band = sp_bandwidth(L)[0]
            print('Original bandwidth ', old_band)
        perm2 = reverse_cuthill_mckee(L)
        rev_perm = np.argsort(perm2)
        L = sp_permute(L, perm2, perm2, 'csc')
        b = b[np.ix_(perm2,)]
        if settings.debug:
            rcm_band = sp_bandwidth(L)[0]
            print('RCM bandwidth ', rcm_band)
            print('Bandwidth reduction factor:', round(old_band/rcm_band, 1))

    L.sort_indices()
    if ss_args['M'] is None and ss_args['use_precond']:
        ss_args['M'] = _iterative_precondition(L, n, ss_args)

    # Select iterative solver type
    if settings.debug:
        _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'],
                         maxiter=ss_args['maxiter'])

    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'])

    elif ss_args['method'] == 'iterative-bicg':
        v, check = bicg(L, b, tol=ss_args['tol'], M=ss_args['M'],
                        x0=ss_args['x0'],
                        maxiter=ss_args['maxiter'])

    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'])
    else:
        raise Exception("Invalid iterative solver method.")

    if settings.debug:
        _iter_end = time.time()
        print('Iteration. time:',_iter_end-_iter_start)

    if check > 0:
        raise Exception("Steadystate solver did not reach tolerance after " +
                        str(check) + " steps.")
    elif check < 0:
        raise Exception(
            "Steadystate solver 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)

    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.
    """
    atol = 1e-12
    rtol = 1e-12
    if settings.debug:
        print('Starting SVD solver...')

    u, s, vh = svd(L.full(), full_matrices=False)
    tol = max(atol, rtol * s[0])
    nnz = (s >= tol).sum()
    ns = vh[nnz:].conj().T

    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())
        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.
    """
    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)

    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')

    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
    return rhoss