"""
This module provides solvers for system-bath evoluation using the
HEOM (hierarchy equations of motion).
See https://en.wikipedia.org/wiki/Hierarchical_equations_of_motion for a very
basic introduction to the technique.
The implementation is derived from the BoFiN library (see
https://github.com/tehruhn/bofin) which was itself derived from an earlier
implementation in QuTiP itself.
"""
from time import time
import numpy as np
import scipy.sparse as sp
from scipy.sparse.linalg import spsolve
from qutip.settings import settings
from qutip import state_number_enumerate
from qutip.core import data as _data
from qutip.core.data import csr as _csr
from qutip.core import Qobj, QobjEvo
from qutip.core.superoperator import liouvillian, spre, spost
from .bofin_baths import (
BathExponent, DrudeLorentzBath,
)
from ..solver_base import Solver
from .. import Result
# Load MKL spsolve if avaiable
if settings.has_mkl:
from qutip._mkl.spsolve import mkl_spsolve
else:
mkl_spsolve = None
__all__ = [
"heomsolve",
"HierarchyADOs",
"HierarchyADOsState",
"HEOMResult",
"HEOMSolver",
"HSolverDL",
]
[docs]class HierarchyADOs:
"""
A description of ADOs (auxilliary density operators) with the
hierarchical equations of motion.
The list of ADOs is constructed from a list of bath exponents
(corresponding to one or more baths). Each ADO is referred to by a label
that lists the number of "excitations" of each bath exponent. The
level of a label within the hierarchy is the sum of the "excitations"
within the label.
For example the label ``(0, 0, ..., 0)`` represents the density matrix
of the system being solved and is the only 0th level label.
The labels with a single 1, i.e. ``(1, 0, ..., 0)``, ``(0, 1, 0, ... 0)``,
etc. are the 1st level labels.
The second level labels all have either two 1s or a single 2, and so on
for the third and higher levels of the hierarchy.
Parameters
----------
exponents : list of BathExponent
The exponents of the correlation function describing the bath or
baths.
max_depth : int
The maximum depth of the hierarchy (i.e. the maximum sum of
"excitations" in the hierarchy ADO labels or maximum ADO level).
Attributes
----------
exponents : list of BathExponent
The exponents of the correlation function describing the bath or
baths.
max_depth : int
The maximum depth of the hierarchy (i.e. the maximum sum of
"excitations" in the hierarchy ADO labels).
dims : list of int
The dimensions of each exponent within the bath(s).
vk : list of complex
The frequency of each exponent within the bath(s).
ck : list of complex
The coefficient of each exponent within the bath(s).
ck2: list of complex
For exponents of type "RI", the coefficient of the exponent within
the imaginary expansion. For other exponent types, the entry is None.
sigma_bar_k_offset: list of int
For exponents of type "+" or "-" the offset within the list of modes
of the corresponding "-" or "+" exponent. For other exponent types,
the entry is None.
labels: list of tuples
A list of the ADO labels within the hierarchy.
"""
def __init__(self, exponents, max_depth):
self.exponents = exponents
self.max_depth = max_depth
self.dims = [exp.dim or (max_depth + 1) for exp in self.exponents]
self.vk = [exp.vk for exp in self.exponents]
self.ck = [exp.ck for exp in self.exponents]
self.ck2 = [exp.ck2 for exp in self.exponents]
self.sigma_bar_k_offset = [
exp.sigma_bar_k_offset for exp in self.exponents
]
self.labels = list(state_number_enumerate(self.dims, max_depth))
self._label_idx = {s: i for i, s in enumerate(self.labels)}
self.idx = self._label_idx.__getitem__
[docs] def idx(self, label):
"""
Return the index of the ADO label within the list of labels,
i.e. within ``self.labels``.
Parameters
----------
label : tuple
The label to look up.
Returns
-------
int
The index of the label within the list of ADO labels.
Notes
-----
This implementation of the ``.idx(...)`` method is just for
reference and documentation. To avoid the cost of a Python
function call, it is replaced with
``self._label_idx.__getitem__`` when the instance is created.
"""
return self._label_idx[label]
[docs] def next(self, label, k):
"""
Return the ADO label with one more excitation in the k'th exponent
dimension or ``None`` if adding the excitation would exceed the
dimension or maximum depth of the hierarchy.
Parameters
----------
label : tuple
The ADO label to add an excitation to.
k : int
The exponent to add the excitation to.
Returns
-------
tuple or None
The next label.
"""
if label[k] >= self.dims[k] - 1:
return None
if sum(label) >= self.max_depth:
return None
return label[:k] + (label[k] + 1,) + label[k + 1:]
[docs] def prev(self, label, k):
"""
Return the ADO label with one fewer excitation in the k'th
exponent dimension or ``None`` if the label has no exciations in the
k'th exponent.
Parameters
----------
label : tuple
The ADO label to remove the excitation from.
k : int
The exponent to remove the excitation from.
Returns
-------
tuple or None
The previous label.
"""
if label[k] <= 0:
return None
return label[:k] + (label[k] - 1,) + label[k + 1:]
[docs] def exps(self, label):
"""
Converts an ADO label into a tuple of exponents, with one exponent
for each "excitation" within the label.
The number of exponents returned is always equal to the level of the
label within the hierarchy (i.e. the sum of the indices within the
label).
Parameters
----------
label : tuple
The ADO label to convert to a list of exponents.
Returns
-------
tuple of BathExponent
A tuple of BathExponents.
Examples
--------
``ados.exps((1, 0, 0))`` would return ``[ados.exponents[0]]``
``ados.exps((2, 0, 0))`` would return
``[ados.exponents[0], ados.exponents[0]]``.
``ados.exps((1, 2, 1))`` would return
``[ados.exponents[0], ados.exponents[1], ados.exponents[1], \
ados.exponents[2]]``.
"""
return sum(
((exp,) * n for (n, exp) in zip(label, self.exponents) if n > 0),
(),
)
[docs] def filter(self, level=None, tags=None, dims=None, types=None):
"""
Return a list of ADO labels for ADOs whose "excitations"
match the given patterns.
Each of the filter parameters (tags, dims, types) may be either
unspecified (None) or a list. Unspecified parameters are excluded
from the filtering.
All specified filter parameters must be lists of the same length.
Each position in the lists describes a particular excitation and
any exponent that matches the filters may supply that excitation.
The level of all labels returned is thus equal to the length of
the filter parameter lists.
Within a filter parameter list, items that are None represent
wildcards and match any value of that exponent attribute
Parameters
----------
level : int
The hierarchy depth to return ADOs from.
tags : list of object or None
Filter parameter that matches the ``.tag`` attribute of
exponents.
dims : list of int
Filter parameter that matches the ``.dim`` attribute of
exponents.
types : list of BathExponent types or list of str
Filter parameter that matches the ``.type`` attribute
of exponents. Types may be supplied by name (e.g. "R", "I", "+")
instead of by the actual type (e.g. ``BathExponent.types.R``).
Returns
-------
list of tuple
The ADO label for each ADO whose exponent excitations
(i.e. label) match the given filters or level.
"""
if types is not None:
types = [
t if t is None or isinstance(t, BathExponent.types)
else BathExponent.types[t]
for t in types
]
filters = [("tag", tags), ("type", types), ("dim", dims)]
filters = [(attr, f) for attr, f in filters if f is not None]
n = max((len(f) for _, f in filters), default=0)
if any(len(f) != n for _, f in filters):
raise ValueError(
"The tags, dims and types filters must all be the same length."
)
if n > self.max_depth:
raise ValueError(
f"The maximum depth for the hierarchy is {self.max_depth} but"
f" {n} levels of excitation filters were given."
)
if level is None:
if not filters:
# fast path for when there are no excitation filters
return self.labels[:]
else:
if not filters:
# fast path for when there are no excitation filters
return [label for label in self.labels if sum(label) == level]
if level != n:
raise ValueError(
f"The level parameter is {level} but {n} levels of"
" excitation filters were given."
)
filtered_dims = [1] * len(self.exponents)
for lvl in range(n):
level_filters = [
(attr, f[lvl]) for attr, f in filters
if f[lvl] is not None
]
for j, exp in enumerate(self.exponents):
if any(getattr(exp, attr) != f for attr, f in level_filters):
continue
filtered_dims[j] += 1
filtered_dims[j] = min(self.dims[j], filtered_dims[j])
return [
label for label in state_number_enumerate(filtered_dims, n)
if sum(label) == n
]
[docs]class HierarchyADOsState:
"""
Provides convenient access to the full hierarchy ADO state at a particular
point in time, ``t``.
Parameters
----------
rho : :class:`.Qobj`
The current state of the system (i.e. the 0th component of the
hierarchy).
ados : :class:`HierarchyADOs`
The description of the hierarchy.
ado_state : numpy.array
The full state of the hierarchy.
Attributes
----------
rho : Qobj
The system state.
In addition, all of the attributes of the hierarchy description,
i.e. ``HierarchyADOs``, are provided directly on this class for
convenience. E.g. one can access ``.labels``, or ``.exponents`` or
call ``.idx(label)`` directly.
See :class:`HierarchyADOs` for a full list of the available attributes
and methods.
"""
def __init__(self, rho, ados, ado_state):
self.rho = rho
self._ado_state = ado_state
self._ados = ados
def __getattr__(self, name):
return getattr(self._ados, name)
[docs]class HEOMResult(Result):
def _post_init(self):
super()._post_init()
self.store_ados = self.options["store_ados"]
if self.store_ados:
self._final_ado_state = None
self.ado_states = []
def _e_op_func(self, e_op):
""" Convert an e_op into a function ``f(t, ado_state)``. """
if isinstance(e_op, Qobj):
return lambda t, ado_state: (ado_state.rho * e_op).tr()
elif isinstance(e_op, QobjEvo):
return lambda t, ado_state: e_op.expect(t, ado_state.rho)
elif callable(e_op):
return e_op
raise TypeError(f"{e_op!r} has unsupported type {type(e_op)!r}.")
def _pre_copy(self, state):
return state
def _store_state(self, t, ado_state):
self.states.append(ado_state.rho)
if self.store_ados:
self.ado_states.append(ado_state)
def _store_final_state(self, t, ado_state):
self._final_state = ado_state.rho
if self.store_ados:
self._final_ado_state = ado_state
@property
def final_ado_state(self):
if self._final_ado_state is not None:
return self._final_state
if self.ado_states:
return self.ado_states[-1]
return None
[docs]def heomsolve(
H, bath, max_depth, state0, tlist, *, e_ops=None, args=None, options=None,
):
"""
Hierarchical Equations of Motion (HEOM) solver that supports multiple
baths.
The baths must be all either bosonic or fermionic baths.
If you need to run many evolutions of the same system and bath, consider
using :class:`HEOMSolver` directly to avoid having to continually
reconstruct the equation hierarchy for every evolution.
Parameters
----------
H : :obj:`.Qobj`, :obj:`.QobjEvo`
Possibly time-dependent system Liouvillian or Hamiltonian as a Qobj or
QobjEvo. list of [:obj:`.Qobj`, :obj:`.Coefficient`] or callable that
can be made into :obj:`.QobjEvo` are also accepted.
bath : Bath or list of Bath
A :obj:`Bath` containing the exponents of the expansion of the
bath correlation funcion and their associated coefficients
and coupling operators, or a list of baths.
If multiple baths are given, they must all be either fermionic
or bosonic baths.
max_depth : int
The maximum depth of the heirarchy (i.e. the maximum number of bath
exponent "excitations" to retain).
state0 : :obj:`.Qobj` or :class:`~HierarchyADOsState` or array-like
If ``rho0`` is a :obj:`.Qobj` the it is the initial state
of the system (i.e. a :obj:`.Qobj` density matrix).
If it is a :class:`~HierarchyADOsState` or array-like, then
``rho0`` gives the initial state of all ADOs.
Usually the state of the ADOs would be determine from a previous
call to ``.run(...)`` with the solver results option ``store_ados``
set to True. For example, ``result = solver.run(...)`` could be
followed by ``solver.run(result.ado_states[-1], tlist)``.
If a numpy array-like is passed its shape must be
``(number_of_ados, n, n)`` where ``(n, n)`` is the system shape
(i.e. shape of the system density matrix) and the ADOs must
be in the same order as in ``.ados.labels``.
tlist : list
An ordered list of times at which to return the value of the state.
e_ops : Qobj / QobjEvo / callable / list / dict / None, optional
A list or dictionary of operators as :obj:`.Qobj`,
:obj:`.QobjEvo` and/or callable functions (they can be mixed) or
a single operator or callable function. For an operator ``op``, the
result will be computed using ``(state * op).tr()`` and the state
at each time ``t``. For callable functions, ``f``, the result is
computed using ``f(t, ado_state)``. The values are stored in the
``expect`` and ``e_data`` attributes of the result (see the return
section below).
args : dict, optional
Change the ``args`` of the RHS for the evolution.
options : dict, optional
Generic solver options.
- | store_final_state : bool
| Whether or not to store the final state of the evolution in the
result class.
- | store_states : bool, None
| Whether or not to store the state vectors or density matrices.
On `None` the states will be saved if no expectation operators are
given.
- | store_ados : bool
| Whether or not to store the HEOM ADOs.
- | normalize_output : bool
| Normalize output state to hide ODE numerical errors.
- | progress_bar : str {'text', 'enhanced', 'tqdm', ''}
| How to present the solver progress.
'tqdm' uses the python module of the same name and raise an error
if not installed. Empty string or False will disable the bar.
- | progress_kwargs : dict
| kwargs to pass to the progress_bar. Qutip's bars use `chunk_size`.
- | state_data_type: str {'dense', 'CSR', 'Dia', }
| Name of the data type of the state used during the ODE evolution.
Use an empty string to keep the input state type. Many integrator
can only work with `Dense`.
- | method : str ["adams", "bdf", "lsoda", "dop853", "vern9", etc.]
| Which differential equation integration method to use.
- | atol, rtol : float
| Absolute and relative tolerance of the ODE integrator.
- | nsteps : int
| Maximum number of (internally defined) steps allowed in one
``tlist`` step.
- | max_step : float,
| Maximum lenght of one internal step. When using pulses, it should
be less than half the width of the thinnest pulse.
Returns
-------
:class:`~HEOMResult`
The results of the simulation run, with the following important
attributes:
* ``times``: the times ``t`` (i.e. the ``tlist``).
* ``states``: the system state at each time ``t`` (only available
if ``e_ops`` was ``None`` or if the solver option
``store_states`` was set to ``True``).
* ``ado_states``: the full ADO state at each time (only available
if the results option ``ado_return`` was set to ``True``).
Each element is an instance of :class:`HierarchyADOsState`.
The state of a particular ADO may be extracted from
``result.ado_states[i]`` by calling :meth:`extract`.
* ``expect``: a list containing the values of each ``e_ops`` at
time ``t``.
* ``e_data``: a dictionary containing the values of each ``e_ops``
at tme ``t``. The keys are those given by ``e_ops`` if it was
a dict, otherwise they are the indexes of the supplied ``e_ops``.
See :class:`~HEOMResult` and :class:`.Result` for the complete
list of attributes.
"""
H = QobjEvo(H, args=args, tlist=tlist)
solver = HEOMSolver(H, bath=bath, max_depth=max_depth, options=options)
return solver.run(state0, tlist, e_ops=e_ops)
[docs]class HEOMSolver(Solver):
"""
HEOM solver that supports multiple baths.
The baths must be all either bosonic or fermionic baths.
Parameters
----------
H : :obj:`.Qobj`, :obj:`.QobjEvo`
Possibly time-dependent system Liouvillian or Hamiltonian as a Qobj or
QobjEvo. list of [:obj:`.Qobj`, :obj:`.Coefficient`] or callable that
can be made into :obj:`.QobjEvo` are also accepted.
bath : Bath or list of Bath
A :obj:`Bath` containing the exponents of the expansion of the
bath correlation funcion and their associated coefficients
and coupling operators, or a list of baths.
If multiple baths are given, they must all be either fermionic
or bosonic baths.
max_depth : int
The maximum depth of the heirarchy (i.e. the maximum number of bath
exponent "excitations" to retain).
options : dict, optional
Generic solver options.
If set to None the default options will be used. Keyword only.
Default: None.
Attributes
----------
ados : :obj:`HierarchyADOs`
The description of the hierarchy constructed from the given bath
and maximum depth.
rhs : :obj:`.QobjEvo`
The right-hand side (RHS) of the hierarchy evolution ODE. Internally
the system and bath coupling operators are converted to
:class:`qutip.data.CSR` instances during construction of the RHS,
so the operators in the ``rhs`` will all be sparse.
"""
name = "heomsolver"
_resultclass = HEOMResult
_avail_integrators = {}
solver_options = {
"progress_bar": "text",
"progress_kwargs": {"chunk_size": 10},
"store_final_state": False,
"store_states": None,
"normalize_output": False,
"method": "adams",
"store_ados": False,
"state_data_type": "dense",
}
def __init__(self, H, bath, max_depth, *, options=None):
_time_start = time()
if not isinstance(H, (Qobj, QobjEvo)):
raise TypeError("The Hamiltonian (H) must be a Qobj or QobjEvo")
H = QobjEvo(H)
self.L_sys = (
liouvillian(H) if H.type == "oper" # hamiltonian
else H # already a liouvillian
)
self._sys_shape = int(np.sqrt(self.L_sys.shape[0]))
self._sup_shape = self.L_sys.shape[0]
self._sys_dims = self.L_sys.dims[0]
self.ados = HierarchyADOs(
self._combine_bath_exponents(bath), max_depth,
)
self._n_ados = len(self.ados.labels)
self._n_exponents = len(self.ados.exponents)
self._init_ados_time = time() - _time_start
_time_start = time()
# pre-calculate identity matrix required by _grad_n
self._sId = _data.identity(self._sup_shape, dtype="csr")
# pre-calculate superoperators required by _grad_prev and _grad_next:
Qs = [exp.Q.to("csr") for exp in self.ados.exponents]
self._spreQ = [spre(op).data for op in Qs]
self._spostQ = [spost(op).data for op in Qs]
self._s_pre_minus_post_Q = [
_data.sub(self._spreQ[k], self._spostQ[k])
for k in range(self._n_exponents)
]
self._s_pre_plus_post_Q = [
_data.add(self._spreQ[k], self._spostQ[k])
for k in range(self._n_exponents)
]
self._spreQdag = [spre(op.dag()).data for op in Qs]
self._spostQdag = [spost(op.dag()).data for op in Qs]
self._s_pre_minus_post_Qdag = [
_data.sub(self._spreQdag[k], self._spostQdag[k])
for k in range(self._n_exponents)
]
self._s_pre_plus_post_Qdag = [
_data.add(self._spreQdag[k], self._spostQdag[k])
for k in range(self._n_exponents)
]
self._init_superop_cache_time = time() - _time_start
_time_start = time()
rhs = self._calculate_rhs()
self._init_rhs_time = time() - _time_start
super().__init__(rhs, options=options)
@property
def sys_dims(self):
"""
Dimensions of the space that the system use, excluding any environment:
``qutip.basis(sovler.dims)`` will create a state with proper dimensions
for this solver.
"""
return self._sys_dims
def _initialize_stats(self):
stats = super()._initialize_stats()
stats.update({
"init time": sum([
stats["init time"], self._init_ados_time,
self._init_superop_cache_time, self._init_rhs_time,
]),
"init ados time": self._init_ados_time,
"init superop cache time": self._init_superop_cache_time,
"init rhs time": self._init_rhs_time,
"solver": "Hierarchical Equations of Motion Solver",
"max_depth": self.ados.max_depth,
})
return stats
def _combine_bath_exponents(self, bath):
""" Combine the exponents for the specified baths. """
if not isinstance(bath, (list, tuple)):
exponents = bath.exponents
else:
exponents = []
for b in bath:
exponents.extend(b.exponents)
if not all(exp.Q.dims == exponents[0].Q.dims for exp in exponents):
raise ValueError(
"All bath exponents must have system coupling operators"
" with the same dimensions but a mixture of dimensions"
" was given."
)
return exponents
def _grad_n(self, he_n):
""" Get the gradient for the hierarchy ADO at level n. """
vk = self.ados.vk
vk_sum = sum(he_n[i] * vk[i] for i in range(len(vk)))
op = _data.mul(self._sId, -vk_sum)
return op
def _grad_prev(self, he_n, k):
""" Get the previous gradient. """
if self.ados.exponents[k].fermionic:
return self._grad_prev_fermionic(he_n, k)
else:
return self._grad_prev_bosonic(he_n, k)
def _grad_prev_bosonic(self, he_n, k):
if self.ados.exponents[k].type == BathExponent.types.R:
op = _data.mul(
self._s_pre_minus_post_Q[k],
-1j * he_n[k] * self.ados.ck[k],
)
elif self.ados.exponents[k].type == BathExponent.types.I:
op = _data.mul(
self._s_pre_plus_post_Q[k],
-1j * he_n[k] * 1j * self.ados.ck[k],
)
elif self.ados.exponents[k].type == BathExponent.types.RI:
term1 = _data.mul(
self._s_pre_minus_post_Q[k],
he_n[k] * -1j * self.ados.ck[k],
)
term2 = _data.mul(
self._s_pre_plus_post_Q[k],
he_n[k] * self.ados.ck2[k],
)
op = _data.add(term1, term2)
else:
raise ValueError(
f"Unsupported type {self.ados.exponents[k].type}"
f" for exponent {k}"
)
return op
def _grad_prev_fermionic(self, he_n, k):
ck = self.ados.ck
he_fermionic_n = [
i * int(exp.fermionic)
for i, exp in zip(he_n, self.ados.exponents)
]
n_excite = sum(he_fermionic_n)
sign1 = (-1) ** (n_excite + 1)
n_excite_before_m = sum(he_fermionic_n[:k])
sign2 = (-1) ** (n_excite_before_m)
sigma_bar_k = k + self.ados.sigma_bar_k_offset[k]
if self.ados.exponents[k].type == BathExponent.types["+"]:
op = _data.sub(
_data.mul(self._spreQdag[k], -1j * sign2 * ck[k]),
_data.mul(
self._spostQdag[k],
-1j * sign2 * sign1 * np.conj(ck[sigma_bar_k]),
),
)
elif self.ados.exponents[k].type == BathExponent.types["-"]:
op = _data.sub(
_data.mul(self._spreQ[k], -1j * sign2 * ck[k]),
_data.mul(
self._spostQ[k],
-1j * sign2 * sign1 * np.conj(ck[sigma_bar_k]),
),
)
else:
raise ValueError(
f"Unsupported type {self.ados.exponents[k].type}"
f" for exponent {k}"
)
return op
def _grad_next(self, he_n, k):
""" Get the previous gradient. """
if self.ados.exponents[k].fermionic:
return self._grad_next_fermionic(he_n, k)
else:
return self._grad_next_bosonic(he_n, k)
def _grad_next_bosonic(self, he_n, k):
op = _data.mul(self._s_pre_minus_post_Q[k], -1j)
return op
def _grad_next_fermionic(self, he_n, k):
he_fermionic_n = [
i * int(exp.fermionic)
for i, exp in zip(he_n, self.ados.exponents)
]
n_excite = sum(he_fermionic_n)
sign1 = (-1) ** (n_excite + 1)
n_excite_before_m = sum(he_fermionic_n[:k])
sign2 = (-1) ** (n_excite_before_m)
if self.ados.exponents[k].type == BathExponent.types["+"]:
if sign1 == -1:
op = _data.mul(self._s_pre_minus_post_Q[k], -1j * sign2)
else:
op = _data.mul(self._s_pre_plus_post_Q[k], -1j * sign2)
elif self.ados.exponents[k].type == BathExponent.types["-"]:
if sign1 == -1:
op = _data.mul(self._s_pre_minus_post_Qdag[k], -1j * sign2)
else:
op = _data.mul(self._s_pre_plus_post_Qdag[k], -1j * sign2)
else:
raise ValueError(
f"Unsupported type {self.ados.exponents[k].type}"
f" for exponent {k}"
)
return op
def _rhs(self):
""" Make the RHS for the HEOM. """
ops = _GatherHEOMRHS(
self.ados.idx, block=self._sup_shape, nhe=self._n_ados
)
for he_n in self.ados.labels:
op = self._grad_n(he_n)
ops.add_op(he_n, he_n, op)
for k in range(len(self.ados.dims)):
next_he = self.ados.next(he_n, k)
if next_he is not None:
op = self._grad_next(he_n, k)
ops.add_op(he_n, next_he, op)
prev_he = self.ados.prev(he_n, k)
if prev_he is not None:
op = self._grad_prev(he_n, k)
ops.add_op(he_n, prev_he, op)
return ops.gather()
def _calculate_rhs(self):
""" Make the full RHS required by the solver. """
rhs_mat = self._rhs()
rhs_dims = [
[self._sup_shape * self._n_ados], [self._sup_shape * self._n_ados]
]
h_identity = _data.identity(self._n_ados, dtype="csr")
if self.L_sys.isconstant:
# For the constant case, we just add the Liouvillian to the
# diagonal blocks of the RHS matrix.
rhs_mat += _data.kron(h_identity, self.L_sys(0).to("csr").data)
rhs = QobjEvo(Qobj(rhs_mat, dims=rhs_dims))
else:
# In the time dependent case, we construct the parameters
# for the ODE gradient function under the assumption that
#
# RHSmat(t) = RHSmat + time dependent terms that only affect the
# diagonal blocks of the RHS matrix.
#
# This assumption holds because only _grad_n dependents on
# the system Liouvillian (and not _grad_prev or _grad_next) and
# the bath coupling operators are not time-dependent.
rhs = QobjEvo(Qobj(rhs_mat, dims=rhs_dims))
def _kron(x):
return Qobj(
_data.kron(h_identity, x.data),
dims=rhs_dims,
).to("csr")
rhs += self.L_sys.linear_map(_kron)
# The assertion that rhs_mat has data type CSR is just a sanity
# check on the RHS creation. The base solver class will still
# convert the RHS to the type required by the ODE integrator if
# needed.
assert isinstance(rhs_mat, _csr.CSR)
assert isinstance(rhs, QobjEvo)
assert rhs.dims == rhs_dims
return rhs
[docs] def steady_state(
self,
use_mkl=True, mkl_max_iter_refine=100, mkl_weighted_matching=False
):
"""
Compute the steady state of the system.
Parameters
----------
use_mkl : bool, default=False
Whether to use mkl or not. If mkl is not installed or if
this is false, use the scipy splu solver instead.
mkl_max_iter_refine : int
Specifies the the maximum number of iterative refinement steps that
the MKL PARDISO solver performs.
For a complete description, see iparm(7) in
https://www.intel.com/content/www/us/en/docs/onemkl/developer-reference-c/2023-0/pardiso-iparm-parameter.html
mkl_weighted_matching : bool
MKL PARDISO can use a maximum weighted matching algorithm to
permute large elements close the diagonal. This strategy adds an
additional level of reliability to the factorization methods.
For a complete description, see iparm(12) in
https://www.intel.com/content/www/us/en/docs/onemkl/developer-reference-c/2023-0/pardiso-iparm-parameter.html
Returns
-------
steady_state : Qobj
The steady state density matrix of the system.
steady_ados : :class:`HierarchyADOsState`
The steady state of the full ADO hierarchy. A particular ADO may be
extracted from the full state by calling
:meth:`extract`.
"""
if not self.L_sys.isconstant:
raise ValueError(
"A steady state cannot be determined for a time-dependent"
" system"
)
n = self._sys_shape
b_mat = np.zeros(n ** 2 * self._n_ados, dtype=complex)
b_mat[0] = 1.0
L = self.rhs(0).data.copy().as_scipy()
L = L.tolil()
L[0, 0: n ** 2 * self._n_ados] = 0.0
L = L.tocsr()
L += sp.csr_matrix((
np.ones(n),
(np.zeros(n), [num * (n + 1) for num in range(n)])
), shape=(n ** 2 * self._n_ados, n ** 2 * self._n_ados))
if mkl_spsolve is not None and use_mkl:
L.sort_indices()
solution = mkl_spsolve(
L,
b_mat,
perm=None,
verbose=False,
max_iter_refine=mkl_max_iter_refine,
scaling_vectors=True,
weighted_matching=mkl_weighted_matching,
)
else:
L = L.tocsc()
solution = spsolve(L, b_mat)
data = _data.Dense(solution[:n ** 2].reshape((n, n), order='F'))
data = _data.mul(_data.add(data, data.adjoint()), 0.5)
steady_state = Qobj(data, dims=self._sys_dims)
solution = solution.reshape((self._n_ados, n, n))
steady_ados = HierarchyADOsState(steady_state, self.ados, solution)
return steady_state, steady_ados
[docs] def run(self, state0, tlist, *, args=None, e_ops=None):
"""
Solve for the time evolution of the system.
Parameters
----------
state0 : :obj:`.Qobj` or :class:`~HierarchyADOsState` or array-like
If ``rho0`` is a :obj:`.Qobj` the it is the initial state
of the system (i.e. a :obj:`.Qobj` density matrix).
If it is a :class:`~HierarchyADOsState` or array-like, then
``rho0`` gives the initial state of all ADOs.
Usually the state of the ADOs would be determine from a previous
call to ``.run(...)`` with the solver results option ``store_ados``
set to True. For example, ``result = solver.run(...)`` could be
followed by ``solver.run(result.ado_states[-1], tlist)``.
If a numpy array-like is passed its shape must be
``(number_of_ados, n, n)`` where ``(n, n)`` is the system shape
(i.e. shape of the system density matrix) and the ADOs must
be in the same order as in ``.ados.labels``.
tlist : list
An ordered list of times at which to return the value of the state.
args : dict, optional {None}
Change the ``args`` of the RHS for the evolution.
e_ops : Qobj / QobjEvo / callable / list / dict / None, optional
A list or dictionary of operators as :obj:`.Qobj`,
:obj:`.QobjEvo` and/or callable functions (they can be mixed) or
a single operator or callable function. For an operator ``op``, the
result will be computed using ``(state * op).tr()`` and the state
at each time ``t``. For callable functions, ``f``, the result is
computed using ``f(t, ado_state)``. The values are stored in the
``expect`` and ``e_data`` attributes of the result (see the return
section below).
Returns
-------
:class:`~HEOMResult`
The results of the simulation run, with the following important
attributes:
* ``times``: the times ``t`` (i.e. the ``tlist``).
* ``states``: the system state at each time ``t`` (only available
if ``e_ops`` was ``None`` or if the solver option
``store_states`` was set to ``True``).
* ``ado_states``: the full ADO state at each time (only available
if the results option ``ado_return`` was set to ``True``).
Each element is an instance of :class:`HierarchyADOsState`.
The state of a particular ADO may be extracted from
``result.ado_states[i]`` by calling :meth:`extract`.
* ``expect``: a list containing the values of each ``e_ops`` at
time ``t``.
* ``e_data``: a dictionary containing the values of each ``e_ops``
at tme ``t``. The keys are those given by ``e_ops`` if it was
a dict, otherwise they are the indexes of the supplied ``e_ops``.
See :class:`~HEOMResult` and :class:`.Result` for the complete
list of attributes.
"""
return super().run(state0, tlist, args=args, e_ops=e_ops)
def _prepare_state(self, state):
n = self._sys_shape
rho_dims = self._sys_dims
hierarchy_shape = (self._n_ados, n, n)
rho0 = state
ado_init = not isinstance(rho0, Qobj)
if ado_init:
if isinstance(rho0, HierarchyADOsState):
rho0_he = rho0._ado_state
elif hasattr(rho0, "shape"): # array-like
rho0_he = rho0
else:
raise TypeError(
f"Initial ADOs passed have type {type(rho0)}"
" but a HierarchyADOsState or a numpy array-like instance"
" was expected"
)
if rho0_he.shape != hierarchy_shape:
raise ValueError(
f"Initial ADOs passed have shape {rho0_he.shape}"
f" but the solver hierarchy shape is {hierarchy_shape}"
)
rho0_he = rho0_he.reshape(n ** 2 * self._n_ados)
rho0_he = _data.create(rho0_he)
else:
if rho0.dims != rho_dims:
raise ValueError(
f"Initial state rho has dims {rho0.dims}"
f" but the system dims are {rho_dims}"
)
rho0_he = np.zeros([n ** 2 * self._n_ados], dtype=complex)
rho0_he[:n ** 2] = rho0.full().ravel('F')
rho0_he = _data.create(rho0_he)
if self.options["state_data_type"]:
rho0_he = _data.to(self.options["state_data_type"], rho0_he)
return rho0_he
def _restore_state(self, state, *, copy=True):
n = self._sys_shape
rho_shape = (n, n)
rho_dims = self._sys_dims
hierarchy_shape = (self._n_ados, n, n)
rho = Qobj(
state.to_array()[:n ** 2].reshape(rho_shape, order='F'),
dims=rho_dims,
)
ado_state = HierarchyADOsState(
rho, self.ados, state.to_array().reshape(hierarchy_shape)
)
return ado_state
[docs] def start(self, state0, t0):
"""
Set the initial state and time for a step evolution.
Parameters
----------
state0 : :obj:`.Qobj`
Initial state of the evolution. This may provide either just the
initial density matrix of the system, or the full set of ADOs
for the hierarchy. See the documentation for ``rho0`` in the
``.run(...)`` method for details.
t0 : double
Initial time of the evolution.
"""
super().start(state0, t0)
@property
def options(self):
"""
Options for HEOMSolver:
store_final_state: bool, default: False
Whether or not to store the final state of the evolution in the
result class.
store_states: bool, default: None
Whether or not to store the state vectors or density matrices.
On `None` the states will be saved if no expectation operators are
given.
normalize_output: bool, default: False
Normalize output state to hide ODE numerical errors.
progress_bar: str {'text', 'enhanced', 'tqdm', ''}, default: "text"
How to present the solver progress.
'tqdm' uses the python module of the same name and raise an error
if not installed. Empty string or False will disable the bar.
progress_kwargs: dict, default: {"chunk_size": 10}
Arguments to pass to the progress_bar. Qutip's bars use
``chunk_size``.
method: str, default: "adams"
Which ordinary differential equation integration method to use.
state_data_type: str, default: "dense"
Name of the data type of the state used during the ODE evolution.
Use an empty string to keep the input state type. Many integrators
support only work with `Dense`.
store_ados : bool, default: False
Whether or not to store the HEOM ADOs. Only relevant when using
the HEOM solver.
"""
return self._options
@options.setter
def options(self, new_options):
Solver.options.fset(self, new_options)
[docs]class HSolverDL(HEOMSolver):
"""
A helper class for creating an :class:`HEOMSolver` that is backwards
compatible with the ``HSolverDL`` provided in ``qutip.nonmarkov.heom``
in QuTiP 4.6 and below.
See :class:`HEOMSolver` and :class:`DrudeLorentzBath` for more
descriptions of the underlying solver and bath construction.
An exact copy of the QuTiP 4.6 HSolverDL is provided in
``qutip.nonmarkov.dlheom_solver`` for cases where the functionality of
the older solver is required. The older solver will be completely
removed in QuTiP 5.
.. note::
Unlike the version of ``HSolverDL`` in QuTiP 4.6, this solver
supports supplying a time-dependent or Liouvillian ``H_sys``.
.. note::
For compatibility with ``HSolverDL`` in QuTiP 4.6 and below, the
parameter ``N_exp`` specifying the number of exponents to keep in
the expansion of the bath correlation function is one more than
the equivalent ``Nk`` used in the :class:`DrudeLorentzBath`. I.e.,
``Nk = N_exp - 1``. The ``Nk`` parameter in the
:class:`DrudeLorentzBath` does not count the zeroeth exponent in
order to better match common usage in the literature.
.. note::
The ``stats`` and ``renorm`` arguments accepted in QuTiP 4.6 and below
are no longer supported.
Parameters
----------
H_sys : Qobj or QobjEvo or list
The system Hamiltonian or Liouvillian. See :class:`HEOMSolver` for
a complete description.
coup_op : Qobj
Operator describing the coupling between system and bath.
See parameter ``Q`` in :class:`BosonicBath` for a complete description.
coup_strength : float
Coupling strength. Referred to as ``lam`` in :class:`DrudeLorentzBath`.
temperature : float
Bath temperature. Referred to as ``T`` in :class:`DrudeLorentzBath`.
N_cut : int
The maximum depth of the hierarchy. See ``max_depth`` in
:class:`HEOMSolver` for a full description.
N_exp : int
Number of exponential terms used to approximate the bath correlation
functions. The equivalent ``Nk`` in :class:`DrudeLorentzBath` is one
less than ``N_exp`` (see note above).
cut_freq : float
Bath spectral density cutoff frequency. Referred to as ``gamma`` in
:class:`DrudeLorentzBath`.
bnd_cut_approx : bool
Use boundary cut off approximation. If true, the Matsubara
terminator is added to the system Liouvillian (and H_sys is
promoted to a Liouvillian if it was a Hamiltonian). Keyword only.
Default: False.
options : dict, optional
Generic solver options.
If set to None the default options will be used. Keyword only.
Default: None.
combine : bool, default: True
Whether to combine exponents with the same frequency (and coupling
operator). See :meth:`BosonicBath.combine` for details.
Keyword only. Default: True.
"""
def __init__(
self, H_sys, coup_op, coup_strength, temperature,
N_cut, N_exp, cut_freq, *, bnd_cut_approx=False, options=None,
combine=True,
):
H_sys = QobjEvo(H_sys)
bath = DrudeLorentzBath(
Q=coup_op,
lam=coup_strength,
gamma=cut_freq,
Nk=N_exp - 1,
T=temperature,
combine=combine,
)
if bnd_cut_approx:
# upgrade H_sys to a Liouvillian if needed and add the
# bath terminator
if H_sys.type == "oper": # H_sys is a Hamiltonian
H_sys = liouvillian(H_sys)
_, terminator = bath.terminator()
H_sys = H_sys + terminator
super().__init__(H_sys, bath=bath, max_depth=N_cut, options=options)
# store input parameters as attributes for politeness and compatibility
# with HSolverDL in QuTiP 4.6 and below.
self.coup_strength = coup_strength
self.cut_freq = cut_freq
self.temperature = temperature
self.N_exp = N_exp
self.bnd_cut_approx = bnd_cut_approx
class _GatherHEOMRHS:
""" A class for collecting elements of the right-hand side matrix
of the HEOM.
Parameters
----------
f_idx: function(he_state) -> he_idx
A function that returns the index of a hierarchy state
(i.e. an ADO label).
block : int
The size of a single ADO Liovillian operator in the hierarchy.
nhe : int
The number of ADOs in the hierarchy.
"""
def __init__(self, f_idx, block, nhe):
self._block_size = block
self._n_blocks = nhe
self._f_idx = f_idx
self._ops = []
def add_op(self, row_he, col_he, op):
""" Add an block operator to the list. """
self._ops.append(
(self._f_idx(row_he), self._f_idx(col_he), op)
)
def gather(self):
""" Create the HEOM liouvillian from a sorted list of smaller sparse
matrices.
.. note::
The list of operators contains tuples of the form
``(row_idx, col_idx, op)``. The row_idx and col_idx give the
*block* row and column for each op. An operator with
block indices ``(N, M)`` is placed at position
``[N * block: (N + 1) * block, M * block: (M + 1) * block]``
in the output matrix.
Returns
-------
rhs : :obj:`Data`
A combined matrix of shape ``(block * nhe, block * ne)``.
"""
self._ops.sort()
ops = np.array(self._ops, dtype=[
("row", _data.base.idxint_dtype),
("col", _data.base.idxint_dtype),
("op", _data.CSR),
])
return _csr._from_csr_blocks(
ops["row"], ops["col"], ops["op"],
self._n_blocks, self._block_size,
)