### Verify Installation

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/install.rst.txt

Import OQuPy in an interactive session to check the installed version.

```python
>>> import oqupy
>>> oqupy.__version__
```

```none
'0.5.0'
```

--------------------------------

### Install OQuPy

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/install.rst.txt

Install the OQuPy package using pip.

```bash
$ python3 -m pip install oqupy
```

--------------------------------

### Import OQuPy and dependencies

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/pt_tebd.rst.txt

Initial setup for the OQuPy environment and plotting tools.

```python
import sys
sys.path.insert(0,'..')

import oqupy
import numpy as np
import matplotlib.pyplot as plt
```

--------------------------------

### Importing Required Packages

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/pt_gradient/pt_gradient.rst.txt

Initial setup for the OQuPy environment, including path configuration and necessary library imports.

```python
import sys
sys.path.insert(0,'..')
import numpy as np
import oqupy
from oqupy import operators as op
import matplotlib.pyplot as plt
```

--------------------------------

### Initialize Process Tensor and Backend

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/pt_tempo.html

Handles the setup of process tensors and backend configuration based on provided parameters and bath properties.

```python
        self._process_tensor = None
        if process_tensor_file or isinstance(process_tensor_file, Text):
            if isinstance(process_tensor_file, Text):
                filename = process_tensor_file
            else:
                filename = None
            self._init_file_process_tensor(filename, overwrite)
        else:
            self._init_simple_process_tensor()

        if backend_config is None:
            self._backend_config = PT_TEMPO_BACKEND_CONFIG
        else:
            self._backend_config = backend_config

        self._coupling_comm = self._bath._coupling_comm
        self._coupling_acomm = self._bath._coupling_acomm

        tmp_num_steps = int((end_time - self._start_time)/self._parameters.dt)
        assert tmp_num_steps >= 2, \
            "Parameter `end_time` must be more than two times steps " \
            + "larger than the parameter `start_time`!"
        self._num_steps = tmp_num_steps

        self._backend_instance = None
        self._init_pt_tempo_backend()
```

--------------------------------

### Define Spin-Boson Model

Source: https://oqupy.readthedocs.io/en/latest/pages/tutorials/parameters.html

Setup the system, bath, and environment parameters for a spin-boson model.

```python
sigma_x = oqupy.operators.sigma('x')
sigma_y = oqupy.operators.sigma('y')
sigma_z = oqupy.operators.sigma('z')
sigma_m = oqupy.operators.sigma('-')

omega_cutoff = 2.5
alpha = 0.8
T = 0.2
correlations = oqupy.PowerLawSD(alpha=alpha,
                                zeta=1,
                                cutoff=omega_cutoff,
                                cutoff_type='exponential',
                                temperature=T)
bath = oqupy.Bath(0.5 * sigma_z, correlations)
Omega = 2.0
Gamma = 0.02
system = oqupy.System(0.5 * Omega * sigma_x,
                      gammas=[Gamma],
                      lindblad_operators=[sigma_m], # incoherent dissipation
                     )

t_start = 0.0
t_end = 5.0
```

--------------------------------

### Spin-boson model simulation script

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/quickstart.rst.txt

A complete example script demonstrating the definition of system parameters, bath correlations, and the execution of a TEMPO computation.

```python
Omega = 1.0
omega_cutoff = 5.0
alpha = 0.3

system = oqupy.System(0.5 * Omega * sigma_x)
correlations = oqupy.PowerLawSD(alpha=alpha,
                                zeta=1,
                                cutoff=omega_cutoff,
                                cutoff_type='exponential')
bath = oqupy.Bath(0.5 * sigma_z, correlations)
tempo_parameters = oqupy.TempoParameters(dt=0.1, tcut=3.0, epsrel=10**(-4))

dynamics = oqupy.tempo_compute(system=system,
                               bath=bath,
                               initial_state=up_density_matrix,
                               start_time=0.0,
                               end_time=15.0,
                               parameters=tempo_parameters,
                               unique=False)
t, s_z = dynamics.expectations(0.5*sigma_z, real=True)

plt.plot(t, s_z, label=r'$\alpha=0.3$')
plt.xlabel(r'$t\,\Omega$')
plt.ylabel(r'$\langle\sigma_z\rangle$')
plt.legend()
```

--------------------------------

### Define Bath and Parameters for PT-TEMPO

Source: https://oqupy.readthedocs.io/en/latest/pages/tutorials/parameters.html

Setup the environment and bath parameters required for a PT-TEMPO computation.

```python
alpha = 0.25 # Doesn't affect PT computation
nu_c = 227.9
T = 39.3
start_time = 0.0

dt = 1e-3
epsrel = 1e-8
end_time = 250 * dt # 251 steps starting from t=0.0

correlations = oqupy.PowerLawSD(alpha=alpha,
                                zeta=1,
                                cutoff=nu_c,
                                cutoff_type='gaussian')
bath = oqupy.Bath(oqupy.operators.sigma("z")/2.0, correlations)
```

--------------------------------

### Check Python Version

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/install.rst.txt

Verify that Python 3.10 or higher is installed on the system.

```bash
$ python3 --version
Python 3.10.14
```

--------------------------------

### Initialize open system state

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/pt_tebd.rst.txt

Create an AugmentedMPS for the open spin chain starting in a fully mixed state.

```python
initial_augmented_mps_B = oqupy.AugmentedMPS([mixed_dm, mixed_dm, mixed_dm, mixed_dm, mixed_dm])
```

--------------------------------

### Get Unitary Propagators

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/system.html

Prepares propagator functions for the system based on the Hamiltonian.

```python
def get_unitary_propagators(self, dt, start_time, subdiv_limit, epsrel):
    """Prepare propagator functions for the system. """
    first_step = expm(-1j*self._hamiltonian*dt/2.0)
    second_step = expm(-1j*self._hamiltonian*dt/2.0)
    def propagators(step: int):
        """Create the system propagators (first and second half) for
        the time step `step`  """
        return first_step, second_step
    return propagators
```

--------------------------------

### Compute System Dynamics

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/system_dynamics.html

Example of calculating system dynamics using a process tensor and system operators.

```python
    dynamics = compute_dynamics(
        system=system,
        process_tensor=process_tensor,
        control=control,
        start_time=start_time,
        initial_state=initial_state,
        dt=dt,
        num_steps=max_step,
        progress_type='silent')
    _, corr = dynamics.expectations(operators[-1])
    ret_correlations = corr[last_times]
    return ret_correlations
```

--------------------------------

### Prepare TEMPO Backend

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/tempo.html

Internal method to initialize the backend instance with system propagators and correlation coefficients.

```python
    def _prepare_backend(self):
        """Create and initialize the TEMPO backend. """
        dim = self._dimension

        def coeffs(k):
            shape = "upper-triangle" if k==0 else "square"
            return self._correlations.correlation_2d_integral(
                self._dt, k * self._dt, shape=shape, matsubara=True)

        operators = (-self._bath.coupling_operator.diagonal(),
                     self._bath.coupling_operator.diagonal(),
                     np.zeros((dim,)))


        unitary_transform = self._bath.unitary_transform
        # ToDo: Unitary transform is not used. Check that GibbsTempo also works
        #       with non-diagonal coupling operators!!!
        if not np.allclose(unitary_transform,np.identity(self.dimension)):
            raise NotImplementedError(
                "GibbsTempo is currently implemented only for diagonal "\
                + "bath coupling operators. See GitHub Issue #???.")


        epsrel = self._parameters.epsrel
        max_step = self._parameters.n_steps
        propagators = self._system.get_unitary_propagators(
            - 1j * self._dt, 0, 0, 0)
        self._backend_instance = TIBaseBackend(
                dim,
                epsrel,
                propagators(1)[0],
                coeffs,
                operators,
                max_step=max_step,
                config=self._backend_config)
```

--------------------------------

### Fidelity output

Source: https://oqupy.readthedocs.io/en/latest/pages/tutorials/pt_gradient/pt_gradient.html

Example output of the fidelity calculation.

```text
The fidelity is 0.9012528539245532.
```

--------------------------------

### Check OQuPy version

Source: https://oqupy.readthedocs.io/en/latest/pages/tutorials/pt_tempo.html

Verify the currently installed version of the OQuPy package.

```python
oqupy.__version__

```

--------------------------------

### Import OQuPy and dependencies

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/mf_tempo.rst.txt

Initializes the environment by importing necessary libraries and setting the path.

```python
import sys
sys.path.insert(0,'..')
import oqupy
import numpy as np
import matplotlib.pyplot as plt
```

--------------------------------

### Check Pip Version

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/install.rst.txt

Verify that pip version 23.0 or higher is installed.

```bash
$ python3 -m pip --version
pip 23.3.1 from /home/gefux/anaconda3/envs/oqupy-ci/lib/python3.10/site-packages/pip (python 3.10)
```

--------------------------------

### Plot Label Output

Source: https://oqupy.readthedocs.io/en/latest/pages/tutorials/bath_dynamics.html

Example of the return value from a matplotlib label command.

```text
Text(0, 0.5, '$\Delta Q ( \Omega, t)$')
```

--------------------------------

### Initialize PT-TEMPO Backend

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/pt_tempo.html

Configures the PtTempoBackend instance with degeneracy maps and influence parameters.

```python
    def _init_pt_tempo_backend(self):
        """Create and initialize the pt-tempo backend. """
        if self._unique:
            sum_north = np.ones(np.max(self._bath.north_degeneracy_map)+1,
                                dtype=float)
            sum_west = np.ones(np.max(self._bath.west_degeneracy_map)+1,
                               dtype=float)
            degeneracy_maps = [self._bath.north_degeneracy_map,
                               self._bath.west_degeneracy_map]
        else:
            sum_north =  np.ones(self._dimension**2, dtype=float)
            sum_west = np.ones(self._dimension**2, dtype=float)
            degeneracy_maps = None
        dkmax = self._parameters.dkmax
        if dkmax is None:
            dkmax = self._num_steps
        self._backend_instance = PtTempoBackend(
                dimension=self._dimension,
                influence=self._influence,
                process_tensor=self._process_tensor,
                sum_north=sum_north,
                sum_west=sum_west,
                num_steps=self._num_steps,
                dkmax=dkmax,
                epsrel=self._parameters.epsrel,
                config=self._backend_config,
                degeneracy_maps=degeneracy_maps)
```

--------------------------------

### oqupy.pt_tebd.PtTebd.initialize

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/pt_tebd.html

Prepares the propagator, backend, and results structure for the PT-TEBD computation.

```APIDOC
## oqupy.pt_tebd.PtTebd.initialize

### Description
Initializes the propagator, the PT-TEBD backend, and the results dictionary to prepare for the simulation steps.
```

--------------------------------

### Get Initial Tensor

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/process_tensor.html

Retrieves the initial system state, which may be correlated.

```python
    def get_initial_tensor(self) -> ndarray:
        """
        Get the (possibly correlated) initial system state.
        """
        return _get_data_and_shape(step=0,
                                   data=self._initial_tensor_data,
                                   shape=self._initial_tensor_shape)
```

--------------------------------

### Parse system parameters and initialize backend

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/tempo.html

Parses input system tuples and prepares the MeanFieldTempo backend instance.

```python
        # List of tuples, one for each system: (system, initial_state,
        # bath, hs_dim)
        parsed_system_tuple_list = [_tempo_physical_input_parse(
            True, system, initial_state, bath)
            for system, initial_state, bath in zip(
                mean_field_system.system_list, initial_state_list,
                bath_list)]

        parsed_parameter_names = ["system", "initial_state", "bath", "hs_dim"]
        # Dictionary of keys from parsed_parameter_names. The items are
        # a lists of systems (Key = "system") and corresponding lists of
        # values (Key = "initial_state", "bath", "hs_dim"). Considered to be
        # 'parameters' for the overall (mean-field system dynamics) computation.
        self._parsed_parameters_dict = {}
        for i, parsed_parameter_name in enumerate(parsed_parameter_names):
            self._parsed_parameters_dict[parsed_parameter_name] = []
            for parsed_parameter_tuple in parsed_system_tuple_list:
                self._parsed_parameters_dict[parsed_parameter_name].append(
                    parsed_parameter_tuple[i])

        # Check initial field can be cast to complex, and start time float
        self._initial_field = check_convert(initial_field, complex,
                                            "initial_field")
        self._start_time = check_convert(start_time, float, "start_time")
        # Prepare the MeanFieldTempo backend
        self._prepare_backend()
```

```python
    def _prepare_backend(self):
        """Create and initialize the MeanFieldTempo backend. """
        initial_state_list = \
                self._parsed_parameters_dict["initial_state"]
        hs_dim_list = \
                self._parsed_parameters_dict["hs_dim"]
        initial_field = self._initial_field
        influence_list = [self._get_influence(bath)
                for bath in self._parsed_parameters_dict["bath"]]
        unitary_transform_list = [bath.unitary_transform
                for bath in self._parsed_parameters_dict["bath"]]
        propagators_list = [system.get_propagators(
                self._parameters.dt,
                self._start_time,
                self._parameters.subdiv_limit,
                self._parameters.liouvillian_epsrel)
            for system in self._parsed_parameters_dict["system"]]
        compute_field = self._compute_field
        compute_field_derivative = self._compute_field_derivative
        if self._unique:
            sum_north_list = [np.ones(np.max(bath.north_degeneracy_map)+1,
                                      dtype=float)
                for bath in self._parsed_parameters_dict["bath"]]
            sum_west_list = [np.ones(np.max(bath.west_degeneracy_map)+1,
                                      dtype=float)
                for bath in self._parsed_parameters_dict["bath"]]
            degeneracy_maps_list = [[bath.north_degeneracy_map,
                                     bath.west_degeneracy_map]
                for bath in self._parsed_parameters_dict["bath"]]
        else:
            sum_north_list = [np.ones(dim**2, dtype=float)
                    for dim in self._parsed_parameters_dict["hs_dim"]]
            sum_west_list = [np.ones(dim**2, dtype=float)
                    for dim in self._parsed_parameters_dict["hs_dim"]]
            degeneracy_maps_list = [None
                for bath in self._parsed_parameters_dict["bath"]]
        # N.B. For now all baths constrained to have same memory length
        dkmax = self._parameters.dkmax
        epsrel = self._parameters.epsrel
        self._backend_instance = MeanFieldTempoBackend(
                initial_state_list,
                initial_field,
                influence_list,
                unitary_transform_list,
                propagators_list,
                compute_field,
                compute_field_derivative,
                sum_north_list,
                sum_west_list,
                dkmax,
                epsrel,
                config=self._backend_config,
                degeneracy_maps_list=degeneracy_maps_list,
                dim_list=hs_dim_list,
                )
```

--------------------------------

### Get Cap Tensor

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/process_tensor.html

Retrieves the cap tensor for a specific time step.

```python
def get_cap_tensor(self, step: int) -> ndarray:
    """
    Get the cap tensor (vector) to terminate the PT-MPO at time step `step`.
    """
    length = len(self._cap_tensors)
    if step >= length or step < 0:
        return None
    return self._cap_tensors[step]
```

--------------------------------

### Initialize TEMPO backend

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/tempo.html

Configures and initializes the backend instance required for tensor network propagation.

```python
def _prepare_backend(self):
    """Create and initialize the TEMPO backend. """
    dim = self._dimension
    initial_state = self._initial_state.reshape(dim**2)
    influence = self._influence
    unitary_transform = self._bath.unitary_transform
    propagators = self._system.get_propagators(
            self._parameters.dt,
            self._start_time,
            self._parameters.subdiv_limit,
            self._parameters.liouvillian_epsrel)
    if self._unique:
        sum_north = np.ones(np.max(self._bath.north_degeneracy_map)+1,
                            dtype=float)
        sum_west = np.ones(np.max(self._bath.west_degeneracy_map)+1,
                           dtype=float)
        degeneracy_maps = [self._bath.north_degeneracy_map,
                           self._bath.west_degeneracy_map]
    else:
        sum_north = np.ones(dim**2, dtype=float)
        sum_west = np.ones(dim**2, dtype=float)
        degeneracy_maps = None
    dkmax = self._parameters.dkmax
    epsrel = self._parameters.epsrel
    self._backend_instance = TempoBackend(
            initial_state,
            influence,
            unitary_transform,
            propagators,
            sum_north,
            sum_west,
            dkmax,
            epsrel,
            config=self._backend_config,
            degeneracy_maps=degeneracy_maps,
            dim=dim)
```

--------------------------------

### Get Initial Tensor

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/process_tensor.html

Retrieves the initial system state, which may be correlated.

```python
def get_initial_tensor(self) -> ndarray:
    """
    Get the (possibly correlated) initial system state.
    """
    return self._initial_tensor
```

--------------------------------

### Computation Progress Output

Source: https://oqupy.readthedocs.io/en/latest/pages/tutorials/bath_dynamics.html

Example of the console output generated during the correlation computation process.

```text
--> Compute correlations:
100.0%  100 of  100 [########################################] 00:00:07
Elapsed time: 7.7s
```

--------------------------------

### Prepare the PT-TEBD simulation

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/pt_tebd.rst.txt

Initialize the PtTebd object with the system chain and parameters.

```python
pt_tebd_closed = oqupy.PtTebd(
    initial_augmented_mps=initial_augmented_mps_A,
    system_chain=system_chain,
    process_tensors=[None, None, None, None, None],
    parameters=pt_tebd_params,
    dynamics_sites=[0, 1, 2, 3, 4])
```

--------------------------------

### Initialize SystemChain

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/system.html

Constructor for a 1D chain of systems, initializing site Liouvillians and nearest-neighbor Liouvillians based on Hilbert space dimensions.

```python
def __init__(
            self,
            hilbert_space_dimensions: List[int],
            name: Optional[Text] = None,
            description: Optional[Text] = None) -> None:
        """Create a SystemChain object. """
        tmp_hs_dims = np.array(hilbert_space_dimensions, int)
        assert len(tmp_hs_dims.shape) == 1
        assert len(hilbert_space_dimensions) >= 1
        assert np.all(tmp_hs_dims > 0)
        self._hs_dims = tmp_hs_dims

        self._site_liouvillians = []
        for hs_dim in self._hs_dims:
            self._site_liouvillians.append(
                np.zeros((hs_dim**2, hs_dim**2), dtype=NpDtype))

        self._nn_liouvillians = []
        for hs_dim_l, hs_dim_r in zip(self._hs_dims[:-1], self._hs_dims[1:]):
            self._nn_liouvillians.append(
                np.zeros((hs_dim_l**2 * hs_dim_r**2, hs_dim_l**2 * hs_dim_r**2),
                dtype=NpDtype))

        super().__init__(name, description)
```

--------------------------------

### Define system matrices

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/n_time_correlations.rst.txt

Setup of the matrices representing the three-level chromophore system operators.

```python
P_1 = np.array([[0., 0., 0.],
                [0., 1., 0.],
                [0., 0., 0.]], dtype=complex)

P_2 = np.array([[0., 0., 0.],
                [0., 0., 0.],
                [0., 0., 1.]], dtype=complex)

sigma_min = np.array([[0., 0., 0.],
                      [0., 0., 1.],
                      [0., 0., 0.]], dtype=complex)

sigma_plus = np.array([[0., 0., 0.],
                       [0., 0., 0.],
                       [0., 1., 0.]], dtype=complex)

dip_02 = np.array([[0., 0., 1.],
                   [0., 0., 0.],
                   [1., 0., 0.]], dtype=complex)
```

--------------------------------

### Tempo.__init__

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/tempo.html

Initializes a new Tempo object with the specified system, bath, and computation parameters.

```APIDOC
## Tempo.__init__(system, bath, parameters, backend_config=None, name=None, description=None)

### Description
Creates a Tempo object to manage the simulation of an open quantum system.

### Parameters
- **system** (System) - Required - The quantum system to simulate.
- **bath** (Bath) - Required - The bath object, initiated with a CustomSD or derived correlations object.
- **parameters** (GibbsParameters) - Required - The parameters for the GibbsTEMPO computation.
- **backend_config** (dict) - Optional - The configuration of the backend. Defaults to None.
- **name** (str) - Optional - An optional name for the tempo object.
- **description** (str) - Optional - An optional description of the tempo object.
```

--------------------------------

### Get Bond Dimensions

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/process_tensor.html

Returns the bond dimensions of the MPS form of the process tensor.

```python
def get_bond_dimensions(self) -> ndarray:
    """
    Return the bond dimensions of the MPS form of the process tensor.
    """
    bond_dims = []
    for mpo in self._mpo_tensors:
        if mpo is None:
            bond_dims.append(0)
        else:
            bond_dims.append(mpo.shape[0])
    bond_dims.append(self._mpo_tensors[-1].shape[1])
    return np.array(bond_dims)
```

--------------------------------

### Define Bath and Simulation Parameters

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/parameters.rst.txt

Sets up the harmonic environment and simulation parameters for a PT-TEMPO computation.

```python
alpha = 0.25 # Doesn't affect PT computation
nu_c = 227.9
T = 39.3
start_time = 0.0

dt = 1e-3
epsrel = 1e-8 
end_time = 250 * dt # 251 steps starting from t=0.0

correlations = oqupy.PowerLawSD(alpha=alpha,
                                zeta=1,
                                cutoff=nu_c,
                                cutoff_type='gaussian')
bath = oqupy.Bath(oqupy.operators.sigma("z")/2.0, correlations)
```

--------------------------------

### Configure Bath and TEMPO Parameters

Source: https://oqupy.readthedocs.io/en/latest/pages/tutorials/pt_gradient/pt_gradient.html

Sets up the spectral density, bath coupling, and numerical TEMPO parameters for the process tensor computation.

```python
# spectral density parameters
alpha = 0.126
omega_cutoff = 3.04
temperature = 5 * 0.1309 # 1K = 0.1309/ps in natural units

# numerical tempo parameters
tcut  = 2.0
esprel = 10**(-4)

correlations = oqupy.PowerLawSD(
    alpha=alpha,
    zeta=3,
    cutoff=omega_cutoff,
    cutoff_type='gaussian',
    temperature=temperature)
bath = oqupy.Bath(op.sigma("z")/2, correlations)

tempo_params = oqupy.TempoParameters(dt=dt, tcut=tcut, epsrel=esprel)
```

--------------------------------

### Initialize GibbsTempo Object

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/tempo.html

Constructor for the GibbsTempo class, requiring a system, bath, and Gibbs parameters.

```python
    def __init__(
            self,
            system: System,
            bath: Bath,
            parameters: GibbsParameters,
            backend_config: Optional[Dict] = None,
            name: Optional[Text] = None,
            description: Optional[Text] = None) -> None:
        """Create a Tempo object. """
        super().__init__(name, description)

        check_isinstance(system, System, name='system')
        if len(system.gammas) > 0:
            raise Warning('Markovian decay ignored for GibbsTempo')
        check_isinstance(bath, Bath, name='bath')
        check_isinstance(bath.correlations, CustomSD, name='bath.correlations')
        check_isinstance(parameters, GibbsParameters, name='parameters')

        self._system, self._initial_state, self._bath, self._dimension = \
            _tempo_physical_input_parse(False, system, None, bath)

        self._parameters = parameters
        self._correlations = self._bath.correlations
        self._temperature = self._correlations.temperature
        self._dt = self._parameters.time_step_length(self._temperature)

        if backend_config is None:
            self._backend_config = TEMPO_BACKEND_CONFIG
        else:
            self._backend_config = backend_config

        self._dynamics = None
        self._backend_instance = None

        self._prepare_backend()
```

--------------------------------

### Get Bond Dimensions

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/process_tensor.html

Calculates and returns the bond dimensions of the MPS form of the process tensor.

```python
    def get_bond_dimensions(self) -> ndarray:
        """Return the bond dimensions of the MPS form of the process tensor."""
        bond_dims = []
        for tensor_shape in self._mpo_tensors_shape:
            bond_dims.append(tensor_shape[0])
        bond_dims.append(self._mpo_tensors_shape[-1][1])
        return np.array(bond_dims)
```

--------------------------------

### Get MPO Tensor

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/process_tensor.html

Retrieves the MPO tensor for a given time step, optionally applying transformations.

```python
    def get_mpo_tensor(
            self,
            step: int,
            transformed: Optional[bool] = True) -> ndarray:
        """
        Get the MPO tensor for time step `step`.

        The axes correspond to the following legs:
            [0] ... past bond leg,
            [1] ... future bond leg,
            [2] ... input (from system) leg,
            [3] ... output (to system) leg.

        Applies the transformation (stored in `.transform_in` and
        `.transform_out`) when `transformed` is true.
        """
        tensor = _get_data_and_shape(step,
                                     data=self._mpo_tensors_data,
                                     shape=self._mpo_tensors_shape)
        if transformed:
            if len(tensor.shape) == 3:
                tensor = util.create_delta(tensor, [0, 1, 2, 2])
            if self._transform_in is not None:
                tensor = np.dot(np.moveaxis(tensor, -2, -1),
                                self._transform_in.T)
                tensor = np.moveaxis(tensor, -1, -2)
            if self._transform_out is not None:
                tensor = np.dot(tensor, self._transform_out)
        return tensor
```

--------------------------------

### Define multiple system configurations

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/pt_tempo.rst.txt

Create a list of system objects with varying detuning parameters for batch processing.

```python
deltas = [-10.0, -5.0, 0.0, 5.0, 10.0]
systems = []
for delta in deltas:
    # NOTE: omitting "delta=delta" in the parameter definition below
    #       would lead to all systems having the same detuning.
    #       This is a common python pitfall. Check out 
    #       https://docs.python-guide.org/writing/gotchas/#late-binding-closures
    #       for more information on this.
    def hamiltonian_t(t, delta=delta): 
        return delta/2.0 * sigma_z \
            + gaussian_shape(t, area = np.pi/2.0, tau = 0.245)/2.0 * sigma_x 
    system = oqupy.TimeDependentSystem(hamiltonian_t)
    systems.append(system)
```

--------------------------------

### Get Cap Tensor

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/process_tensor.html

Retrieves the cap tensor for a specific time step, returning None if the index is out of bounds.

```python
    def get_cap_tensor(self, step: int) -> ndarray:
        """
        Get the cap tensor (vector) to terminate the PT-MPO at time step `step`.
        """
        try:
            tensor = _get_data_and_shape(step,
                                         data=self._cap_tensors_data,
                                         shape=self._cap_tensors_shape)
        except IndexError:
            tensor = None
        return tensor
```

--------------------------------

### Initialize results dictionary

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/pt_tebd.html

Internal method to set up the results storage dictionary, including bond dimensions and dynamics tracking.

```python
def _init_results(self) -> None:
    """Initialise the results dictionary. """
    pt_bond_dimensions = {}
    for site, pt in enumerate(self._process_tensors):
        if pt is not None:
            pt_bond_dimensions[site] = pt.get_bond_dimensions()

    self._results = {
        'time':[],
        'norm': [],
        'bond_dimensions': [],
        'dynamics': {},
        'pt_bond_dimensions': pt_bond_dimensions,
    }
    for sites in self._dynamics_sites:
        self._results['dynamics'][sites] = Dynamics(name=f"site{sites}")
```

--------------------------------

### Get MPO Tensor

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/process_tensor.html

Retrieves the MPO tensor for a given time step, optionally applying input/output transformations.

```python
def get_mpo_tensor(
        self,
        step: int,
        transformed: Optional[bool] = True) -> ndarray:
    """
    Get the MPO tensor for time step `step`.

    The axes correspond to the following legs:
        [0] ... past bond leg,
        [1] ... future bond leg,
        [2] ... input (from system) leg,
        [3] ... output (to system) leg.

    Applies the transformation (stored in `.transform_in` and
    `.transform_out`) when `transformed` is true.
    """
    length = len(self._mpo_tensors)
    if step >= length or step < 0:
        raise IndexError("Process tensor index out of bound. ")
    tensor = self._mpo_tensors[step]
    if len(tensor.shape) == 3:
        tensor = util.create_delta(tensor, [0, 1, 2, 2])
    if transformed is False:
        return tensor
    if self._transform_in is not None:
        tensor = np.dot(np.moveaxis(tensor, -2, -1),
                        self._transform_in.T)
        tensor = np.moveaxis(tensor, -1, -2)
    if self._transform_out is not None:
        tensor = np.dot(tensor, self._transform_out)
    return tensor
```

--------------------------------

### Initialize Tempo Object

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/quickstart.rst.txt

Create a Tempo object to manage the computation of system dynamics.

```python
tempo = oqupy.Tempo(system=system,
                      bath=bath,
                      parameters=tempo_parameters,
                      initial_state=up_density_matrix,
                      start_time=0.0,
                      unique=False)
```

--------------------------------

### Calculate number of steps

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/tempo.html

Determines the required number of steps to reach a target end time from a starting step.

```python
def _get_num_step(self,
        start_step: int,
        end_time: float) -> Tuple[int, int]:
    """Return the number of steps required from start_step to reach
    end_time"""
    end_step = int((end_time - self._start_time)/self._parameters.dt)
    num_step = max(0, end_step - start_step)
    return num_step
```

--------------------------------

### Compute dynamics with TEMPO

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/pt_tempo.rst.txt

Initialize TEMPO parameters and compute the system dynamics over a specified time interval.

```python
tempo_parameters = oqupy.TempoParameters(dt=0.1, tcut=2.0, epsrel=10**(-4))

tempo_sys = oqupy.Tempo(system=system,
                        bath=bath,
                        initial_state=initial_state,
                        start_time=-2.0,
                        parameters=tempo_parameters)
dynamics = tempo_sys.compute(end_time=3.0)
```

--------------------------------

### Initialize PT-TEBD backend

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/pt_tebd.html

Method to initialize the propagator, backend, and results structure for the simulation.

```python
def initialize(self) -> None:
    """Initialize propagator, the PT-TEBD backend and results. """
    self._step = self._start_step
    self._tebd_propagator = compute_tebd_propagator(
            system_chain=self._system_chain,
            time_step=self._parameters.dt/2.0,
            epsrel=self._parameters.epsrel,
            order=self._parameters.order)
    self._results = {}
    self._t_mps = PtTebdBackend(
            gammas=self._initial_augmented_mps.gammas,
            lambdas=self._initial_augmented_mps.lambdas,
            epsrel=self._parameters.epsrel,
            config=self._backend_config)
    self._init_results()
    self._apply_controls(step=self.step, post=False)
    self._append_results()
```

--------------------------------

### Configure bosonic environment

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/pt_tebd.rst.txt

Define a PowerLawSD spectral density and create a Bath object coupled to the first spin site.

```python
alpha = 0.32
omega_cutoff = 4.0
temperature = 1.6

correlations = oqupy.PowerLawSD(
    alpha=alpha,
    zeta=1,
    cutoff=omega_cutoff,
    cutoff_type='exponential',
    temperature=temperature)
bath = oqupy.Bath(sy, correlations)
```

--------------------------------

### PowerLawSD.__init__

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/bath_correlations.html

Initializes a new PowerLawSD object representing a spectral density with a power-law form.

```APIDOC
## PowerLawSD(alpha, zeta, cutoff, cutoff_type='exponential', temperature=0.0, name=None, description=None)

### Description
Creates a PowerLawSD object to model spectral density correlations of the form J(omega) = 2 * alpha * (omega^zeta / omega_c^(zeta-1)) * X(omega, omega_c).

### Parameters
- **alpha** (float) - Required - The coupling strength.
- **zeta** (float) - Required - The exponent (dimensionality of the environment).
- **cutoff** (float) - Required - The cutoff frequency.
- **cutoff_type** (str) - Optional - The cutoff type: 'hard', 'exponential', or 'gaussian'. Defaults to 'exponential'.
- **temperature** (float) - Optional - The environment's temperature. Defaults to 0.0.
- **name** (str) - Optional - An optional name for the correlations.
- **description** (str) - Optional - An optional description of the correlations.
```

--------------------------------

### SystemChain Constructor

Source: https://oqupy.readthedocs.io/en/latest/pages/modules.html

Initializes a new 1D chain of systems with specified Hilbert space dimensions.

```APIDOC
## SystemChain(hilbert_space_dimensions, name=None, description=None)

### Description
Represents a 1D chain of systems with nearest neighbor interactions.

### Parameters
- **hilbert_space_dimensions** (List[int]) - Required - Hilbert space dimension for each chain site.
- **name** (str) - Optional - An optional name for the system chain.
- **description** (str) - Optional - An optional description of the system chain.
```

--------------------------------

### Initialize MeanFieldSystem

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/mf_tempo.rst.txt

Wraps a list of TimeDependentSystemWithField objects into a MeanFieldSystem for collective dynamics.

```python
system_list = [system] # a list of TimeDependentiSystemWithField objects
mean_field_system = oqupy.MeanFieldSystem(
                    system_list=system_list,
                    field_eom=field_eom)
```

--------------------------------

### TimeDependentSystem.__init__

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/system.html

Initializes a new TimeDependentSystem object representing a system with time-dependent Hamiltonian, gammas, and Lindblad operators.

```APIDOC
## TimeDependentSystem(hamiltonian, gammas=None, lindblad_operators=None, name=None, description=None)

### Description
Creates a TimeDependentSystem object for systems without coupling to a non-Markovian bath, allowing for time-dependent Hamiltonian and Lindblad terms.

### Parameters
- **hamiltonian** (Callable[[float], ndarray]) - Required - System-only Hamiltonian H(t).
- **gammas** (List[Callable[[float], float]]) - Optional - The rates gamma_n(t).
- **lindblad_operators** (List[Callable[[float], ndarray]]) - Optional - The Lindblad operators A_n(t).
- **name** (str) - Optional - An optional name for the system.
- **description** (str) - Optional - An optional description of the system.
```

--------------------------------

### Import OQuPy and dependencies

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/n_time_correlations.rst.txt

Initial imports required for OQuPy functionality, Fourier transforms, and plotting.

```python
import sys
sys.path.insert(0,'..')

import oqupy
from oqupy.system_dynamics import compute_correlations_nt
import numpy as np
import matplotlib.pyplot as plt
from scipy.fft import fftfreq, fftshift, fft2
```

--------------------------------

### Define spin operators and density matrices

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/pt_tempo.rst.txt

Initialize common Pauli operators and standard spin density matrices for quantum dot simulations.

```python
sigma_x = oqupy.operators.sigma("x")
sigma_y = oqupy.operators.sigma("y")
sigma_z = oqupy.operators.sigma("z")
up_density_matrix = oqupy.operators.spin_dm("z+")
down_density_matrix = oqupy.operators.spin_dm("z-")
mixed_density_matrix = oqupy.operators.spin_dm("mixed")
```

--------------------------------

### oqupy.pt_tebd.PtTebd.__init__

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/pt_tebd.html

Initializes the PT-TEBD computation object with the necessary system parameters and configuration.

```APIDOC
## oqupy.pt_tebd.PtTebd.__init__

### Description
Initializes a new PtTebd instance to manage the time-evolution of an augmented Matrix Product State (MPS) using process tensors.

### Parameters
- **initial_augmented_mps** (AugmentedMPS) - Required - The initial augmented MPS state.
- **system_chain** (SystemChain) - Required - The system chain Liouvillians.
- **process_tensors** (List[BaseProcessTensor]) - Required - List of process tensors for each site.
- **parameters** (PtTebdParameters) - Required - Computation parameters.
- **chain_control** (ChainControl) - Optional - Control operations.
- **start_time** (float) - Optional - Starting time stamp.
- **start_step** (int) - Optional - Starting time step.
- **dynamics_sites** (List[Union[int, tuple]]) - Optional - Sites to record dynamics for.
- **backend_config** (dict) - Optional - Backend configuration (e.g., 'parallel': 'multiprocess' or 'multithread').
```

--------------------------------

### PtTempo.__init__

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/pt_tempo.html

Initializes a new PtTempo object to facilitate a PT-TEMPO computation.

```APIDOC
## PtTempo(bath, start_time, end_time, parameters, unique=False, process_tensor_file=None, overwrite=False, backend_config=None, name=None, description=None)

### Description
Creates a PtTempo object to manage the computation of a process tensor using the PT-TEMPO algorithm.

### Parameters
- **bath** (Bath) - Required - The Bath object including the coupling operator to the system.
- **start_time** (float) - Required - The start time of the computation.
- **end_time** (float) - Required - The end time of the computation.
- **parameters** (TempoParameters) - Required - The parameters for the PT-TEMPO computation.
- **unique** (bool) - Optional - Whether to use degeneracy checking to reduce the dimension of bath tensors.
- **process_tensor_file** (Union[Text, bool]) - Optional - File path or boolean for process tensor storage.
- **overwrite** (bool) - Optional - Whether to overwrite existing process tensor files.
- **backend_config** (Dict) - Optional - Configuration for the backend; defaults to PT_TEMPO_BACKEND_CONFIG if None.
- **name** (Text) - Optional - An optional name for the object.
- **description** (Text) - Optional - An optional description of the object.
```

--------------------------------

### Configure PT-TEMPO Parameters

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/n_time_correlations.rst.txt

Sets the physical bath parameters and numerical convergence settings for the PT-TEMPO computation.

```python
omega_cutoff = 3.04
alpha =0.1
temperature = 13.09

dt=0.2
dkmax=200
epsrel=10**(-2)

tempo_parameters = oqupy.TempoParameters(dt=dt, dkmax=dkmax, epsrel=epsrel)
```

--------------------------------

### Configure Time Parameters

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/pt_gradient/pt_gradient.rst.txt

Set the simulation duration and discretization steps.

```python
max_time = 5.0
N = 50 # number of time steps
dt = max_time/N
```

--------------------------------

### Iterate and Plot TEMPO Convergence for Varying Timesteps

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/parameters.rst.txt

Compares simulation results across different timesteps by adjusting epsrel to maintain convergence. Requires pre-defined system, bath, and initial state objects.

```python
for dt, epsrel in zip([0.0625,0.125, 0.25],[6.9e-06,6.9e-05,6.9e-05]):
    params = oqupy.TempoParameters(dt=dt, epsrel=epsrel, tcut=2.5)
    dynamics = oqupy.tempo_compute(system=system,
                                   bath=bath,
                                   initial_state=initial_state,
                                   start_time=t_start,
                                   end_time=t_end,
                                   parameters=params)
    t, s_z = dynamics.expectations(sigma_z, real=True)
    plt.plot(t, s_z, label=r'${}$, ${:.2g}$'.format(dt,epsrel))
plt.xlabel(r'$t$')
plt.ylabel(r'$\langle\sigma_z\rangle$')
plt.legend(title=r'$dt, \epsilon_{rel}$')

```

--------------------------------

### Define System and Dipole Operators

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/n_time_correlations.rst.txt

Sets up the system Hamiltonian, dipole operators, and the initial density matrix for the correlation calculation.

```python
eps = 5.
omeg= 2.
reorg = 2.0*alpha*omega_cutoff
system = oqupy.System((eps+reorg)*(P_1 + P_2)
                      + omeg * (sigma_plus + sigma_min))

dip_v = np.array([[0., 0., 1.],
       [0., 0., 0.],[1., 0., 0.]], dtype=complex)
dipole_ops = [dip_v, dip_v, dip_v, dip_v]

initial_state = np.array([[1., 0., 0.],
       [0., 0., 0.],[0., 0., 0.]], dtype=complex)
```

--------------------------------

### Initialize MeanFieldSystem

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/system.html

Constructor for a system interacting with a common field, requiring a list of systems and a field equation of motion.

```python
def __init__(self,
            system_list: List[TimeDependentSystemWithField],
            field_eom: Callable[[float, List[ndarray], complex], complex],
            name: Optional[Text] = None,
            description: Optional[Text] = None) -> None:

        super().__init__(name, description)
        tmp_system_list = _check_mean_field_system_list(system_list)
        self._system_list = tmp_system_list

        # input check for field equation of motion
        tmp_dimension_list = [system.hamiltonian(1.0, 1.0+1.0j).shape[0]
                              for system in self.system_list]
        tmp_field_eom = _check_mean_field_system_eom(tmp_dimension_list,
                                                     field_eom)
        self._field_eom = tmp_field_eom
```

--------------------------------

### Visualize Custom Parameters

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/quickstart.rst.txt

Verify the manually defined parameters against the bath correlations.

```python
fig, ax = plt.subplots(1,1)
oqupy.helpers.plot_correlations_with_parameters(bath.correlations, tempo_parameters, ax=ax)
```

--------------------------------

### Initialize dynamics metadata

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/tempo.html

Creates a MeanFieldDynamics object using metadata from the current MeanFieldTempo instance.

```python
    def _init_dynamics(self):
        """Create a MeanFieldDynamics object with metadata from the
        MeanFieldTempo object. """
        name = None
        description = "computed from '{}' MeanFieldTempo".format(self.name)
        self._dynamics = MeanFieldDynamics(name=name,
                                  description=description)
```

--------------------------------

### preparation(density_matrix)

Source: https://oqupy.readthedocs.io/en/latest/_modules/oqupy/operators.html

Constructs a state preparation superoperator.

```APIDOC
## preparation(density_matrix)

### Description
Constructs the superoperator that prepares the given density matrix state.

### Parameters
- **density_matrix** (ndarray) - Required - The density matrix to prepare.
```

--------------------------------

### Set physical parameters for the quantum dot

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/pt_tempo.rst.txt

Define the cutoff frequency, coupling strength, temperature, and initial state for the spin-boson model.

```python
omega_cutoff = 3.04 
alpha = 0.126
temperature = 0.1309
initial_state=down_density_matrix
```

--------------------------------

### Configure and run open PT-TEBD

Source: https://oqupy.readthedocs.io/en/latest/_sources/pages/tutorials/pt_tebd.rst.txt

Initialize the PtTebd object with the computed process tensor and run the computation for the open spin chain.

```python
pt_tebd_open = oqupy.PtTebd(
    initial_augmented_mps=initial_augmented_mps_B,
    system_chain=system_chain,
    process_tensors=[pt, None, None, None, None],
    parameters=pt_tebd_params,
    dynamics_sites=[0, 1, 2, 3, 4],
    chain_control=None)

print("PT-TEBD computation (open spin chain):")
results_open = pt_tebd_open.compute(num_steps, progress_type="bar")
```