### Install Example Dependencies

Source: https://squlearn.github.io/index.html

Install the required packages to run the examples provided with sQUlearn.

```bash
pip install .[examples]
```

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

### Install sQUlearn with Example Dependencies

Source: https://squlearn.github.io/_sources/install/install.rst.txt

Install sQUlearn along with packages required to run all examples.

```bash
pip install squlearn[examples]
```

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

### Install sQUlearn with Documentation Dependencies

Source: https://squlearn.github.io/_sources/install/install.rst.txt

Install sQUlearn along with packages necessary for building the documentation.

```bash
pip install squlearn[docs]
```

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

### Install sQUlearn from GitHub

Source: https://squlearn.github.io/index.html

Install sQUlearn directly from its GitHub repository using SSH.

```bash
pip install git+ssh://git@github.com:sQUlearn/squlearn.git
```

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

### Install sQUlearn with Development Dependencies

Source: https://squlearn.github.io/_sources/install/install.rst.txt

Install sQUlearn along with packages useful for development.

```bash
pip install squlearn[dev]
```

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

### Executor Initialization Examples

Source: https://squlearn.github.io/modules/generated/squlearn.Executor.html

Demonstrates various ways to initialize the Executor class with different quantum computing frameworks and backends.

```APIDOC
## Executor Initialization Examples

### PennyLane Initialization

```python
from squlearn import Executor
import pennylane as qml

# Executor with a PennyLane device (statevector)
executor = Executor(qml.device("default.qubit"))

# Executor with a PennyLane device (shot-based)
executor = Executor(qml.device("default.qubit", shots=1000))

# Executor with a PennyLane lightning device
executor = Executor(qml.device("lightning.qubit"))

# Executor with an AWS Braket device with 4 qubits
# (requires a valid AWS credential to be set)
dev = qml.device(
    "braket.aws.qubit",
    device_arn="arn:aws:braket:::device/quantum-simulator/amazon/sv1",
    wires=4
)
executor = Executor(dev)
```

### Qulacs Initialization

```python
from squlearn import Executor

# Executor with Qulacs backend
executor = Executor("qulacs")
```

### Qiskit Initialization

```python
from squlearn import Executor
from qiskit_ibm_runtime import QiskitRuntimeService

# Executor with a ideal simulator backend
exec = Executor("statevector_simulator")

# Executor with a shot-based simulator backend and 1000 shots
exec = Executor("qasm_simulator")
exec.set_shots(1000)

# Executor with a IBM Quantum backend (with context manager - recommended)
# Session is automatically closed after the with block
with Executor(service.backend('ibm_kingston'), caching=True,
             cache_dir='cache', log_file="log.log") as executor:
    # Your quantum computations here
    pass

# Executor with a IBM Quantum backend (without context manager - not recommended)
service = QiskitRuntimeService(channel="ibm_quantum_platform", token="INSERT_YOUR_TOKEN_HERE")
executor = Executor(service.backend('ibm_kingston'))

# Make sure to close the session when done
try:
    # Your code here
    pass
finally:
    executor.close_session()
```
```

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

### QGPR Example

Source: https://squlearn.github.io/modules/generated/squlearn.kernel.QGPR.html

Demonstrates how to initialize and use the QGPR class for regression. This involves setting up an encoding circuit, a quantum kernel, and then fitting and predicting using the QGPR model.

```python
from squlearn import Executor
from squlearn.encoding_circuit import HubregtsenEncodingCircuit
from squlearn.kernel.lowlevel_kernel import FidelityKernel
from squlearn.kernel import QGPR
enc_circ = HubregtsenEncodingCircuit(num_qubits=num_qubits, num_layers=2)
q_kernel = FidelityKernel(encoding_circuit=enc_circ, executor=Executor())
q_kernel.assign_parameters(np.random.rand(enc_circ.num_parameters))
qgpr_ansatz = QGPR(quantum_kernel=q_kernel)
qgpr_ansatz.fit(sample_train,label_train)
qgpr_ansatz.predict(sample_test)
```

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

### Simple Kernel Method Example

Source: https://squlearn.github.io/user_guide/kernel_methods.html

This snippet shows how to initialize and fit a kernel method model with sample training data. Ensure the 'qkrr' object is properly instantiated before use.

```python
x_train = [[0.1], [0.2], [0.3], [0.4], [0.5]]
y_train = [0.1, 0.2, 0.3, 0.4, 0.5]
qkrr.fit(X=x_train, y=y_train)
```

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

### QGPC Example with Quantum Kernel

Source: https://squlearn.github.io/modules/generated/squlearn.kernel.QGPC.html

Demonstrates how to use the QGPC class with a FidelityKernel for classification. This involves setting up an encoding circuit, a quantum kernel, and then fitting and predicting with the QGPC model.

```python
from sklearn.datasets import load_iris
from squlearn import Executor
from squlearn.encoding_circuit import HubregtsenEncodingCircuit
from squlearn.kernel.lowlevel_kernel import FidelityKernel
from squlearn.kernel import QGPC
X, y = load_iris(return_X_y=True)

enc_circ = HubregtsenEncodingCircuit(num_qubits=X.shape[1], num_layers=2)
q_kernel = FidelityKernel(encoding_circuit=enc_circ, executor=Executor())
q_kernel.assign_parameters(np.random.rand(enc_circ.num_parameters))
qgpc_ansatz = QGPC(quantum_kernel=q_kernel)
qgpc_ansatz.fit(X, y)
qgpc_ansatz.score(X, y)
    0.98...
qgpc_ansatz.predict_proba(X[:2,:])
    array([[0.85643716, 0.07037611, 0.07318673],
    [0.80314475, 0.09988938, 0.09696586]])

```

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

### Install Development Version of sQUlearn

Source: https://squlearn.github.io/_sources/install/install.rst.txt

Install the latest version from the 'develop' branch of the GitHub repository.

```bash
pip install git+https://github.com/sQUlearn/squlearn.git@develop
```

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

### QSVC Example: Quantum Support Vector Classification

Source: https://squlearn.github.io/modules/generated/squlearn.kernel.QSVC.html

Demonstrates how to use QSVC for classification with a ProjectedQuantumKernel. This example involves setting up an encoding circuit, defining a quantum kernel, preparing data, training the QSVC model, and evaluating its performance on a test set.

```python
import numpy as np

from sklearn.datasets import make_moons
from sklearn.model_selection import train_test_split

from squlearn import Executor
from squlearn.encoding_circuit import HubregtsenEncodingCircuit
from squlearn.kernel.qsvc import QSVC
from squlearn.kernel.lowlevel_kernel import ProjectedQuantumKernel

encoding_circuit = HubregtsenEncodingCircuit(num_qubits=2, num_layers=2)
kernel = ProjectedQuantumKernel(
    encoding_circuit,
    executor=Executor(),
    initial_parameters=np.random.rand(encoding_circuit.num_parameters)
)

X, y = make_moons(n_samples=100, noise=0.3, random_state=0)
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)

qsvc = QSVC(quantum_kernel=kernel)
qsvc.fit(X_train, y_train)
print(f"The score on the test set is {qsvc.score(X_test, y_test)}")

```

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

### Setup Quantum Circuit and Projected Quantum Kernel

Source: https://squlearn.github.io/examples/example_ode_solver.html

Configure a quantum circuit for encoding and define a Projected Quantum Kernel (PQK) for ODE solving. This involves specifying circuit parameters, observables, and the executor.

```python
num_qubits = 6
circuit = KyriienkoEncodingCircuit(
    num_qubits=num_qubits,
    encoding_style="chebyshev_product",  # other options are chebyshev_tower, chebyshev_sparse, chebyshev_product
    variational_arrangement="HEA",  # other option can be HEA (Hardware Efficient Ansatz), with consecutive entangling layers
    num_features=1,
    num_encoding_layers=2,
    num_variational_layers=1,
)
observable = SummedPaulis(num_qubits, op_str=("XYZ"), include_identity=True)


pqk = ProjectedQuantumKernel(executor=Executor("pennylane"), encoding_circuit=circuit)
loss_ODE_squ = KODELoss(
    eq,
    symbols_involved_in_ode=[x, y, dydx],
    initial_values=initial_values,
    eta=1,
)

circuit.draw("mpl")
```

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

### Executor Initialization with Backend

Source: https://squlearn.github.io/_sources/user_guide/executor.rst.txt

Initialize the Executor with a specific backend from QiskitRuntimeService. This setup is a prerequisite for adjusting primitive options.

```python
from squlearn import Executor
from qiskit_ibm_runtime import QiskitRuntimeService, Options

service = QiskitRuntimeService(channel="ibm_quantum_platform", token="INSERT_YOUR_TOKEN_HERE")

with Executor(service.backend("ibm_kingston")) as executor:
    
```

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

### Importing Libraries for QML and ML Workflows

Source: https://squlearn.github.io/examples/example_kernel_grid_search.html

Imports necessary libraries from squlearn for quantum kernel methods and scikit-learn for data processing and hyperparameter optimization. Ensure these libraries are installed.

```python
import matplotlib.pylab as plt
import numpy as np
import pandas as pd
import seaborn as sns

from sklearn.model_selection import GridSearchCV
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import StandardScaler

from squlearn import Executor
from squlearn.encoding_circuit import YZ_CX_EncodingCircuit
from squlearn.kernel.lowlevel_kernel import FidelityKernel, ProjectedQuantumKernel
from squlearn.kernel import QGPR, QSVR
```

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

### Install Bleeding-edge Version of sQUlearn

Source: https://squlearn.github.io/_sources/install/install.rst.txt

Install the latest version directly from the main branch of the GitHub repository.

```bash
pip install git+https://github.com/sQUlearn/squlearn.git
```

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

### QNNClassifier Example

Source: https://squlearn.github.io/modules/generated/squlearn.qnn.QNNClassifier.html

Demonstrates the usage of QNNClassifier for a binary classification task. It includes data preprocessing, model initialization, training, and prediction.

```python
from squlearn import Executor
from squlearn.encoding_circuit import ChebyshevRx
from squlearn.observables import SummedPaulis
from squlearn.qnn import QNNClassifier, SquaredLoss
from squlearn.optimizers import SLSQP
from sklearn.datasets import make_blobs
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import MinMaxScaler

X, y = make_blobs(60, centers=2, random_state=0)
X = MinMaxScaler((-0.9, 0.9)).fit_transform(X)
X_train, X_test, y_train, y_test = train_test_split(
    X, y, test_size=0.33, random_state=42
)
clf = QNNClassifier(
    ChebyshevRx(4, 2),
    SummedPaulis(4),
    Executor(),
    SquaredLoss(),
    SLSQP(),
    np.random.rand(16),
    np.random.rand(5)
)
clf.fit(X_train, y_train)
y_pred = clf.predict(X_test)

```

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

### Initialize and Use Executor with Context Manager

Source: https://squlearn.github.io/modules/generated/squlearn.Executor.html

Demonstrates the recommended way to initialize and use the Executor with an IBM Quantum backend within a context manager. This ensures proper session closure. Requires Qiskit Runtime service setup.

```python
from squlearn import Executor
from qiskit_ibm_runtime import QiskitRuntimeService

service = QiskitRuntimeService(channel="ibm_quantum_platform", token="INSERT_YOUR_TOKEN_HERE")

# Recommended: Use context manager
with Executor(service.backend('ibm_kingston'), caching=True,
              cache_dir='cache', log_file="log.log") as executor:
    # Your quantum computations here
    pass

```

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

### Install Stable Release of sQUlearn

Source: https://squlearn.github.io/_sources/install/install.rst.txt

Use this command to install the latest stable version of sQUlearn from PyPI.

```bash
pip install squlearn
```

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

### QSVR Example: Quantum Support Vector Regression

Source: https://squlearn.github.io/modules/generated/squlearn.kernel.QSVR.html

Demonstrates how to use QSVR for regression tasks. It involves setting up an encoding circuit, a quantum kernel, splitting data, initializing QSVR with the quantum kernel, fitting the model, and evaluating its performance on a test set.

```python
import numpy as np

from sklearn.model_selection import train_test_split

from squlearn import Executor
from squlearn.encoding_circuit import HubregtsenEncodingCircuit
from squlearn.kernel.qsvr import QSVR
from squlearn.kernel.lowlevel_kernel import ProjectedQuantumKernel

encoding_circuit = HubregtsenEncodingCircuit(num_qubits=2, num_layers=2)
kernel = ProjectedQuantumKernel(
    encoding_circuit,
    executor=Executor(),
    initial_parameters=np.random.rand(encoding_circuit.num_parameters))

X = np.linspace(0, np.pi, 100)
y = np.sin(X)
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)

qsvc = QSVR(quantum_kernel=kernel)
qsvc.fit(X_train, y_train)
print(f"The score on the test set is {qsvc.score(X_test, y_test)}")

```

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

### QNNRegressor Example with ODELoss for Differential Equations

Source: https://squlearn.github.io/modules/generated/squlearn.qnn.QNNRegressor.html

Illustrates using QNNRegressor with ODELoss to solve differential equations. This example sets up a symbolic differential equation, initial conditions, and uses an Adam optimizer with learning rate decay. The QNN is configured with a KyriienkoEncodingCircuit and a SummedPaulis observable.

```python
import numpy as np
import sympy as sp
import matplotlib.pyplot as plt
from squlearn import Executor
from squlearn.encoding_circuit import KyriienkoEncodingCircuit
from squlearn.observables import SummedPaulis
from squlearn.qnn import QNNRegressor, ODELoss
from squlearn.qnn.util import get_lr_decay
from squlearn.optimizers import Adam


t, y, dydt, = sp.symbols("t y dydt")
eq = 20 * sp.exp(-20 * t * 0.1) * sp.sin(20 * t) + 20 * 0.1 * y  + dydt
initial_values = [1.0]

loss_ODE = ODELoss(
    eq,

```

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

### ProjectedQuantumKernel Initialization and Usage

Source: https://squlearn.github.io/modules/generated/squlearn.kernel.lowlevel_kernel.ProjectedQuantumKernel.html

This snippet demonstrates how to initialize and use the ProjectedQuantumKernel to compute a kernel matrix. It shows the basic setup with an encoding circuit and executor, and then calls the evaluate method.

```APIDOC
## ProjectedQuantumKernel

Computes the quantum kernel matrix by projecting quantum states onto a computational basis.

### Parameters

*   **encoding_circuit** (EncodingCircuit): The encoding circuit used to prepare quantum states.
*   **executor** (Executor): The executor to run the quantum circuits.
*   **measurement** (list[Observable], optional): A list of observables to measure. If None, the default measurement is used. Defaults to None.
*   **outer_kernel** (str, optional): The type of outer kernel to use. Can be 'matern', 'rational_quadratic', 'exp_sine_squared', or 'dot_product'. Defaults to None.
*   **hyperparameters** (dict, optional): A dictionary of hyperparameters for the outer kernel. Defaults to None.

### Example

```python
import numpy as np
from squlearn.encoding_circuit import ChebyshevTower
from squlearn.kernel.lowlevel_kernel import ProjectedQuantumKernel
from squlearn.util import Executor

fm = ChebyshevTower(num_qubits=4, num_chebyshev=4)
kernel = ProjectedQuantumKernel(encoding_circuit=fm, executor=Executor())
x = np.random.rand(10)
kernel_matrix = kernel.evaluate(x.reshape(-1, 1), x.reshape(-1, 1))
print(kernel_matrix)
```
```

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

### Initialize Executor with Qiskit Runtime Service and Options

Source: https://squlearn.github.io/_sources/user_guide/executor.rst.txt

Initialize the Executor using a QiskitRuntimeService backend and provide options for the estimator primitive. This example shows how to configure ZNE mitigation.

```python
from squlearn import Executor
from qiskit_ibm_runtime import QiskitRuntimeService

service = QiskitRuntimeService(channel="ibm_quantum_platform", token="INSERT_YOUR_TOKEN_HERE")

options = {"resilience":{"zne_mitigation"  : True,
                         "zne":{"noise_factors" : (1, 3, 5),"extrapolator":"linear" }}
          }

with Executor(service.backend("ibm_kingston"), options_estimator=options) as executor:
    # Your quantum computations here
    pass
```

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

### Initialize ODELoss for a Cosine ODE with Custom Parameters

Source: https://squlearn.github.io/modules/generated/squlearn.kernel.loss.ODELoss.html

Demonstrates initializing ODELoss for the ODE (df(x)/dx) - cos(f(x)) = 0 with initial value f(0) = 0. This example also shows how to adjust the 'eta' parameter, which controls the weight of the initial-value penalty.

```python
import sympy as sp
from squlearn.kernel.loss import ODELoss

x, f, dfdx = sp.symbols("x f dfdx")
eq = dfdx - sp.cos(f)
initial_values = [0]

loss_ode = ODELoss(
    ode_functional=eq,
    initial_values=initial_values,
    symbols_involved_in_ode=[x, f, dfdx],
    eta=1.2,
)
```

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

### Initialize Executor with Options

Source: https://squlearn.github.io/_sources/user_guide/executor.rst.txt

Initialize the Executor with specific execution settings and options for the estimator primitive. This example demonstrates setting restart pauses, retry limits, and estimator options.

```python
with Executor(execution="statevector_simulator",
                   wait_restart=600,  # Set 10 min pause between restarts of Jobs
                   max_jobs_retries=10, # Set maximum number of restarts to 10 before aborting
                   options_estimator=options # Set options for the Estimator primitive
                   ) as executor:
        executor.set_shots(1234) # Shots can be adjusted after initialization
        # Your quantum computations here
        qnn.fit(X_train, y_train)
```

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

### Optimize Quantum Kernels with Kernel Target Alignment

Source: https://squlearn.github.io/_sources/user_guide/kernel_methods.rst.txt

Set up and train a Quantum Kernel Ridge Regression (QKRR) model using a ProjectedQuantumKernel optimized via Kernel Target Alignment (KTA) with the Adam optimizer. This example demonstrates the full workflow from circuit definition to model fitting.

```python
from squlearn.util import Executor
from squlearn.encoding_circuit import ChebyshevPQC
from squlearn.optimizers import Adam
from squlearn.kernel import ProjectedQuantumKernel, KernelOptimizer, QKRR
from squlearn.kernel.loss import TargetAlignment

# set up the encoding circuit
encoding_circuit = ChebyshevPQC(num_qubits=4, num_layers=2)

# set up the quantum kernel
pqk_instance = ProjectedQuantumKernel(encoding_circuit, Executor())

# set up the optimizer
adam_opt = Adam(options={"maxiter":100, "lr": 0.1})

# define KTA loss function
kta_loss = TargetAlignment()

# set up the kernel optimizer
kernel_optimizer = KernelOptimizer(quantum_kernel=pqk_instance, loss=kta_loss, optimizer=adam_opt)

# initialize the QKRR model with the kernel optimizer
qkrr = QKRR(kernel_optimizer)

# Simple example
x_train = [[0.1], [0.2], [0.3], [0.4], [0.5]]
y_train = [0.1, 0.2, 0.3, 0.4, 0.5]
qkrr.fit(X=x_train, y=y_train)
```

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

### ProjectedQuantumKernel with Custom Measurements and Outer Kernel

Source: https://squlearn.github.io/modules/generated/squlearn.kernel.lowlevel_kernel.ProjectedQuantumKernel.html

This example shows how to customize the ProjectedQuantumKernel by providing a list of custom observables for measurement and specifying an 'matern' outer kernel with a given nu hyperparameter.

```APIDOC
## ProjectedQuantumKernel with Customization

This example demonstrates customizing the `ProjectedQuantumKernel` with specific measurement observables and an outer kernel.

### Parameters

*   **encoding_circuit** (EncodingCircuit): The encoding circuit used to prepare quantum states.
*   **executor** (Executor): The executor to run the quantum circuits.
*   **measurement** (list[Observable]): A list of custom observables for measurement.
*   **outer_kernel** (str): Specifies the type of outer kernel, e.g., 'matern'.
*   **nu** (float): Hyperparameter for the 'matern' outer kernel.

### Example

```python
import numpy as np
from squlearn.encoding_circuit import ChebyshevTower
from squlearn.kernel.lowlevel_kernel import ProjectedQuantumKernel
from squlearn.util import Executor
from squlearn.observables import CustomObservable
from squlearn.kernel import QKRR

fm = ChebyshevTower(num_qubits=4, num_chebyshev=4)

# Create custom observables
measurements = []
measurements.append(CustomObservable(4,"ZZZZ"))
measurements.append(CustomObservable(4,"YYYY"))
measurements.append(CustomObservable(4,"XXXX"))

# Use Matern Outer kernel with nu=0.5 as a outer kernel hyperparameter
kernel = ProjectedQuantumKernel(encoding_circuit=fm,
                                executor=Executor(),
                                measurement=measurements,
                                outer_kernel="matern",
                                nu=0.5)
ml_method = QKRR(quantum_kernel=kernel)
```
```

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

### Initialize ODELoss for a first-order ODE with custom eta

Source: https://squlearn.github.io/modules/generated/squlearn.qnn.loss.ODELoss.html

Example of initializing ODELoss for the ODE df(x)/dx - cos(f(x)) = 0 with initial values f(0) = 0. Demonstrates setting a custom 'eta' value to adjust the weight of the initial condition penalty.

```python
import sympy as sp
from squlearn.qnn.loss import ODELoss

x, f, dfdx = sp.symbols("x f dfdx")
eq = dfdx - sp.cos(f)
initial_values = [0]

loss_ode = ODELoss(
    eq,
    symbols_involved_in_ode=[x, f, dfdx],
    initial_values=initial_values,
    eta=1.2,
)
```

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

### QNNRegressor Example with ChebyshevRx and IsingHamiltonian

Source: https://squlearn.github.io/modules/generated/squlearn.qnn.QNNRegressor.html

Demonstrates the basic usage of QNNRegressor for a regression task. It initializes the regressor with a ChebyshevRx encoding circuit, an IsingHamiltonian observable, an Executor, SquaredLoss, and an SLSQP optimizer. The model is then trained on split data and used for prediction.

```python
import numpy as np
from squlearn import Executor
from squlearn.encoding_circuit import ChebyshevRx
from squlearn.observables import IsingHamiltonian
from squlearn.qnn import QNNRegressor, SquaredLoss
from squlearn.optimizers import SLSQP
from sklearn.model_selection import train_test_split

X, y = np.arange(0.1, 0.9, 0.01), np.log(np.arange(0.1, 0.9, 0.01))
X_train, X_test, y_train, y_test = train_test_split(
    X, y, test_size=0.33, random_state=42
)
reg = QNNRegressor(
    ChebyshevRx(4, 1),
    IsingHamiltonian(4, I="S", Z="S", ZZ="S"),
    Executor(),
    SquaredLoss(),
    SLSQP(),
    np.random.rand(16),
    np.random.rand(5)
)
reg.fit(X_train, y_train)
y_pred = reg.predict(X_test[:5])

```

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

### Get Squared Operator for Summed Pauli Operator

Source: https://squlearn.github.io/modules/generated/squlearn.observables.observable_derivatives.ObservableDerivatives.html

Shows how to obtain the squared form (O^2) of a SummedPaulis observable. This is useful when the observable's square is needed, for example, in variance calculations.

```python
from squlearn.observables import SummedPaulis
from squlearn.observables.observable_derivatives import ObservableDerivatives
op = SummedPaulis(num_qubits=3)
print(ObservableDerivatives(op).get_operator_squared())
```

```text
SparsePauliOp(['III', 'IIZ', 'IZI', 'ZII', 'IZZ', 'ZIZ', 'ZZI'],
              coeffs=[<qiskit._accelerate.circuit.ParameterExpression object at 0x7f404ba78210>,
 <qiskit._accelerate.circuit.ParameterExpression object at 0x7f404ba78270>,
 <qiskit._accelerate.circuit.ParameterExpression object at 0x7f404ba782d0>,
 <qiskit._accelerate.circuit.ParameterExpression object at 0x7f404ba78330>,
 <qiskit._accelerate.circuit.ParameterExpression object at 0x7f404ba78390>,
 <qiskit._accelerate.circuit.ParameterExpression object at 0x7f404ba783f0>,
 <qiskit._accelerate.circuit.ParameterExpression object at 0x7f404ba78450>])
```

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

### Initialize ProjectedQuantumKernel

Source: https://squlearn.github.io/examples/example_kernel_digit_classification.html

Sets up a ProjectedQuantumKernel for calculating kernel matrices. This involves specifying the encoding circuit, an executor for running quantum computations, the measurement strategy, the type of outer kernel, initial trainable parameters, and the gamma parameter for the Gaussian outer kernel.

```python
kernel = ProjectedQuantumKernel(
    encoding_circuit=encoding_circuit,
    executor=Executor(),
    measurement="XYZ",
    outer_kernel="gaussian",
    initial_parameters=np.random.rand(encoding_circuit.num_parameters),
    gamma=0.5,
)
```

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

### get_derivative Method Examples

Source: https://squlearn.github.io/modules/generated/squlearn.observables.observable_derivatives.ObservableDerivatives.html

Examples demonstrating the usage of the `get_derivative` method for calculating different types of derivatives.

```APIDOC
## Method: get_derivative

### Description
Determine the derivative of the observable.

### Parameters
* **derivative** (_str_ _or_ _tuple_) – String or tuple of parameters for specifying the derivation. See `ObservableDerivatives` for more information.

### Returns
Differentiated observable in OpTree format

### Example: first-order derivative of the Ising Hamiltonian
```python
from squlearn.observables import IsingHamiltonian
from squlearn.observables.observable_derivatives import ObservableDerivatives
op = IsingHamiltonian(num_qubits=3)
print(ObservableDerivatives(op).get_derivative("dop"))
```

### Example: Squared summed Pauli Operator
```python
from squlearn.observables import SummedPaulis
from squlearn.observables.observable_derivatives import ObservableDerivatives
op = SummedPaulis(num_qubits=3)
print(ObservableDerivatives(op).get_operator_squared())
```
```

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

### Select Backend for QNN Execution

Source: https://squlearn.github.io/examples/example_qnn_backend_mitigation.html

Configures the backend for QNN execution, choosing between a local simulator (FakeBelemV2) or a real IBM Quantum backend based on the 'local' flag. Error mitigation is disabled on local simulators.

```python
local = True
if local:
    backend = FakeBelemV2()
else:
    service = QiskitRuntimeService()
    backend = service.least_busy(operational=True, simulator=False)
    print("Selected Backend:", backend.name)
```

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

### QRCClassifier Example: Classification of Moon example with Quantum Reservoir Computing

Source: https://squlearn.github.io/modules/generated/squlearn.qrc.QRCClassifier.html

This example demonstrates how to use QRCClassifier for a classification task on the moon dataset. It involves splitting the data, initializing the QRCClassifier with a specified encoding circuit and executor, and then fitting the model to the training data before making predictions on the test data.

```python
from squlearn import Executor
from squlearn.encoding_circuit import HubregtsenEncodingCircuit
from squlearn.qrc import QRCClassifier
from sklearn.datasets import make_moons
from sklearn.model_selection import train_test_split

X, y = make_moons(n_samples=1000, noise=0.2, random_state=42)

X_train, X_test, y_train, y_test = train_test_split(
    X, y, test_size=0.33, random_state=42)

clf = QRCClassifier(HubregtsenEncodingCircuit(num_qubits=4),
                    Executor(),
                    ml_model="linear",
                    operators="random_paulis",
                    num_operators=300,
                    )

clf.fit(X_train, y_train)
y_pred = clf.predict(X_test)

```

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

### Pruned Hubregtsen Encoding Circuit Example

Source: https://squlearn.github.io/modules/generated/squlearn.encoding_circuit.PrunedEncodingCircuit.html

Demonstrates how to create and draw a pruned encoding circuit using the HubregtsenEncodingCircuit. This example shows initializing a HubregtsenEncodingCircuit and then creating a PrunedEncodingCircuit from it, specifying which parameters to prune.

```python
from squlearn.encoding_circuit import HubregtsenEncodingCircuit,PrunedEncodingCircuit
fm = HubregtsenEncodingCircuit(4,2)
PrunedEncodingCircuit(fm,[4,7,11,15]).draw("mpl", num_features=2)
```

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

### Calculate Kernel Matrix with ProjectedQuantumKernel

Source: https://squlearn.github.io/modules/generated/squlearn.kernel.lowlevel_kernel.ProjectedQuantumKernel.html

Demonstrates how to initialize and use ProjectedQuantumKernel to compute a kernel matrix. Ensure necessary imports are included.

```python
import numpy as np
from squlearn.encoding_circuit import ChebyshevTower
from squlearn.kernel.lowlevel_kernel import ProjectedQuantumKernel
from squlearn.util import Executor

fm = ChebyshevTower(num_qubits=4, num_chebyshev=4)
kernel = ProjectedQuantumKernel(encoding_circuit=fm, executor=Executor())
x = np.random.rand(10)
kernel_matrix = kernel.evaluate(x.reshape(-1, 1), x.reshape(-1, 1))
print(kernel_matrix)
```

```text
[[1.00000000e+00 4.06794475e-05 9.49463805e-03 9.32251830e-02
  1.00951692e-03 6.45442026e-05 1.24839015e-02 1.72501703e-05
  1.60224819e-01 9.31212263e-01]
 [4.06794475e-05 1.00000000e+00 3.00823782e-02 2.68739824e-03
  2.03679424e-01 9.39710740e-01 2.31250462e-02 5.28290705e-01
  1.38032573e-03 6.69069973e-05]
 [9.49463805e-03 3.00823782e-02 1.00000000e+00 5.67749051e-01
  6.15479006e-01 6.28136328e-02 9.92685083e-01 2.30564806e-03
  4.02050742e-01 2.20990267e-02]
 [9.32251830e-02 2.68739824e-03 5.67749051e-01 1.00000000e+00
  1.36399426e-01 6.00665133e-03 6.39323079e-01 2.38149509e-04
  9.56977901e-01 1.84058756e-01]
 [1.00951692e-03 2.03679424e-01 6.15479006e-01 1.36399426e-01
  1.00000000e+00 3.47250768e-01 5.44185409e-01 2.17769276e-02
  7.81319173e-02 2.34075959e-03]
 [6.45442026e-05 9.39710740e-01 6.28136328e-02 6.00665133e-03
  3.47250768e-01 1.00000000e+00 4.90627670e-02 3.40590318e-01
  3.05334315e-03 1.16250527e-04]
 [1.24839015e-02 2.31250462e-02 9.92685083e-01 6.39323079e-01
  5.44185409e-01 4.90627670e-02 1.00000000e+00 1.75549208e-03
  4.67125096e-01 2.87692102e-02]
 [1.72501703e-05 5.28290705e-01 2.30564806e-03 2.38149509e-04
  2.17769276e-02 3.40590318e-01 1.75549208e-03 1.00000000e+00
  1.38671329e-04 2.05412131e-05]
 [1.60224819e-01 1.38032573e-03 4.02050742e-01 9.56977901e-01
  7.81319173e-02 3.05334315e-03 4.67125096e-01 1.38671329e-04
  1.00000000e+00 2.93890695e-01]
 [9.31212263e-01 6.69069973e-05 2.20990267e-02 1.84058756e-01
  2.34075959e-03 1.16250527e-04 2.87692102e-02 2.05412131e-05
  2.93890695e-01 1.00000000e+00]]
```

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

### Setting up Quantum Kernels with Encoding Circuit

Source: https://squlearn.github.io/examples/example_kernel_grid_search.html

Initializes an encoding circuit and quantum kernels (FidelityKernel, ProjectedQuantumKernel) using Qiskit's StatevectorSimulator. The encoding circuit parameters will be varied during grid search.

```python
encoding_circuit = YZ_CX_EncodingCircuit(num_qubits=4, num_layers=2, closed=False)

executor = Executor()

fidelity_kernel = FidelityKernel(
    encoding_circuit=encoding_circuit, executor=executor, parameter_seed=0
)
projected_kernel = ProjectedQuantumKernel(
    encoding_circuit=encoding_circuit, executor=executor, parameter_seed=0
)

encoding_circuit.draw("mpl", num_features=1)
```

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

### get_shots

Source: https://squlearn.github.io/modules/generated/squlearn.qnn.lowlevel_qnn.lowlevel_qnn_qiskit.LowLevelQNNQiskit.html

Gets the number of shots configured for the QNN execution.

```APIDOC
## get_shots

### Description
Gets the number of shots configured for the QNN execution.

### Returns
Number of shots.
```

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

### FidelityKernel

Source: https://squlearn.github.io/modules/generated/squlearn.kernel.lowlevel_kernel.FidelityKernel.html

Initializes the FidelityQuantum Kernel.

```APIDOC
## FidelityKernel

### Description
Initializes the FidelityQuantum Kernel.

### Parameters
* **encoding_circuit** (*EncodingCircuitBase*) – PQC encoding circuit.
* **executor** (*Executor*) – Executor object.
* **evaluate_duplicates** (*str*) – Option for evaluating duplicates (‘all’, ‘off_diagonal’, ‘none’).
* **mit_depol_noise** (*Union_[str, None]*) – Option for mitigating depolarizing noise ("msplit" or "mmean") after Ref. [4]. Only meaningful for FQKs computed on a real backend.
* **initial_parameters** (*Union_[np.ndarray, None]*) – Initial parameters for the encoding circuit.
* **parameter_seed** (*Union_[int, None]*) – Seed for the random number generator for the parameter initialization, if initial_parameters is None.
* **regularization** (*Union_[str, None]*) – Option for choosing different regularization techniques ("thresholding" or "tikhonov") after Ref. [4] for the training kernel matrix, prior to solving the linear system in the `fit()`-procedure.
* **use_expectation** (*bool*) – Option for using the expectation value of a QNN circuit that computes the fidelity as the expectation value of the observable P0=|0⟩⟨0|⊗n. This is not very efficient in simulation, but allows the computation of derivatives of the kernel.
```

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

### Get Hessian from Tuple

Source: https://squlearn.github.io/user_guide/observables.html

Calculate the second-order derivative (Hessian) of an observable with respect to a specific parameter. Requires initializing ObservableDerivatives with the operator.

```python
from squlearn.observables.observable_derivatives import ObservableDerivatives

# Assuming 'op3' is a defined observable operator
# op3 = ... 
deriv = ObservableDerivatives(op3)
print("Second-order derivative w.r.t. p[0]:\n",
         deriv.get_derivative((deriv.parameter_vector[0],deriv.parameter_vector[0])))
```

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

### Select Best IBM Quantum Backend with Executor

Source: https://squlearn.github.io/user_guide/executor.html

Initializes the Executor with a list of simulated IBM backends and trains a QNNRegressor. The 'quality' mode is used by default to select the best backend. The chosen backend is printed after training.

```python
import warnings
import numpy as np
from qiskit_ibm_runtime.fake_provider import FakeManilaV2, FakeBelemV2, FakeAthensV2
from squlearn.util import Executor
from squlearn.qnn import QNNRegressor
from squlearn.observables import SummedPaulis
from squlearn.encoding_circuit import ChebyshevPQC
from squlearn.optimizers import Adam
from squlearn.qnn.loss import SquaredLoss

backends = [FakeBelemV2(), FakeAthensV2(), FakeManilaV2()]
executor = Executor(backends, shots=10000)
qnn = QNNRegressor(
    ChebyshevPQC(2),
    SummedPaulis(2),
    executor,
    SquaredLoss(),
    Adam({'maxiter':2}), # Two iteration for demonstration purposes only
    callback=None # Remove print of the progress bar for cleaner output
)
with warnings.catch_warnings():
   warnings.simplefilter("ignore")
   qnn.fit(np.array([[0.25],[0.75]]),np.array([0.25,0.75]))
print("Chosen backend:", executor.backend)
```

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

### Getting Qiskit Primitives from Executor

Source: https://squlearn.github.io/modules/generated/squlearn.Executor.html

Demonstrates how to obtain Estimator and Sampler objects from an initialized Executor, ensuring all operations are routed through the executor.

```APIDOC
## Getting Qiskit Primitives from Executor

```python
from squlearn import Executor

# Initialize the Executor
executor = Executor("statevector_simulator")

# Get the Executor based Estimator with all executions routed through the Executor
estimator = executor.get_estimator()

# Get the Executor based Sampler with all executions routed through the Executor
sampler = executor.get_sampler()

# Run a circuit with the Executor based Sampler
from qiskit.circuit.random import random_circuit
circuit = random_circuit(2, 2, seed=1, measure=True).decompose(reps=1)
job = sampler.run([(circuit,)])
result = job.result()
```
```

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

### Setup Encoding Circuit for Regression

Source: https://squlearn.github.io/user_guide/quantum_reservoir_computing.html

Initialize an encoding circuit for regression tasks. The HubregtsenEncodingCircuit is used here with specified number of qubits and layers.

```python
from squlearn.encoding_circuit import HubregtsenEncodingCircuit

pqc = HubregtsenEncodingCircuit(num_qubits=10, num_layers=2)
pqc.draw("mpl", num_features=1)

```

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

### Executor Initialization and Usage

Source: https://squlearn.github.io/modules/generated/squlearn.Executor.html

Demonstrates how to initialize the Executor class and use it within a context manager for proper session handling, especially when interacting with IBM Quantum backends.

```APIDOC
## `squlearn`.Executor

### Description
A class for executing quantum jobs on IBM Quantum systems or simulators. The Executor class is the central component of sQUlearn, responsible for running quantum jobs. Both high- and low-level methods utilize the Executor class to execute jobs seamlessly. It for example automatically creates the necessary Qiskit primitives when they are required in the sQUlearn sub-program or takes care of the execution of PennyLane circuits. The Executor takes care about Qiskit Runtime session handling, result caching, and automatic restarts of failed jobs.

### Parameters
* **_execution** (str | Backend | List[Backend] | QiskitRuntimeService | Session | BaseEstimatorV1 | BaseSamplerV1 | BaseEstimatorV2 | BaseSamplerV2 | Device) - The execution environment specification.
* **_options_estimator** (Any | None) - Options for the estimator.
* **_options_sampler** (Any | None) - Options for the sampler.
* **_log_file** (str) - Path to a log file. Defaults to '' (no logging).
* **_caching** (bool | None) - Whether to enable caching. Defaults to None.
* **_cache_dir** (str) - Directory for caching results. Defaults to '_cache'.
* **_max_session_time** (str) - Maximum session time. Defaults to '8h'.
* **_max_jobs_retries** (int) - Maximum number of retries for failed jobs. Defaults to 10.
* **_wait_restart** (int) - Time to wait before restarting a failed job. Defaults to 1.
* **_shots** (int | None) - Number of shots for the execution. Defaults to None.
* **_seed** (int | None) - Seed for random number generation. Defaults to None.
* **_qpu_parallelization** (int | str | None) - QPU parallelization setting. Defaults to None.
* **_auto_backend_mode** (str) - Automatic backend mode. Defaults to 'quality'.

### Important: Session Management
When using the Executor with IBM Quantum backends or Sessions, it is **strongly recommended** to use the Executor within a context manager (`with` statement). This ensures that sessions are properly closed when you are done with the executor, avoiding unnecessary open sessions and preventing charges for unused sessions.

### Request Example
```python
from squlearn import Executor
from qiskit_ibm_runtime import QiskitRuntimeService

service = QiskitRuntimeService(channel="ibm_quantum_platform", token="INSERT_YOUR_TOKEN_HERE")

# Recommended: Use context manager
with Executor(service.backend('ibm_kingston'), caching=True,
              cache_dir='cache', log_file="log.log") as executor:
    # Your quantum computations here
    pass
```
```

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

### Initialize SLSQP Optimizer

Source: https://squlearn.github.io/_sources/user_guide/quantum_neural_networks.rst.txt

Instantiate the SLSQP optimizer wrapper from squlearn.optimizers. This optimizer can be configured with specific options, such as the maximum number of iterations.

```python
from squlearn.optimizers import SLSQP

...

slsqp = SLSQP(options={"maxiter": 100})
...
```

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

### Adjust Sampler Options Dynamically

Source: https://squlearn.github.io/_sources/user_guide/executor.rst.txt

Modify sampler options after initializing the Executor. This example enables and configures dynamical decoupling for the sampler primitive.

```python
from squlearn import Executor
from qiskit_ibm_runtime import QiskitRuntimeService

service = QiskitRuntimeService(channel="ibm_quantum_platform", token="INSERT_YOUR_TOKEN_HERE")

with Executor(service.backend("ibm_kingston")) as executor:
    executor.sampler_options.dynamical_decoupling.enable = True
    executor.sampler_options.dynamical_decoupling.sequence_type = "XpXm"
    # Your quantum computations here
    pass
```

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

### Get Executor-based Qiskit Primitives

Source: https://squlearn.github.io/modules/generated/squlearn.Executor.html

Retrieve the Estimator and Sampler primitives from an initialized Executor. These primitives route all operations through the Executor's backend.

```python
from squlearn import Executor

# Initialize the Executor
executor = Executor("statevector_simulator")

# Get the Executor based Estimator with all execusions routed through the Executor
estimator = executor.get_estimator()

# Get the Executor based Sampler with all execusions routed through the Executor
sampler = executor.get_sampler()

# Run a circuit with the Executor based Sampler
from qiskit.circuit.random import random_circuit
circuit = random_circuit(2, 2, seed=1, measure=True).decompose(reps=1)
job = sampler.run([(circuit,)])
result = job.result()
```