### Basic Hello World with QuantumString

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Examples/HelloWorld.rst

Initialize a QuantumString, assign the string 'hello world' to it, and print the result. This is a fundamental example for getting started with QRISP.

```python
from qrisp import QuantumString

q_str = QuantumString()
q_str[:] = "hello world"
print(q_str)
```

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

### QAOA M$\kappa$CS Example Implementation

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Algorithms/qaoa/implementations/maxKColorableSubgraph.rst

This snippet demonstrates the full setup and execution of QAOA for the Max-K-Colorable Subgraph problem. It includes graph creation, color definition, QAOA problem initialization, setting an initial state, running the QAOA algorithm, and processing the results.

```python
from qrisp import QuantumArray
from qrisp.qaoa import QAOAProblem, RX_mixer, apply_XY_mixer, QuantumColor, create_coloring_cl_cost_function, create_coloring_operator
import networkx as nx
import random
from operator import itemgetter

G = nx.Graph()
G.add_edges_from([[0,1],[0,4],[1,2],[1,3],[1,4],[2,3],[3,4]])
color_list = ["red", "blue", "orange", "green"]

qarg = QuantumArray(qtype = QuantumColor(color_list, one_hot_enc=True), shape = G.number_of_nodes())
#qarg = QuantumArray(qtype = QuantumColor(color_list, one_hot_enc=False), shape = G.number_of_nodes()) # use one_hot_enc=False for binary encoding

qaoa_coloring = QAOAProblem(cost_operator=create_coloring_operator(G),
                        mixer=apply_XY_mixer, # use RX_mixer for binary encoding
                        cl_cost_function=create_coloring_cl_cost_function(G))

init_state = [random.choice(color_list) for _ in range(len(G))]
qaoa_coloring.set_init_function(lambda x : x.encode(init_state))

result = qaoa_coloring.run(qarg, depth=3, max_iter=50)
```

```python
cl_cost =create_coloring_cl_cost_function(G)

print("5 most likely solutions")
max_five = sorted(result.items(), key=lambda item: item[1], reverse=True)[:5]
for res, prob in max_five:
    print(res, prob, cl_cost({res : 1}))
```

```python
best_of_five = min([(cl_cost({item[0]:1}),item[0]) for item in max_five], key=itemgetter(0))[1]
x.draw(G, with_labels=True, font_color='white', node_size=1000, font_size=22,
        node_color=[best_of_five[node] for node in G.nodes()],
        edge_color='#D3D3D3')
```

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

### Run Documentation Server with sphinx-autobuild

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/README.md

Navigate to the documentation folder and execute this command to start a local server for live previewing documentation changes. Ensure sphinx-autobuild is installed.

```bash
sphinx-autobuild source build/html --open-browser
```

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

### Initialize IQM Backend

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/tutorial.ipynb

Connect to an IQM device using the IQM interface. Ensure qrisp is installed with the IQM extra: pip install qrisp[iqm]. Replace 'IQM_RESONANCE_API_TOKEN' with your actual API token.

```python
# Make sure to install qrisp with the IQM interface: pip install qrisp[iqm]
# from qrisp.interface import IQMBackend
# iqm_garnet = IQMBackend(api_token = "IQM_RESONANCE_API_TOKEN",
#                           device_instance = "garnet")
# results = qch.get_measurement(backend = iqm_garnet)
# print(results)
```

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

### Install Documentation Dependencies and Build HTML

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/README.md

These commands are used to install the necessary Python packages and system dependencies for documentation, and then to generate the final HTML output for the documentation pages.

```bash
make docs-install
```

```bash
make html
```

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

### BackendServer.start

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Backend Interface/generated/qrisp.interface.qunicorn.BackendServer.start.rst

Starts the backend server. This method is part of the qrisp.interface.qunicorn.BackendServer class.

```APIDOC
## BackendServer.start

### Description
Starts the backend server. This method is part of the qrisp.interface.qunicorn.BackendServer class.

### Method Signature
`start()`
```

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

### Install Qrisp with IQM Dependencies

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/README.md

Install Qrisp with additional dependencies for working with IQM quantum computers.

```bash
pip install qrisp[iqm]
```

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

### Install Qrisp using pip

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/setup.rst

Use this command to install Qrisp. Ensure you have Python 3.11 or 3.12 installed.

```bash
pip install qrisp
```

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

### BackendServer.start

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Backend Interface/BackendServer.rst

Starts the BackendServer. This method is used to initiate the server's operations.

```APIDOC
## BackendServer.start

### Description
Starts the BackendServer.

### Method
This is a method call within the BackendServer class.

### Parameters
This method does not explicitly list any parameters in the documentation.

### Request Example
```python
backend_server.start()
```

### Response
This method does not explicitly document a return value or response structure.
```

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

### QPE Test Case Setup

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/tutorial.ipynb

Sets up a test case for the QPE function with a custom unitary U and a QuantumVariable psi. Requires numpy.

```python
from qrisp import p, QuantumVariable, multi_measurement
import numpy as np


def U(psi):
    phi_1 = 0.5
    phi_2 = 0.125

    p(phi_1 * 2 * np.pi, psi[0])
    p(phi_2 * 2 * np.pi, psi[1])


psi = QuantumVariable(2)

h(psi)

res = QPE(psi, U, 3)
```

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

### Full Code Example for Hamiltonian Simulation

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/H2.ipynb

A complete example demonstrating Hamiltonian simulation, including loading molecular data, initializing the quantum state, performing the simulation, and evaluating the number operator's expectation value.

```python
# Loading molecular data
from qrisp.operators import a, c, FermionicOperator
from pyscf import gto

mol = gto.M(atom="""H 0 0 0; H 0 0 0.74""", basis="sto-3g")
H_ferm = FermionicOperator.from_pyscf(mol)

# Initializing the quantum state and performing Hamiltonian simulation
from qrisp import QuantumVariable

U = H_ferm.trotterization()


def state_prep():
    electron_state = QuantumVariable(4)
    electron_state[:] = {"1100": 2**-0.5, "0001": 2**-0.5}

    U(electron_state, t=100, steps=20)

    return electron_state


# Evaluating the number operator
N = sum(c(i) * a(i) for i in range(4))
expectation_value = N.expectation_value(state_prep, precision=0.01)()
print(expectation_value)
```

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

### E3Lin2 QAOA Problem Setup and Run

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Algorithms/qaoa/implementations/eThrLinTwo.rst

Sets up the E3Lin2 problem with a specified number of variables and clauses, then runs the QAOA algorithm. Consider using 'init_type="tqa"' for potential performance improvements.

```python
from qrisp import QuantumVariable
from qrisp.qaoa import QAOAProblem, RX_mixer, create_e3lin2_cl_cost_function, create_e3lin2_cost_operator

clauses = [[0,1,2,1],[1,2,3,0],[0,1,4,0],[0,2,4,1],[2,4,5,1],[1,3,5,1],[2,3,4,0]]
num_variables = 6
qarg = QuantumVariable(num_variables)

qaoa_e3lin2 = QAOAProblem(cost_operator=create_e3lin2_cost_operator(clauses),
                           mixer=RX_mixer,
                           cl_cost_function=create_e3lin2_cl_cost_function(clauses))
results = qaoa_e3lin2.run(qarg, depth=5)
```

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

### Example QAOA MinSetCover Implementation

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Algorithms/qaoa/implementations/minSetCover.rst

This snippet demonstrates how to set up and run the QAOA for the Minimum Set Cover problem. It defines the sets, universe, and initializes the QAOAProblem with specific functions for mixer, cost, and initial state.

```python
from qrisp import QuantumVariable
from qrisp.qaoa import QAOAProblem, RZ_mixer, create_min_set_cover_mixer, create_min_set_cover_cl_cost_function, min_set_cover_init_function

sets = [{0,1,2,3},{1,5,6,4},{0,2,6,3,4,5},{3,4,0,1},{1,2,3,0},{1}]
universe = set.union(*sets)
qarg = QuantumVariable(len(sets))

qaoa_min_set_cover = QAOAProblem(cost_operator=RZ_mixer, 
                                mixer= create_min_set_cover_mixer(sets, universe), 
                                cl_cost_function=create_min_set_cover_cl_cost_function(sets, universe),
                                init_function=min_set_cover_init_function)
results = qaoa_min_set_cover.run(qarg, depth=5)
```

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

### Use MQTSim with QuantumFloat

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Backend Interface/DockerSimulators.rst

Example demonstrating the use of `MQTSim` backend with `QuantumFloat` for quantum computations. Ensure to provide the `shots` argument in `get_measurement`.

```python
from qrisp import QuantumFloat
a = QuantumFloat(3)
a[:] = 3
b = QuantumFloat(3)
b[:] = 4
c = a*b
from qrisp.interface import MQTSim
c.get_measurement(backend = MQTSim(), shots = 1000)
```

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

### BackendServer

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Backend Interface/index.rst

This class is a wrapper for convenient setup of a server capable of processing quantum circuits. Providers need to create a 'run_func' that accepts a QuantumCircuit, shots, and an optional token, returning measurement results.

```APIDOC
## BackendServer

### Description
This class wraps the setup for a server that can process quantum circuits. It requires a Python function `run_func` which takes a :ref:`QuantumCircuit`, an integer `shots`, and an optional string `token`. The function should return measurement results as a dictionary of bitstrings.

### Usage
```python
from qrisp.interface import BackendServer
from backend_provider import run_func, ping_func

server = BackendServer( run_func, port = 8080)
server.start()
```
```

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

### Example: Quantum-Quantum Comparison Evaluation

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/Sudoku.ipynb

Demonstrates the usage of `eval_qq_checks` with a QuantumArray and a list of comparison pairs. Prints the resulting QuantumBools indicating equality.

```python
q_assigments = QuantumArray(qtype=QuantumFloat(2), shape=(3,))

q_assigments[:] = [3, 2, 3]

comparison_bools = eval_qq_checks([(0, 1), (0, 2), (1, 2)], q_assigments)

for qbl in comparison_bools:
    print(qbl)

# Yields
# {False: 1.0}
# {True: 1.0}
# {False: 1.0}
```

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

### Connect to a Backend Server

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Backend Interface/index.rst

Connect to a BackendServer using BackendClient to send requests for processing quantum circuits. This example shows connecting to a server and then querying a remote backend with a quantum algorithm.

```python
from qrisp.interface import BackendClient
backend = BackendClient(server_ip, port)

from some_library import some_quantum_algorithm

qv = some_quantum_algorithm()

res = qv.get_measurement(backend = backend)
```

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

### Start a Backend Server

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Backend Interface/index.rst

Use BackendServer to wrap a Python function for processing quantum circuits. The function should accept a QuantumCircuit, shots, and an optional token, returning measurement results.

```python
from qrisp.interface import BackendServer
from backend_provider import run_func, ping_func

server = BackendServer( run_func, port = 8080)
                   
server.start()
```

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

### Classical Matrix Arithmetic Example

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/index.rst

Demonstrates how classical matrix operations can be translated to QRISp's Block Encoding syntax for operator manipulation.

```python
# Classical: C = I + A - 2*A^2 + B^(-1)
C = np.eye(4) + A - 2 * A @ A + np.linalg.inv(B)
```

```python
# Qrisp:
B_C = B_A.poly([1., 1., -2.]) + B_B.inv(epsilon, kappa)
```

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

### Example Usage: Compute Average Cut

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/JaspQAOAtutorial.ipynb

Demonstrates how to create a sample graph, define the post-processor, and compute the average cut for a given set of sample bipartitions.

```python
# Create a sample graph
G = nx.Graph()
G.add_edges_from([(0, 1), (1, 2), (2, 0), (1, 3)])

# Create the post processor function
post_processor = create_sample_array_post_processor(G)

# Sample input array representing different cuts
sample_array = jnp.array(
    [0b0001, 0b0010, 0b0100, 0b1000]
)  # Example binary representations

# Compute the average cut
average_cut = post_processor(sample_array)
print("Average Cut:", average_cut)
```

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

### Run QAOA for QUBO and Post-process

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/QAOAtutorial/QUBO.ipynb

Example of running the QAOA algorithm for a QUBO problem and performing classical post-processing. This includes defining the QUBO matrix, quantum arguments, running the instance, and sorting solutions by cost.

```python
from qrisp.default_backend import def_backend
from qrisp import QuantumVariable, QuantumArray
from operator import itemgetter

Q = np.array(
    [
        [-17, 20, 20, 20, 0, 40],
        [0, -18, 20, 20, 20, 40],
        [0, 0, -29, 20, 40, 40],
        [0, 0, 0, -19, 20, 20],
        [0, 0, 0, 0, -17, 20],
        [0, 0, 0, 0, 0, -28],
    ]
)

qarg = QuantumArray(qtype=QuantumVariable(1), shape=len(Q))

QUBO_instance = QUBO_problem(Q)

depth = 1
res = QUBO_instance.run(qarg, depth, mes_kwargs={"backend": def_backend}, max_iter=50)

costs_and_solutions = [(QUBO_obj(bitstring, Q), bitstring) for bitstring in res.keys()]

sorted_costs_and_solutions = sorted(costs_and_solutions, key=itemgetter(0))

for i in range(5):
    print(
```

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

### Initialize QuantumBacktrackingTree and Plotting Setup

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/Sudoku.ipynb

Initializes a `QuantumBacktrackingTree` with a specified maximum depth and branch variable type, using the previously defined `accept` and `reject` functions. It also sets up a matplotlib figure for plotting.

```python
tree = QuantumBacktrackingTree(
    max_depth=3,
    branch_qv=QuantumFloat(1), # Use 1 qubit variable for a binary tree.
    accept=accept,
    reject=reject
)
tree.init_node([]) # Initialize tree in the |root> = |[0,0,0]>|3> state.

# Create a figure with 2 rows and 2 columns.
fig, axs = plt.subplots(2, 2, figsize=(12, 8))
axs = axs.flatten()  # Flatten to iterate easily: 0, 1, 2, 3.
```

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

### Example QUBO Matrix (Symmetric)

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/QAOAtutorial/QUBO.ipynb

A sample symmetric QUBO matrix for the set partitioning problem used in the tutorial. This matrix is used to demonstrate QUBO to QAOA conversion.

```python
Q_sym = np.array(
    [
        [-17, 10, 10, 10, 0, 20],
        [10, -18, 10, 10, 10, 20],
        [10, 10, -29, 10, 20, 20],
        [10, 10, 10, -19, 10, 10],
        [0, 10, 20, 10, -17, 10],
        [20, 20, 20, 10, 10, -28],
    ]
)
```

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

### Run Qrisp Simulator Docker Container

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Backend Interface/DockerSimulators.rst

Start the Qrisp simulator collection Docker container, exposing necessary ports for Qrisp to communicate with the simulators. Use `arm-version` if on an ARM-based CPU.

```console
docker run -p 8083:8083 -p 8084:8084 -p 8085:8085 -p 8086:8086 -p 8087:8087 -p 8088:8088 -p 8089:8089 -p 8090:8090 -p 8091:8091 -p 8092:8092 qrisp/qrisp_sim_collection:x86-version
```

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

### QIRO MaxClique Example Implementation

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Algorithms/qiro/implementations/QIROMaxClique.rst

This snippet shows how to initialize and run the QIRO MaxClique algorithm with a defined graph and quantum variable. It sets up the QIRO problem with specific routines and parameters, then executes the QIRO instance.

```python
from qrisp import QuantumVariable
from qrisp.qiro import QIROProblem, create_max_clique_replacement_routine, create_max_clique_cost_operator_reduced, qiro_rx_mixer, qiro_init_function
from qrisp.qaoa import max_clique_problem, create_max_clique_cl_cost_function
import matplotlib.pyplot as plt
import networkx as nx

# Define a random graph via the number of nodes and the QuantumVariable arguments
num_nodes = 15
G = nx.erdos_renyi_graph(num_nodes, 0.7, seed=99)
qarg = QuantumVariable(G.number_of_nodes())

# QIRO
qiro_instance = QIROProblem(problem=G,
                                replacement_routine= reate_max_clique_replacement_routine,
                                cost_operator=create_max_clique_cost_operator_reduced,
                                mixer=qiro_rx_mixer,
                                cl_cost_function=create_max_clique_cl_cost_function,
                                init_function=qiro_init_function
                                )
res_qiro = qiro_instance.run_qiro(qarg=qarg, depth=3, n_recursions=2)

# The final graph that has been adjusted
final_graph = qiro_instance.problem
```

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

### Define QUBO Matrix and Problem Size

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/CD.ipynb

Initializes the QUBO matrix 'Q' and determines the number of qubits 'N' required for the problem. This setup is crucial for defining the optimization problem.

```python
import numpy as np

Q = np.array([[-1.1, 0.6, 0.4, 0.0, 0.0, 0.0],
              [0.6, -0.9,  0.5, 0.0, 0.0, 0.0],
              [0.4, 0.5, -1.0, -0.6, 0.0, 0.0],
              [0.0, 0.0, -0.6, -0.5, 0.6, 0.0],
              [0.0, 0.0, 0.0, 0.6, -0.3, 0.5],
              [0.0, 0.0, 0.0, 0.0, 0.5, -0.4]])

N = Q.shape[0]
```

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

### Quantum Modulo Division Example

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Examples/QuantumModDivision.rst

Use this snippet to perform quantum modulo division. It requires importing QuantumFloat, q_divmod, and multi_measurement from qrisp. Ensure numbers are defined and encoded before division.

```python
from qrisp import QuantumFloat, q_divmod, multi_measurement
numerator = QuantumFloat(5)
numerator[:] = 13
divisor = QuantumFloat(5)
divisor[:] = 4
quotient, remainder = q_divmod(numerator, divisor)
print(multi_measurement([quotient, remainder]))
```

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

### QAOA MaxClique Problem Setup and Execution

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Algorithms/qaoa/implementations/maxClique.rst

This snippet demonstrates how to set up and run a QAOA problem for finding the maximum clique. It requires importing necessary components from `qrisp` and `networkx`. The MaxClique problem is solved by applying QAOA to the complement graph's Max Independent Set formulation. Ensure `networkx` is installed for graph operations.

```python
from qrisp import QuantumVariable
from qrisp.qaoa import QAOAProblem, RZ_mixer, create_max_indep_set_cl_cost_function, create_max_indep_set_mixer, max_indep_set_init_function
import networkx as nx

G = nx.erdos_renyi_graph(9, 0.7, seed = 133)
G_complement = nx.complement(G)
qarg = QuantumVariable(G.number_of_nodes())

qaoa_max_clique = QAOAProblem(cost_operator=RZ_mixer, 
                                mixer=create_max_indep_set_mixer(G_complement),
                                cl_cost_function=create_max_indep_set_cl_cost_function(G_complement), 
                                init_function=max_indep_set_init_function)
results = qaoa_max_clique.run(qarg, depth=5)
```

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

### Benchmark QAOA with TQA Warm-Start Initialization

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/QAOAtutorial/MaxCut.ipynb

Utilize the `benchmark` method with `init_type='tqa'` to leverage TQA warm-start initialization for QAOA performance evaluation. This method also requires the `optimal_solution` for approximation ratio calculation and provides detailed performance metrics.

```python
print("TQA")

benchmark_data = maxcut_instance.benchmark(
    qarg=QuantumVariable(len(G)),
    depth_range=[1, 2, 3, 4, 5],
    shot_range=[1000, 10000],
    iter_range=[100, 200],
    optimal_solution="00011",
    repetitions=1,
    init_type="tqa",
)

temp = benchmark_data.rank(print_res=True)

_, tqaFO = benchmark_data.evaluate()

print("Approximation ratio: ", sum(tqaFO) / len(tqaFO))
print("Variance: ", np.var(tqaFO))
```

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

### Compare Block Encoding Resource Estimates

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/BE_tutorial/BE_vol1.ipynb

This example shows two different resource estimates for block encodings, likely representing different construction methods. It is used to illustrate the efficiency differences between methods.

```python
{'gate counts': {'u3': 6, 's': 6, 'p': 2, 't': 18, 'h': 22, 't_dg': 20, 'cx': 72, 'x': 5, 'gphase': 2, 'measure': 6}, 'depth': 60, 'qubits': 11}
```

```python
{'gate counts': {'x': 15, 'p': 7, 't': 48, 'cy': 14, 'gphase': 2, 't_dg': 64, 'u3': 30, 'cx': 156, 'h': 32}, 'depth': 220, 'qubits': 12}
```

```python
{'gate counts': {'x': 5, 's': 6, 'p': 2, 't': 18, 'gphase': 2, 't_dg': 20, 'u3': 6, 'cx': 72, 'h': 22, 'measure': 6}, 'depth': 60, 'qubits': 11}
```

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

### Install Catalyst

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/Jasp.ipynb

Install the pennylane-catalyst package if it's not already available. This is typically needed for compiling Jasp to QIR.

```python
try:
    import catalyst
except:
    !pip install pennylane-catalyst
```

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

### Instantiate and Apply the Oracle

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/tutorial.ipynb

Create a QuantumFloat, apply the `sqrt_oracle` to it, and then print the resulting quantum circuit representation using the `.qs` attribute.

```python
qf = QuantumFloat(3, -1, signed=True)
sqrt_oracle(qf)
print(qf.qs)
```

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

### Initialize and Run QAOA for Portfolio Rebalancing

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/QAOAtutorial/PortfolioRebalancing.ipynb

Initializes the QAOAProblem and QuantumArray instances, then runs the optimization process with a specified depth.

```python
# assign QuantumArray to operate on
from qrisp import QuantumVariable, QuantumArray

qv = QuantumVariable(n_assets)
q_array = QuantumArray(qtype=qv, shape=(2))

# run the problem!
theproblem = QAOAProblem(
    cost_operator=cost_op, mixer=mixer_op, cl_cost_function=cl_cost
)
theproblem.set_init_function(init_fun)
theNiceQAOA = theproblem.run(q_array, depth=3)

```

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

### Generate Python Bindings for JASP MLIR Dialect

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/src/qrisp/jasp/mlir/dialect_definition/README.md

Command to generate Python bindings for JASP MLIR dialect operations using `mlir-tblgen`. Ensure MLIR is installed and `mlir-tblgen` is in your PATH. Replace the MLIR include path with your specific installation.

```bash
mlir-tblgen JaspPythonOps.td \
    -gen-python-op-bindings \
    -bind-dialect=jasp \
    -I /path/to/llvm-project/mlir/include \
    -I . \
    -o ../dialect_implementation/_jasp_ops_gen.py
```

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

### QAOA MaxIndepSet Problem Setup and Execution

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Algorithms/qaoa/implementations/maxIndepSet.rst

Sets up and runs the QAOA algorithm for the Maximum Independent Set problem. Requires importing necessary components from qrisp and networkx. The results are then processed to display the most likely solutions and their costs.

```python
from qrisp import QuantumVariable
from qrisp.qaoa import QAOAProblem, RZ_mixer, create_max_indep_set_cl_cost_function, create_max_indep_set_mixer, max_indep_set_init_function
import networkx as nx

G = nx.erdos_renyi_graph(9, 0.5, seed = 133)
qarg = QuantumVariable(G.number_of_nodes())

qaoa_max_indep_set = QAOAProblem(cost_operator=RZ_mixer, 
                                    mixer=create_max_indep_set_mixer(G), 
                                    cl_cost_function=create_max_indep_set_cl_cost_function(G), 
                                    init_function=max_indep_set_init_function)
results = qaoa_max_indep_set.run(qarg, depth=5)
```

```python
cl_cost = create_max_indep_set_cl_cost_function(G)

print("5 most likely solutions")
max_five = sorted(results.items(), key=lambda item: item[1], reverse=True)[:5]
for res, prob in max_five:
    print([index for index, value in enumerate(res) if value == '1'], prob, cl_cost({res : 1}))
```

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

### QuantumVariable Measurement

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Core/QuantumVariable.rst

Gets the measurement outcome of the QuantumVariable.

```APIDOC
## QuantumVariable.get_measurement

### Description
Retrieves the measurement outcome of the QuantumVariable.

### Method
get_measurement
```

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

### MaxSetPacking QAOA Implementation

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Algorithms/qaoa/implementations/maxSetPack.rst

This snippet demonstrates how to set up and run the QAOA for the MaxSetPacking problem. It involves creating the constraint graph, initializing the QAOA problem with appropriate cost and mixer functions, and running the algorithm.

```python
from qrisp import QuantumVariable
from qrisp.qaoa import QAOAProblem, RZ_mixer, create_max_indep_set_cl_cost_function, create_max_indep_set_mixer, max_indep_set_init_function
import networkx as nx
from itertools import combinations

sets = [{0,7,1},{6,5},{2,3},{5,4},{8,7,0},{1}]

def non_empty_intersection(sets):
    return [(i, j) for (i, s1), (j, s2) in combinations(enumerate(sets), 2) if s1.intersection(s2)]

# create constraint graph
G = nx.Graph()
G.add_nodes_from(range(len(sets)))
G.add_edges_from(non_empty_intersection(sets))
qarg = QuantumVariable(G.number_of_nodes())

qaoa_max_indep_set = QAOAProblem(cost_operator=RZ_mixer,
                                mixer=create_max_indep_set_mixer(G),
                                cl_cost_function=create_max_indep_set_cl_cost_function(G),
                                init_function=max_indep_set_init_function)
results = qaoa_max_indep_set.run(qarg, depth=5)
```

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

### Import Matplotlib

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/BigInteger.ipynb

Imports the matplotlib library for plotting. This is a common setup for data visualization in Python.

```python
import matplotlib.pyplot as plt
```

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

### BackendClient.__init__

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Backend Interface/BackendClient.rst

Initializes the BackendClient.

```APIDOC
## BackendClient.__init__

### Description
Initializes the BackendClient.

### Method
__init__

### Parameters
This method does not take any explicit parameters beyond `self`.
```

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

### Example Usage of eval_cq_checks

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/Sudoku.ipynb

Demonstrates how to use the eval_cq_checks function with sample quantum assignments and comparison data.

```python
q_assigments = QuantumArray(qtype=QuantumFloat(2), shape=(3,))
q_assigments[:] = np.arange(3)

res_qbls = eval_cq_checks({0: [1, 2, 3], 1: [1, 2, 3], 2: [1, 2, 3]}, q_assigments)

for qbl in res_qbls:
    print(qbl)
```

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

### Instantiate and Run QAOAProblem for Max Cut

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/QAOAtutorial/MaxCut.ipynb

Instantiates the QAOAProblem class with the Max Cut cost operator and mixer, then runs the simulation. Requires importing QAOAProblem and RX_mixer.

```python
from qrisp.qaoa import QAOAProblem, RX_mixer

maxcut_instance = QAOAProblem(apply_cost_operator, RX_mixer, maxcut_cost_funct)

res = maxcut_instance.run(qarg, depth, max_iter=50)  # runs the simulation
```

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

### Get Number of Qubits

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Core/Uncomputation.rst

Retrieves the total number of qubits currently allocated in the compiled quantum circuit.

```python
compiled_qc.num_qubits()
6
```

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

### Compile QuantumSession and Get Depth

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/TSP.ipynb

Compile a QuantumSession to a QuantumCircuit and retrieve its depth. This is used for performance evaluation.

```python
qpe_compiled_qc = perm_specifiers[0].qs.compile()
qpe_compiled_qc.depth()
```

```text
Result:
1989
```

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

### Run QAOA Optimization and Visualize Solution

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/QAOAtutorial/MaxCut.ipynb

Initializes QAOA parameters, uses COBYLA optimizer to find optimal angles, retrieves the best solution, and visualizes the graph coloring based on the best cut.

```python
import numpy as np
from scipy.optimize import minimize
from operator import itemgetter

init_point = np.pi * np.random.rand(2 * p)

res_sample = minimize(
    quantum_objective, init_point, method="COBYLA", options={"maxiter": 50}
)

optimal_theta = res_sample["x"]
qv_p = QuantumVariable(N)
qv = apply_p_layers(qv_p, optimal_theta[:p], optimal_theta[p:])
results = qv_p.get_measurement()

best_cut, best_solution = min(
    [(maxcut_obj(x), x) for x in results.keys()], key=itemgetter(0)
)
print(f"Best string: {best_solution} with cut: {-best_cut}")

colors = ["r" if best_solution[node] == "0" else "b" for node in G]
x.draw(
    G,
    with_labels=True,
    font_color="white",
    node_size=1000,
    font_size=22,
    node_color=colors,
    pos=nx.bipartite_layout(G, [node for node in G if best_solution[node] == "0"]),
)
```

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

### QuantumBacktrackingTree.init_phi

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Algorithms/generated/qrisp.quantum_backtracking.QuantumBacktrackingTree.init_phi.rst

Initializes the phi parameter for the QuantumBacktrackingTree. This method is part of the internal setup for the quantum backtracking algorithm.

```APIDOC
## QuantumBacktrackingTree.init_phi

### Description
Initializes the phi parameter for the QuantumBacktrackingTree. This method is part of the internal setup for the quantum backtracking algorithm.

### Method
`init_phi()`

### Parameters
This method does not accept any parameters.
```

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

### QuantumVariable.get_measurement

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Core/generated/qrisp.QuantumVariable.get_measurement.rst

Retrieves the measurement of the QuantumVariable. This method is used to get the classical outcome of a quantum measurement performed on the variable.

```APIDOC
## QuantumVariable.get_measurement

### Description
Retrieves the measurement of the QuantumVariable. This method is used to get the classical outcome of a quantum measurement performed on the variable.

### Method
`get_measurement()`

### Parameters
This method does not take any parameters.

### Returns
- The classical outcome of the measurement (e.g., 0 or 1 for a qubit).
```

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

### Initialize and Operate on QuantumVariable

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/index.rst

Demonstrates the basic usage of QuantumVariable, including initialization, applying gates, and printing the variable's state. Ensure QuantumVariable, h, and cx are imported.

```python
from qrisp import QuantumVariable, h, cx

qv = QuantumVariable(3)
h(qv[0])
cx(qv[0], qv[1])
cx(qv[1], qv[2])
print(qv)
# Outcome: {'000': 0.5, '111': 0.5}
```

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

### Import necessary libraries

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/QAOAtutorial/PortfolioRebalancing.ipynb

Imports libraries required for the example implementation, including scikit-learn for preprocessing and numpy for numerical operations.

```python
# imports
import sklearn
from sklearn import preprocessing  # Add explicit import for preprocessing
import numpy as np
```

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

### QuantumSession Management and Backend Integration

Source: https://context7.com/eclipse-qrisp/qrisp/llms.txt

Shows how to set up a QuantumSession with a backend, attach variables, perform operations, and observe automatic session merging.

```python
from qrisp import QuantumSession, QuantumFloat, merge
from qrisp.interface import QiskitBackend
from qiskit_aer import AerSimulator

# Create a session targeting the Aer simulator
backend = QiskitBackend(AerSimulator())
qs = QuantumSession(backend=backend)

# Attach variables to a specific session
qf = QuantumFloat(4, qs=qs)
qf[:] = 5
res = qf * qf

# Evaluate on the bound backend
print(res.get_measurement())
# {25: 1.0}

# Sessions from different variables auto-merge on entanglement
qa = QuantumFloat(3)
qb = QuantumFloat(3)
print(qa.qs == qb.qs)   # False
from qrisp import cx
cx(qa[0], qb[0])
print(qa.qs == qb.qs)   # True  — sessions merged automatically

# Compile to an optimised QuantumCircuit
qc = qa.qs.compile()
print(f"Qubits: {qc.num_qubits()}, Depth: {qc.depth()}")
```

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

### Get QuantumSession Statevector

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/tutorial.ipynb

Retrieve the statevector of the quantum circuit managed by the QuantumSession. This provides a detailed view of the quantum state.

```python
qch.qs.statevector()
```

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

### Example Usage of check_sudoku_assignments (Invalid Case)

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/Sudoku.ipynb

Demonstrates checking a Sudoku board with invalid quantum assignments, expecting a False result.

```python
# Another check

q_assignments = QuantumArray(qtype=QuantumFloat(2), shape=(4,))
q_assignments[:] = [1, 2, 1, 0]

sudoku_check = check_sudoku_assignments(sudoku_board, q_assignments)
print(sudoku_check)
# Yields {False: 1.0}
```

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

### QAOA MaxCut Problem Setup and Execution

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Algorithms/qaoa/implementations/maxCut.rst

Sets up and runs the QAOA algorithm for the MaxCut problem. Requires `networkx` for graph creation and `qrisp` for quantum components. The `depth` and `max_iter` parameters control the QAOA circuit depth and classical optimization iterations.

```python
from qrisp import QuantumVariable
from qrisp.qaoa import QAOAProblem, RX_mixer, create_maxcut_cl_cost_function, create_maxcut_cost_operator
import networkx as nx

G = nx.erdos_renyi_graph(6, 0.7, seed = 133)
qarg = QuantumVariable(G.number_of_nodes())

qaoa_maxcut = QAOAProblem(cost_operator=create_maxcut_cost_operator(G),
                        mixer=RX_mixer, 
                        cl_cost_function=create_maxcut_cl_cost_function(G))
results = qaoa_maxcut.run(qarg, depth=5, max_iter=50)
```

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

### Example Usage of check_sudoku_assignments (Valid Case)

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/Sudoku.ipynb

Demonstrates checking a Sudoku board with valid quantum assignments, expecting a True result.

```python
q_assignments = QuantumArray(qtype=QuantumFloat(2), shape=(4,))
q_assignments[:] = [1, 1, 1, 2]

sudoku_check = check_sudoku_assignments(sudoku_board, q_assignments)
print(sudoku_check)
# Yields {True: 1.0}
```

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

### Get Number of Qubits in Compiled Circuit

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/TSP.ipynb

Determine the number of qubits utilized in a compiled QuantumCircuit. This is a fundamental measure of resource usage.

```python
qdict_compiled_qc.num_qubits()
```

```text
Result:
18
```

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

### Run COLD Algorithm and Print Results

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/CD.ipynb

Executes the COLD algorithm using the configured DCQOProblem instance. It involves setting up quantum variables, simulation parameters, and optimization bounds. The results, including probabilities and costs of different states, are then printed.

```python
from qrisp import QuantumVariable

qarg = QuantumVariable(N)
N_steps = 4
T = 8
method = "COLD"
N_opt = 1
bounds = (-3, 3)

result = cold_problem.run(qarg, N_steps, T, method, N_opt, bounds=bounds)

print(result)
```

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

### Set Up Hamiltonians for COLD Algorithm

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/CD.ipynb

Defines the initial, problem, and control Hamiltonians required for the COLD algorithm. These Hamiltonians are constructed using Qrisp's operator library.

```python
from qrisp.operators.qubit import X, Y, Z

h = -0.5 * np.diag(Q) - 0.5 * np.sum(Q, axis=1)
J = 0.5 * Q

H_init = 1 * sum([X(i) for i in range(N)])

H_prob = (sum([sum([J[i][j]*Z(i)*Z(j) for j in range(i)]) for i in range(N)]) 
          + sum([h[i]*Z(i) for i in range(N)]))

H_control = sum([Z(i) for i in range(N)])
```

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

### Grover's Algorithm Implementation

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/general/tutorial/tutorial.ipynb

Embeds a constructed oracle into Grover's algorithm. Requires prior setup of QuantumFloat and superposition.

```python
from math import pi
from qrisp.grover import diffuser

qf = QuantumFloat(3, -1, signed=True)

n = qf.size
iterations = int(0.25 * pi * (2**n / 2) ** 0.5)

h(qf)

for i in range(iterations):
    sqrt_oracle(qf)
    diffuser(qf)

print(qf)
```

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

### MaxSat QAOA Problem Setup and Execution

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Algorithms/qaoa/implementations/maxSat.rst

Sets up a MaxSat problem with specified clauses and variables, then initializes and runs a QAOAProblem object. This snippet requires the qrisp library and its QAOA components. The `run` method returns a dictionary of solutions and their probabilities.

```python
from qrisp import QuantumVariable
from qrisp.qaoa import QAOAProblem, RX_mixer, create_maxsat_cl_cost_function, create_maxsat_cost_operator

clauses = [[1,2,-3],[1,4,-6],[4,5,6],[1,3,-4],[2,4,5],[1,3,5],[-2,-3,6]]
num_vars = 6
problem = (num_vars, clauses)
qarg = QuantumVariable(num_vars)

qaoa_max_indep_set = QAOAProblem(cost_operator=create_maxsat_cost_operator(problem),
                                 mixer=RX_mixer,
                                 cl_cost_function=create_maxsat_cl_cost_function(problem))
results = qaoa_max_indep_set.run(qarg, depth=5)
```

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

### QIRO Max Independent Set Example Implementation

Source: https://github.com/eclipse-qrisp/qrisp/blob/main/documentation/source/reference/Algorithms/qiro/implementations/QIROMaxIndep.rst

This code demonstrates how to set up and run the QIRO algorithm for the maximum independent set problem using qRISP. It defines a graph, initializes the QIROProblem with specific routines, and runs the algorithm. It also prints the top solutions and compares them with a NetworkX solution.

```python
from qrisp import QuantumVariable
from qrisp.qiro import QIROProblem, create_max_indep_replacement_routine, create_max_indep_cost_operator_reduced, qiro_rx_mixer, qiro_init_function
from qrisp.qaoa import create_max_indep_set_cl_cost_function
import matplotlib.pyplot as plt
import networkx as nx

# Define a random graph via the number of nodes and the QuantumVariable arguments
num_nodes = 13
G = nx.erdos_renyi_graph(num_nodes, 0.4, seed = 107)
qarg = QuantumVariable(G.number_of_nodes())

qiro_instance = QIROProblem(G,
                                replacement_routine=create_max_indep_replacement_routine,
                                cost_operator=create_max_indep_cost_operator_reduced,
                                mixer=qiro_rx_mixer,
                                cl_cost_function=create_max_indep_set_cl_cost_function,
                                init_function=qiro_init_function
                                )
res_qiro = qiro_instance.run_qiro(qarg=qarg, depth=3, n_recursions=2)
```

```python
cl_cost = create_max_indep_set_cl_cost_function(G)

print("5 most likely QIRO solutions")
max_five_qiro = sorted(res_qiro, key=res_qiro.get, reverse=True)[:5]
for res in max_five_qiro: 
    print([index for index, value in enumerate(res) if value == '1'])
    print(cl_cost({res : 1}))

print("Networkx solution")
print(nx.approximation.maximum_independent_set(G))
```

```python
# The final graph that has been adjusted
final_graph = qiro_instance.problem

# Draw the final graph and the original graph for comparison
plt.figure(1)
nx.draw(final_graph, with_labels=True, node_color='#ADD8E6', edge_color='#D3D3D3')
plt.title('Final QIRO graph')

plt.figure(2)
most_likely = [index for index, value in enumerate(max_five_qiro[0]) if value == '1']
nx.draw(G, with_labels=True, 
        node_color=['#FFCCCB' if node in most_likely else '#ADD8E6' for node in G.nodes()],
        edge_color='#D3D3D3')
plt.title('Original graph with most likely QIRO solution')
plt.show()
```