# QAOA Implementation for various problem instances#

The following problem instances, gathered in the table below with their correspondinghave already been successfully implemented using the Qrisp framework:

PROBLEM INSTANCE

MIXER TYPE

IMPLEMENTED IN QRISP

MaxCut

X mixer

Max-$\ell$-SAT

X mixer

MaxIndependentSet

Controlled X mixer

MaxClique

Controlled X mixer

MaxSetPacking

Controlled X mixer

MinSetCover

Controlled X mixer

Max-$\kappa$-Colorable Subgraph

XY mixer

Our voyage into MaxCut and Max-$\kappa$-Colorable Subgraph problems is detailed in our tutorial section. We decided to build the QAOA module with a focus on modularity, ensuring it can adapt to various problem instances while maintaining independence from the choice of decoding. This design approach makes it straightfowrads to formulate and solve other problem instances taking the steps as we did in the tutorial.

Our team’s experience validates the adage “practice makes perfect”, demonstrating that familiarity with the process leads to more efficient and effective solutions.

Some of the problem instances mentioned in the table above are condensed and presented below.

## MaxCut#

The MaxCut problem instance and its QAOA implementation is heavily discussed in the MaxCut tutorial. All the necessary ingredients and required steps to run QAOA are elaborate in easy to grasp manner.

Here we instead provide the condensed implementation of the algorithm for MaxCut with using all of the predefined functions for this specific instance.

### Problem Definition#

The MaxCut problem is defined as follows:

Given a graph $$G=(V,E)$$, find a subset $$S\subset V$$ such that the number of edgest between $$S$$ and $$V\text{\\} S$$ is the largest.

First we import the necessary functions and packages, create a graph G we will be cutting, define a quantum argument qarg we’ll be acting on, as well as specify the depth of our algorithm.

from qrisp.qaoa import QAOAProblem, maxcut_obj,create_maxcut_cl_cost_function,create_maxcut_cost_operator, RX_mixer
from qrisp import QuantumArray, QuantumVariable
import networkx as nx
from operator import itemgetter

G = nx.Graph()

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

depth = 5


### QAOA instanciation#

Next we follow the recipe to run the algorithm with QAOAProblem, feeding it the cost_operator, a mixer and a cl_cost_function.

import time
maxcut_instance = QAOAProblem(create_maxcut_cost_operator(G), RX_mixer, create_maxcut_cl_cost_function(G))

start_time = time.time()
res = maxcut_instance.run(qarg, depth, max_iter = 50)
print(time.time()-start_time)


### Result analysis#

After running our QAOA on the MaxCut problem instance we can now obtain the QAOA solution and draw the graph with optimally colored nodes.

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

res_str = list(res.keys())[0]
print("QAOA solution: ", res_str)
best_cut, best_solution = (maxcut_obj(res_str,G),res_str)

colors = ['r' if best_solution[node] == '0' else 'b' for node in G]
nx.draw(G,node_color = colors, pos=nx.bipartite_layout(G, [0,1,2]))


## MaxIndependentSet#

In the following example we will demonstrate how to solve the maxIndependentSet problem instance with the QAOA module.

The problem is structured as follows:

Given a graph $$G=(V,E)$$ maximize the size of a clique, i.e. a subset $$V' \subset V$$ of mutually non-adjacent vertices.

The problem shows structural similarities to the MaxClique problem instance and may be implemented in analogy. We will not stick to mathematical assignment of variable names.

Imports:

from qrisp.qaoa import QAOAProblem
from qrisp.qaoa import create_rdm_graph
from qrisp.qaoa import maxIndepSetCostOp, maxIndepSetclCostfct,  init_state
from qrisp.qaoa import RX_mixer
from qrisp import QuantumVariable
import networkx as nx
import matplotlib.pyplot as plt


### Problem Definition#

We begin by specifiying the graph considered for the problem, using the create_rdm_graph-function.

Additionally, we define the QuantumVariable to operate on.

giraf = create_rdm_graph(9,0.2, seed = 127)
nx.draw(giraf,with_labels = True) #draw graph
plt.show()
qarg = QuantumVariable(giraf.number_of_nodes())


### QAOA instanciation#

Next we instanciate the QAOAProblem, handing over a cost_operator, a mixer and a cl_cost_function. We then set the the init_function and run the instance.

cost_operator-generator and cl_cost_function-generator have to be called with the problem graph giraf.

The problem operator is based on the pennylane unconstrained maxClique QAOA (TODO: link) implementation, which defines the operator as follows: $$H_C = 3 \sum _{(i,j) \in E(G)} Z_i Z_j - Z_i - Z_j + \sum _{i \in V(G)} Z_i$$

where $$V(G)$$ is is the set of vertices of the input graph $$G$$, $$E(G)$$ is the set of edges of $$G$$, and $$Z_i$$ is the Pauli-$$Z$$ operator applied to the $$i$$-th vertex.

The mixer operator is a basic X mixer applied to all qubits.

QAOAinstance = QAOAProblem(cost_operator = maxIndepSetCostOp(giraf), mixer = RX_mixer, cl_cost_function = maxIndepSetclCostfct(giraf))
QAOAinstance.set_init_function(init_function = init_state)
theNiceQAOA = QAOAinstance.run(qarg = qarg, depth = 5)


### Result analysis#

Define the classical cost_function for analysis of singular result QuantumStates

import itertools
def aClcostFct(state, G ):
# we assume solution is right
temp = True
energy = 0
#intlist = [int(s) for s in list(state)]
intlist = [s for s in range(len(list(state))) if list(state)[s] == "1"]
# get all combinations of vertices in graph that are marked as |1> by the solution
#combinations = list(itertools.combinations(list(np.nonzero(intlist)[0]), 2))
combinations = list(itertools.combinations(intlist, 2))
# if any combination is found in the list of G.edges(), the solution is wrong, and energy == 0
for combination in combinations:
if combination in G.edges():
temp = False
# else we just add the number of marked as |1> nodes
if temp:
energy = -len(intlist)
#energy = -sum(intlist)
return(energy)


Print the 5 most likely solutions and the associated energy/cost value

print("5 most likely Solutions")
maxfive = sorted(theNiceQAOA, key=theNiceQAOA.get, reverse=True)[:5]
for name, age in theNiceQAOA.items():
if name in maxfive:
print((name, age))
print(aClcostFct(name, giraf))


Print the solution as given by networkx

print("NX solution")
print(nx.max_weight_clique(giraf, weight = None))


## Max-$\kappa$-Colorable Subgraph#

The Max-$$\kappa$$-Colorable Subgraph problem instance and its QAOA implementation is heavily discussed in the Max-$\kappa$-Colorable Subgraph tutorial. All the necessary ingredients and required steps to run QAOA are elaborate in easy to grasp manner.

Here we instead provide the condensed implementation of the algorithm for M$$\kappa$$CS with using all of the predefined functions for this specific instance.

### Problem Definition#

The Max-$$\kappa$$-Colorable Subgraph problem is defined as follows:

Given a graph $$G$$ and $$\kappa$$ colors, maximize the size (number of edges) of a properly colored subgraph.

Similarly to the example of MaxCut above, we import the necessary functions and packages, create a graph G we will be cutting, define the colors we want to use, define a quantum argument qarg we’ll be acting on (we provide options for one-hot and binary encoding schemes), as well as specify the depth of our algorithm.

from qrisp.qaoa import QAOAProblem, mkcs_obj, apply_phase_if_eq, create_coloring_operator, create_coloring_cl_cost_function, QuantumColor, XY_mixer, apply_XY_mixer, RX_mixer
from qrisp import QuantumArray
import random
import networkx as nx
from operator import itemgetter

G = nx.Graph()
num_nodes = len(G.nodes)

color_list = ["red", "blue", "yellow", "green"]

qarg = QuantumArray(qtype = QuantumColor(color_list, one_hot_enc = True), shape = num_nodes)
#qarg = QuantumArray(qtype = QuantumColor(color_list, one_hot_enc = False), shape = num_nodes) # use one_hot_enc = False if you use binary encoding

depth = 3


### QAOA instanciation#

Next we follow the recipe to run the algorithm with QAOAProblem, feeding it the cost_operator, a mixer and a cl_cost_function. In case one prefers to use the binary encoding, adjust the # in the code block below.

coloring_instance = QAOAProblem(create_coloring_operator(G), apply_XY_mixer, create_coloring_cl_cost_function(G))
# coloring_instance = QAOAProblem(create_coloring_operator(G), RX_mixer, create_coloring_cl_cost_function(G)) # use RX mixer if you use binary encoding

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

res = coloring_instance.run(qarg, depth, max_iter = 25)


### Result analysis#

After running our QAOA on the M$$\kappa$$CS problem instance we can now obtain the QAOA solution and draw the graph with optimally colored nodes.

best_coloring, best_solution = min([(mkcs_obj(quantumcolor_array,G),quantumcolor_array) for quantumcolor_array in res.keys()], key=itemgetter(0))
print(f"Best string: {best_solution} with coloring: {-best_coloring}")

best_coloring, res_str = min([(mkcs_obj(quantumcolor_array,G),quantumcolor_array) for quantumcolor_array in list(res.keys())[:5]], key=itemgetter(0))
print("QAOA solution: ", res_str)
best_coloring, best_solution = (mkcs_obj(res_str,G),res_str)

nx.draw(G, node_color=res_str, with_labels=True)