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solving max cut with qaoa in qiskit a practical guide skip to main content quantumcomputing courses com courses all courses course platforms coursera edx udemy brilliant hardware providers google quantum ai ibm quantum ionq quantinuum amazon braket azure quantum quera rigetti d wave tutorials all tutorials hello world qiskit hello world cirq hello world pennylane hello world braket quantum gates grover s algorithm shor s algorithm reference all frameworks qiskit cirq pennylane amazon braket pyquil tket d wave ocean q explore learn learning paths prerequisites programming guide case studies glossary books quantum news podcasts tools bloch sphere quantum pinball guides algorithm guide hardware guide qubit types framework comparison migration guide language timeline cheat sheets career events 2026 jobs careers certifications salary guide universities interview prep faq troubleshooting about company editorial policy search browse courses home tutorials solving max cut with qaoa in qiskit a practical guide qiskit intermediate free 35 61 in series 50 minutes 22 feb 2026 editorial policy solving max cut with qaoa in qiskit a practical guide build a complete max cut solver using qaoa in qiskit optimization covers graph construction with networkx quadraticprogram formulation qaoa via minimumeigenoptimizer and how solution quality changes with circuit depth p max cut qaoa qiskit optimization graph optimization combinatorial optimization prerequisites python proficiency beginner quantum computing concepts superposition entanglement linear algebra basics in this guide 10 sections 01 what is max cut 02 setting up 03 constructing the graph 04 formulating as a quadraticprogram 05 classical exact solution 06 qaoa at p 1 07 comparing p 1 p 2 and p 3 08 visualizing qaoa circuit structure 09 sampling the full solution distribution 10 interpreting the results what is max cut the max cut problem asks given a graph with weighted edges how do you partition the vertices into two sets such that the total weight of edges crossing the partition is maximized it is np hard in general making it a natural target for quantum optimization algorithms max cut is also a canonical benchmark for qaoa quantum approximate optimization algorithm it maps naturally to an ising hamiltonian the cost function is easy to evaluate and classical approximate algorithms like goemans williamson provide a well known comparison baseline this tutorial builds a complete solver using qiskit optimization from graph construction through result visualization you will see why qaoa approximates rather than solves exactly how circuit depth affects solution quality and how to use the classical exact solver as a ground truth reference setting up you need the following packages pip install qiskit qiskit optimization qiskit algorithms networkx matplotlib scipy import everything up front import numpy as np import networkx as nx import matplotlib pyplot as plt from qiskit_optimization applications import maxcut from qiskit_optimization algorithms import minimumeigenoptimizer from qiskit_algorithms import qaoa numpyminimumeigensolver from qiskit_algorithms optimizers import cobyla from qiskit primitives import statevectorsampler as sampler constructing the graph start with a small weighted graph seven nodes is enough to see interesting behavior while keeping circuit depth manageable requires qiskit_optimization def build_weighted_graph create a weighted graph for max cut experiments g nx graph add nodes n_nodes 7 g add_nodes_from range n_nodes add weighted edges edges 0 1 1 0 0 2 2 0 1 3 1 5 2 3 1 0 2 4 2 5 3 5 1 0 4 5 1 5 4 6 2 0 5 6 1 0 1 4 0 5 g add_weighted_edges_from edges return g def visualize_graph g partition none title graph draw the graph optionally coloring nodes by partition pos nx spring_layout g seed 42 fig ax plt subplots figsize 8 6 if partition is not none colors steelblue if partition i 0 else coral for i in range g number_of_nodes else colors steelblue edge_weights nx get_edge_attributes g weight nx draw_networkx g pos node_color colors node_size 600 font_color white font_weight bold ax ax nx draw_networkx_edge_labels g pos edge_labels edge_weights ax ax if partition is not none cut_value sum d weight for u v d in g edges data true if partition u partition v ax set_title f title cut cut_value 1f else ax set_title title plt tight_layout plt savefig f maxcut_ title lower replace _ png dpi 150 plt show g build_weighted_graph visualize_graph g title input graph print f graph g number_of_nodes nodes g number_of_edges edges formulating as a quadraticprogram qiskit optimization provides the maxcut application class that converts a networkx graph directly into a quadraticprogram the decision variable for each node is binary 0 for one partition 1 for the other requires qiskit_optimization maxcut_instance maxcut g qp maxcut_instance to_quadratic_program print qp export_as_lp_string the lp output shows the objective maximize the sum of w i j x_i x_j 2 x_i x_j over all edges for binary variables this equals the edge weight when nodes i and j are in different partitions and zero when they are in the same partition that is exactly the cut value classical exact solution before running qaoa get the optimal solution classically this is your benchmark requires qiskit_optimization exact_solver minimumeigenoptimizer numpyminimumeigensolver exact_result exact_solver solve qp print exact solution print f optimal cut value exact_result fval 4f negated because we minimized print f partition exact_result x decode partition optimal_partition i int exact_result x i for i in range g number_of_nodes visualize_graph g partition optimal_partition title optimal cut note qiskit optimization minimizes by default so the max cut is formulated as a minimization of the negated cut value qaoa at p 1 qaoa uses two parameterized layers a cost layer encoding the problem hamiltonian and a mixing layer of x rotations the number of repetitions p controls the approximation quality requires qiskit_optimization def run_qaoa qp p_layers initial_point none shots 4096 run qaoa with p layers and return the optimizationresult sampler sampler optimizer cobyla maxiter 300 if initial_point is none rng np random default_rng 42 initial_point rng uniform 0 np pi 2 p_layers qaoa qaoa sampler sampler optimizer optimizer reps p_layers initial_point initial_point solver minimumeigenoptimizer qaoa result solver solve qp return result p 1 print running qaoa p 1 result_p1 run_qaoa qp p_layers 1 partition_p1 i int result_p1 x i for i in range g number_of_nodes cut_p1 sum d weight for u v d in g edges data true if partition_p1 u partition_p1 v print f p 1 cut value cut_p1 4f visualize_graph g partition partition_p1 title qaoa p 1 comparing p 1 p 2 and p 3 the approximation ratio r qaoa_cut optimal_cut measures how close qaoa gets to optimal theoretically p 1 qaoa guarantees r 0 6924 for unweighted max cut on 3 regular graphs the farhi goldstone gutmann bound in practice weighted and irregular problems plus finite shots mean no such guarantee applies though results are often comparable requires qiskit_optimization def evaluate_all_p_levels qp g exact_result compare qaoa across p levels optimal_cut exact_result fval results for p in 1 2 3 print f n running qaoa p p res run_qaoa qp p_layers p partition i int res x i for i in range g number_of_nodes cut sum d weight for u v d in g edges data true if partition u partition v ratio cut optimal_cut results p cut cut ratio ratio partition partition print f cut value cut 4f print f approximation ratio ratio 4f visualize_graph g partition partition title f qaoa p p return results results evaluate_all_p_levels qp g exact_result optimal_cut exact_result fval print n summary print f optimal cut value optimal_cut 4f for p data in results items print f qaoa p p cut data cut 4f ratio data ratio 4f visualizing qaoa circuit structure understanding what qaoa builds is important for reasoning about its depth and gate count requires qiskit_optimization from qiskit_algorithms import qaoa from qiskit_algorithms optimizers import cobyla from qiskit primitives import statevectorsampler as sampler build a p 2 qaoa circuit without running it sampler sampler qaoa_p2 qaoa sampler sampler optimizer cobyla reps 2 convert problem to ising operator for circuit inspection from qiskit_optimization converters import quadraticprogramtoqubo from qiskit_optimization translators import to_ising converter quadraticprogramtoqubo qubo converter convert qp operator offset to_ising qubo get the ansatz circuit qaoa_p2 _check_operator_ansatz operator circuit qaoa_p2 ansatz circuit circuit decompose print f circuit depth circuit depth print f gate counts circuit count_ops print circuit draw text fold 100 sampling the full solution distribution qaoa outputs a probability distribution over all 2 n bitstrings not just one solution examining this distribution reveals the landscape requires qiskit_optimization from qiskit primitives import statevectorsampler as primitivesampler from qiskit_algorithms import qaoa from qiskit_algorithms optimizers import cobyla from qiskit_optimization converters import quadraticprogramtoqubo from qiskit_optimization translators import to_ising def get_qaoa_distribution qp g p 2 optimal_params none get the full probability distribution from qaoa converter quadraticprogramtoqubo qubo converter convert qp operator offset to_ising qubo sampler primitivesampler use pre optimized parameters if provided if optimal_params is not none gamma_beta optimal_params else rng np random default_rng 0 gamma_beta rng uniform 0 np pi 2 p qaoa qaoa sampler sampler optimizer cobyla reps p initial_point gamma_beta access the sampler distribution directly qaoa _check_operator_ansatz operator bound_circuit qaoa ansatz assign_parameters gamma_beta bound_circuit measure_all shots 8192 job sampler run bound_circuit shots shots counts job result 0 data meas get_counts convert to cut values n g number_of_nodes cut_distribution for bitstring count in counts items partition int bitstring i 1 for i in range n cut sum d weight for u v d in g edges data true if partition u partition v prob count shots cut_distribution round cut 2 cut_distribution get round cut 2 0 prob return cut_distribution plot distribution of cut values dist get_qaoa_distribution qp g p 2 fig ax plt subplots figsize 10 5 cuts sorted dist keys probs dist c for c in cuts ax bar cuts probs width 0 1 ax axvline x optimal_cut color red linestyle label f optimal optimal_cut 1f ax set_xlabel cut value ax set_ylabel probability ax set_title qaoa p 2 cut value distribution ax legend plt tight_layout plt savefig qaoa_distribution png dpi 150 plt show interpreting the results several practical points emerge from this experiment p 1 often achieves a good approximation ratio on small graphs typically 0 70 0 85 the theoretical guarantee of 0 6924 is a worst case lower bound for 3 regular graphs increasing p helps but with diminishing returns going from p 1 to p 2 often improves the ratio significantly going from p 2 to p 3 typically gives smaller gains while roughly tripling the two qubit gate count shot noise limits small p gains at p 1 the optimal landscape is relatively smooth and cobyla converges reliably at p 3 the parameter space is 6 dimensional and classical optimization becomes harder more sophisticated optimizers like spsa or gradient based methods become necessary the problem size matters more than p for real hardware on current nisq hardware circuit noise dominates for p 3 on graphs with more than 20 nodes hardware efficient approaches that map the problem graph to the device topology are essential for larger instances max cut with qaoa remains an active research area the computational advantage over classical approximation algorithms has not been demonstrated but the framework is a template for applying qaoa to other combinatorial optimization problems portfolio optimization vehicle routing and scheduling all admit similar formulations was this tutorial helpful yes no share ready to go deeper browse structured courses from coursera edx udemy brilliant and more browse courses related tutorials continue learning with these guides build your first quantum circuit in qiskit complete beginner guide beginner 25 min read read the deutsch jozsa algorithm quantum s first speedup explained beginner 35 minutes read fault tolerant quantum gates why t gates need magic states advanced 22 min read read previous quantum state tomography with qiskit next vqe for h2 molecule complete worked example in qiskit on this page 01 what is max cut 02 setting up 03 constructing the graph 04 formulating as a quadraticprogram 05 classical exact solution 06 qaoa at p 1 07 comparing p 1 p 2 and p 3 08 visualizing qaoa circuit structure 09 sampling the full solution distribution 10 interpreting the results at a glance level intermediate read time 50 minutes language python updated jun 2026 related tutorials build your first quantum circuit in qiskit complete beginner guide 25 min read the deutsch jozsa algorithm quantum s first speedup explained 35 minutes fault tolerant quantum gates why t gates need magic states 22 min read courses on this quantum optimization with ibm quantum openhpi free a practical introduction to quantum computing cern free getting started with applied quantum optimization paid get one quantum email a week new tutorials courses worth taking and what changed in qiskit cirq pennylane this week no spam unsubscribe anytime email address subscribe 112 courses 220 tutorials 241 glossary terms 26 framework references 34 case studies quantumcomputing courses com free tutorials curated courses framework references and tools for anyone learning quantum computing an independent catalog of quantum computing courses and tutorials published by hadamard llc learn all courses free tutorials learning paths compare frameworks algorithm guide case studies quantum news 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