[[Approximation algorithm|Approximate optimization]] is a way of finding an approximate solution to an optimization problem, which is often [[NP-hard]]. The approximated solution of the combinatorial optimization problem is a string <math> z </math> that is close to maximizing <math> C(z) </math>.
For combinatorial optimization, the Quantum'''quantum Approximateapproximate Optimizationoptimization Algorithmalgorithm''' (QAOA)<ref>{{cite arxiv|last1=Farhi|first1=Edward|last2=Goldstone|first2=Jeffrey|last3=Gutmann|first3=Sam|title=A Quantum Approximate Optimization Algorithm|eprint=1411.4028|class=quant-ph|year=2014}}</ref> briefly had a better approximation ratio than any known [[polynomial time]] classical algorithm (for a certain problem),<ref>{{cite arxiv|last1=Farhi|first1=Edward|last2=Goldstone|first2=Jeffrey|last3=Gutmann|first3=Sam|title=A Quantum Approximate Optimization Algorithm Applied to a Bounded Occurrence Constraint Problem|eprint=1412.6062|class=quant-ph|year=2014}}</ref> until a more effective classical algorithm was proposed.<ref>{{cite arxiv|last1=Barak|first1=Boaz|last2=Moitra|first2=Ankur|last3=O'Donnell|first3=Ryan|last4=Raghavendra|first4=Prasad|last5=Regev|first5=Oded|last6=Steurer|first6=David|last7=Trevisan|first7=Luca|last8=Vijayaraghavan|first8=Aravindan|last9=Witmer|first9=David|last10=Wright|first10=John|title=Beating the random assignment on constraint satisfaction problems of bounded degree|eprint=1505.03424|class=cs.CC|year=2015}}</ref> The relative speed-up of the quantum algorithm is an open research question.
The heart of the QAOA relies on the use of [[unitary operators]] dependent on <math> 2p </math> [[angle]]s, where <math> p>1 </math> is an input integer. These operators are iteratively applied on a state that is an equal-weighted [[quantum superposition]] of all the possible states in the computational basis. In each iteration, the state is measured in the computational basis and <math> C(z) </math> is calculated. After a sufficient number of repetitions, the value of <math> C(z) </math> is almost optimal, and the state being measured is close to being optimal as well.
