# A Quantum Approximate Optimization Algorithm

@article{Farhi2014AQA, title={A Quantum Approximate Optimization Algorithm}, author={Edward Farhi and Jeffrey Goldstone and Sam Gutmann}, journal={arXiv: Quantum Physics}, year={2014} }

We introduce a quantum algorithm that produces approximate solutions for combinatorial optimization problems. The algorithm depends on a positive integer p and the quality of the approximation improves as p is increased. The quantum circuit that implements the algorithm consists of unitary gates whose locality is at most the locality of the objective function whose optimum is sought. The depth of the circuit grows linearly with p times (at worst) the number of constraints. If p is fixed, that…

## 1,010 Citations

The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: A Typical Case

- Mathematics, PhysicsArXiv
- 2020

This work focuses on finding big independent sets in random graphs with dn/2 edges keeping d fixed and n large, and shows that if p is less than a d-dependent constant times log n, the QAOA cannot do better than finding an independent set of size .854 times the optimal for d large.

Parameters Fixing Strategy for Quantum Approximate Optimization Algorithm

- Computer Science, Physics2021 IEEE International Conference on Quantum Computing and Engineering (QCE)
- 2021

A parameters fixing strategy is proposed which gives high approximation ratio on average, even at large circuit depths, by initializing QAOA with the optimal parameters obtained from the previous depths, and is tested on the Max-cut problem of certain classes of graphs such as the 3-regular graphs and the Erdös-Rényi graphs.

The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: Worst Case Examples

- Mathematics, Physics
- 2020

The Quantum Approximate Optimization Algorithm can be applied to search problems on graphs with a cost function that is a sum of terms corresponding to the edges. When conjugating an edge term, the…

A study of the performance of classical minimizers in the Quantum Approximate Optimization Algorithm

- Mathematics
- 2021

Improving the Quantum Approximate Optimization Algorithm with postselection

- Computer Science, PhysicsArXiv
- 2020

This work considers a well-studied quantum algorithm for combinatorial optimization: the Quantum Approximate Optimization Algorithm (QAOA) applied to the MaxCut problem on 3-regular graphs and derives theoretical upper and lower bounds showing that a constant (though small) increase of the fraction of satisfied edges is indeed achievable.

Quantum Supremacy through the Quantum Approximate Optimization Algorithm

- Mathematics, Physics
- 2016

The Quantum Approximate Optimization Algorithm (QAOA) is designed to run on a gate model quantum computer and has shallow depth. It takes as input a combinatorial optimization problem and outputs a…

Solving Combinatorial Optimization Problems using the Quantum Approximation Optimization Algorithm

- Computer Science
- 2020

A sufficient condition for polynomial implementation in QAOA is shown that generalizes for all combinatorial optimization problems and it is demonstrated that the qubits have a natural tendency to decohere towards the ground state of the system during the lifetime of the algorithm.

Classical algorithms and quantum limitations for maximum cut on high-girth graphs

- Computer Science, PhysicsITCS
- 2022

It is proved that every (quantum or classical) one-local algorithm achieves on D-regular graphs of girth > 5 a maximum cut of at most 1/2+C/√ D for C = 1/ √ 2 ≈ 0.7071, the first such result showing that one- local algorithms achieve a value that is bounded away from the true optimum for random graphs.

Quantum Algorithms, Architecture, and Error Correction

- Mathematics, Physics
- 2018

Quantum algorithms have the potential to provide exponential speedups over some of the best known classical algorithms. These speedups may enable quantum devices to solve currently intractable…

Hybrid quantum-classical algorithms for approximate graph coloring

- Computer Science, Physics
- 2020

It is found that level-$1$ RQAOA is surprisingly competitive: for the ensembles considered, its approximation ratios are often higher than those achieved by the best known generic classical algorithm based on rounding an SDP relaxation.

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