updated 2021-08-01

“Quantum Approximate Optimization Algorithm”

- Protocol for approximately solving an optimization problem on a quantum computer
- Can be implemented on near-term devices!
- Variable parameter \(p\) (depth) – the optimal performance increases with \(p\) and solves the problem as \(p \to \infty\)
- In order to use this algorithm optimally, you have to tune \(2p\) parameters (this gets hard as \(p\) grows!)
- Some tutorials here and here
- This is a ‘local’ algorithm, which may limit its power.

Problem | QAOA depth | Graph | optimal performance (satisfying fraction) | best known algorithm? | Papers |
---|---|---|---|---|---|

MAX-CUT | 1 | triangle-free | \(1/2 + 0.30/\sqrt{D}\) | No | Wang+ 2018, Hastings 2019 |

MAX-CUT | 1 | any | see formula | ? (KM project in progress) | Wang+ 2018, Hastings 2019 |

MAX-CUT | 2 | girth above 5 | \(1/2 + 0.41/\sqrt{D}\) | No | Marwaha 2021 |

MAX-3-XOR (E3LIN2) | 1 | no overlapping constraints, or random signs | at least \(1/2 + O(1/\sqrt{D})\) | ? (KM project in progress) | Farhi+ 2015, Barak+ 2015, Lin+ 2016 |

MAX-3-XOR (E3LIN2) | 1 | any | at least \(1/2 + O(1/\sqrt{D log D})\) | so far, No | Farhi+ 2015, Barak+ 2015 |

SK Model | 1-12 | infinite size | see formula | ? | Farhi+ 2019 |

Max Independent Set | 1 | any | at least \(O(n/d)\) | so far, No (KM project in progress) | Farhi+ 2020 |

At high depth, the algorithm’s success is dependent on choosing the best parameters. Strategies to make this easier include:

- Fourier method and interpolation method (Zhou+ 2018)
- Variational methods that alternate between a computer and quantum computer (Farhi+ 2014)

There are some studies of optimal QAOA parameters for some graphs being transferable to other graphs (instance independence); see Galda+ 2021, Brandao+ 2018, or Wurtz+ 2021.

It would be nice to discuss some results on barren plateaus here, and also mention what happens to error.

Several adjustments to QAOA have been proposed:

- ST-QAOA Wurtz+ 2021a
- CD-QAOA Wurtz+ 2021b
- Alternating Operator Ansatz Hadfield+ 2017, Hadfield+ 2021
- RQAOA Bravyi+ 2019

Some of these extensions are “true extensions” (sampling things from quantum computers), such as penalty proposals and adjusting the mixers; while others are new protocols that use QAOA inside of it (CD-QAOA, RQAOA).

Some of these protocols turn QAOA into a non-local algorithm.

- Which problems and families of instances can we show that QAOA provides advantage over classical algorithms?
- How can we analyze QAOA at higher depth?
- How can we better choose the QAOA parameters?

This is the case because…

It’s worth making clear similarities and differences between VQE (e.g. electronic structure) and QAOA (diagonal eigenvalue problems vs non-diagonal eigenvalue problems)

In VQE – the ansatz holds something I can’t hold classically, so intermediate strings don’t tell me what the state is.

You really want the highest value the QAOA returns; when you run the experiment many times, you take the best one, not the average one. But we use the average because it’s easier to calculate, and chebyshev & chernoff to get bound on tail.

The QAOA may also perform well even if the overlap with optimal value is low.

There are alternate metrics of performance, such as ___.

Although there are many similarities, there are settings where QAOA improves upon adiabatic computation. See Zhou+ 2018