The fixed angle conjecture for QAOA on regular MaxCut graphs

Abstract

The quantum approximate optimization algorithm (QAOA) is a near-term combinatorial optimization algorithm suitable for noisy quantum devices. However, little is known about performance guarantees for p > 2. A recent work [1] computing MaxCut performance guarantees for 3-regular graphs conjectures that any d-regular graph evaluated at particular fixed angles has an approximation ratio greater than some worst-case guarantee. In this work, we provide numerical evidence for this fixed angle conjecture for p < 12. We compute and provide these angles via numerical optimization and tensor networks. These fixed angles serve for an optimization-free version of QAOA, and have universally good performance on any 3 regular graph. Heuristic evidence is presented for the fixed angle conjecture on graph ensembles, which suggests that these fixed angles are "close" to global optimum. Under the fixed angle conjecture, QAOA has a larger performance guarantee than the Goemans Williamson algorithm on 3-regular graphs for p ≥ 11.

Parameter Transfer for Quantum Approximate Optimization of Weighted MaxCut

“…Our results demonstrate that parameter concentration previously established for unweighted MaxCut [18][19][20][21] and the SK model [8] applies more generally in instances of weighted MaxCut as long as the problem is appropriately rescaled. At each p, the parameters from a single QAOA circuit can be rescaled and transferred across tens of thousands of instances of weighted MaxCut with a performance that is near to that of QAOA with exhaustively optimized parameters.…”

“…10-11. Additionally, we have experimented with using rescaled fixed-angle conjecture parameters from [20]. Because the median parameters provided better performance, we do not include the results using these fixedangle conjecture parameters.…”

“…Several recent results show that as a function of the parameters, the QAOA energy land-scape is nearly instance independent for typical problem instances that come from a reasonable distribution. This makes it possible to transfer parameters from a set of preoptimized problem instances to new instances, thereby removing or reducing the cost of parameter optimization [18][19][20][21]. However, these prior results address only unweighted MaxCut problems and the Sherrington-Kirkpatrick (SK) model, which are of limited practical interest.…”

Scaling Quantum Approximate Optimization on Near-term Hardware

“…When the QAOA parameters are optimized using measurements from a quantum computer, this optimization will also be greatly inhibited. Parameter optimization has been addressed in some instances using theoretical approaches [9,19,20,[37][38][39][40][41][42][43][44], though for generic instances it is unclear if such approaches can be applied. However, even with a good set of parameters the circuit must still be run to obtain the final bitstring solution to the problem, and in our model this requires a number of measurements that quickly becomes prohibitive at scales relevant for quantum advantage.…”

“…Classical algorithms have also been developed that outperform QAOA at low p [35,36], further suggesting large p may be necessary to compete with conventional methods. To optimize parameters at large n and p, a variety of computational [37,38] and theoretical [39][40][41][42][43][44] approaches have been developed and in some cases the theoretical performance has been characterized. With parameter setting strategies at hand, what remains to be seen is how the QAOA will perform in experimental implementations.…”