Transferability of optimal QAOA parameters between random graphs

Abstract

The Quantum approximate optimization algorithm (QAOA) is one of the most promising candidates for achieving quantum advantage through quantum-enhanced combinatorial optimization. In a typical QAOA setup, a set of quantum circuit parameters is optimized to prepare a quantum state used to find the optimal solution of a combinatorial optimization problem. Several empirical observations about optimal parameter concentration effects for special QAOA MaxCut problem instances have been made in recent literature, however, a rigorous study of the subject is still lacking. We show that convergence of the optimal QAOA parameters around specific values and, consequently, successful transferability of parameters between different QAOA instances can be explained and predicted based on the local properties of the graphs, specifically the types of subgraphs (lightcones) from which the graphs are composed. We apply this approach to random regular and general random graphs. For example, we demonstrate how optimized parameters calculated for a 6-node random graph can be successfully used without modification as nearly optimal parameters for a 64-node random graph, with less than 1% reduction in approximation ratio as a result. This work presents a pathway to identifying classes of combinatorial optimization instances for which such variational quantum algorithms as QAOA can be substantially accelerated.

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.…”

“…Farhi et al have argued that QAOA recovers the ground state of C as p → ∞ [2], but the primary interest in QAOA is in reaching high performance with a modest number of layers p that could realistically be implemented on a quantum computer. A significant body of theoretical [4][5][6][7][8], computational [9][10][11][12][13], and experimental [14,15] research has focused on understanding QAOA performance at p ≈ 1, mostly on the MaxCut problem with a small number of qubits n, but also for other types of problems [16][17][18]. These studies have shown some promising results, for example, with QAOA outperforming the conventional lower bound of the GW algorithm for MaxCut on some small instances [19,20].…”

Unsupervised strategies for identifying optimal parameters in Quantum Approximate Optimization Algorithm

“…Many previous works have extensively employed the concentration property [7,14,15,16,17,18,19,20]. Among them, a few employed ML or designed strategies for setting good QAOA parameters for diferent objectives.…”

“…[19] present a strategy to find good parameters for QAOA based on topological properties of the problem graph and tensor network techniques. [16] point out that the success of transferability of parameters between different problem instances can be explained and predicted based on the types of subgraphs composing a graph. Finally, meta-learning is used in [17] to learn good initial angles for QAOA.…”

“…Many works have studied or illustrated this concentration property [6,7,14,15,16,17,18,19,20]. However, it comes with an underlying assumption that the instances (most often formulated as Ising problems that can be visualized as graphs) should come from a «reasonable» distribution [6].…”