The Quantum Approximate Optimization Algorithm at High Depth for MaxCut on LargeGirth Regular Graphs and the SherringtonKirkpatrick Model
Abstract
The Quantum Approximate Optimization Algorithm (QAOA) finds approximate solutions to combinatorial optimization problems. Its performance monotonically improves with its depth $p$. We apply the QAOA to MaxCut on largegirth $D$regular graphs. We give an iterative formula to evaluate performance for any $D$ at any depth $p$. Looking at random $D$regular graphs, at optimal parameters and as $D$ goes to infinity, we find that the $p=11$ QAOA beats all classical algorithms (known to the authors) that are free of unproven conjectures. While the iterative formula for these $D$regular graphs is derived by looking at a single tree subgraph, we prove that it also gives the ensembleaveraged performance of the QAOA on the SherringtonKirkpatrick (SK) model defined on the complete graph. We also generalize our formula to Max$q$XORSAT on largegirth regular hypergraphs. Our iteration is a compact procedure, but its computational complexity grows as $O(p^2 4^p)$. This iteration is more efficient than the previous procedure for analyzing QAOA performance on the SK model, and we are able to numerically go to $p=20$. Encouraged by our findings, we make the optimistic conjecture that the QAOA, as $p$ goes to infinity, will achieve the Parisi value. We analyze the performance of the quantum algorithm, but one needs to run it on a quantum computer to produce a string with the guaranteed performance.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 DOI:
 10.48550/arXiv.2110.14206
 arXiv:
 arXiv:2110.14206
 Bibcode:
 2021arXiv211014206B
 Keywords:

 Quantum Physics;
 Computer Science  Data Structures and Algorithms
 EPrint:
 39 pages, 7 figures, 5 tables. Full version of the paper in TQC 2022