Graph decomposition techniques for solving combinatorial optimization problems with variational quantum algorithms
Abstract
The quantum approximate optimization algorithm (QAOA) has the potential to approximately solve complex combinatorial optimization problems in polynomial time. However, current noisy quantum devices cannot solve large problems due to hardware constraints. In this work, we develop an algorithm that decomposes the QAOA input problem graph into a smaller problem and solves MaxCut using QAOA on the reduced graph. The algorithm requires a subroutine that can be classical or quantumin this work, we implement the algorithm twice on each graph. One implementation uses the classical solver Gurobi in the subroutine and the other uses QAOA. We solve these reduced problems with QAOA. On average, the reduced problems require only approximately 1/10 of the number of vertices than the original MaxCut instances. Furthermore, the average approximation ratio of the original MaxCut problems is 0.75, while the approximation ratios of the decomposed graphs are on average of 0.96 for both Gurobi and QAOA. With this decomposition, we are able to measure optimal solutions for ten 100vertex graphs by running singlelayer QAOA circuits on the Quantinuum trappedion quantum computer H11, sampling each circuit only 500 times. This approach is best suited for sparse, particularly $k$regular graphs, as $k$regular graphs on $n$ vertices can be decomposed into a graph with at most $\frac{nk}{k+1}$ vertices in polynomial time. Further reductions can be obtained with a potential tradeoff in computational time. While this paper applies the decomposition method to the MaxCut problem, it can be applied to more general classes of combinatorial optimization problems.
 Publication:

arXiv eprints
 Pub Date:
 June 2023
 DOI:
 10.48550/arXiv.2306.00494
 arXiv:
 arXiv:2306.00494
 Bibcode:
 2023arXiv230600494P
 Keywords:

 Quantum Physics;
 Mathematics  Combinatorics