Recursive Quantum Relaxation for Combinatorial Optimization Problems

Ruho Kondo; Yuki Sato; Rudy Raymond; Naoki Yamamoto

doi:10.22331/q-2025-01-15-1594

Recursive Quantum Relaxation for Combinatorial Optimization Problems

Ruho Kondo^1,2, Yuki Sato^1,2, Rudy Raymond^2,3, and Naoki Yamamoto^2,4

¹Toyota Central R&D Labs., Inc., 41-1, Yokomichi, Nagakute, Aichi 480-1192, Japan
²Quantum Computing Center, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan
³Department of Computer Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
⁴Department of Applied Physics and Physico-Informatics, Keio University, Hiyoshi 3-14-1, Kohoku-ku, Yokohama 223-8522, Japan

Published:	2025-01-15, volume 9, page 1594
Editor:	Mohan Sarovar
Eprint:	arXiv:2403.02045v3
Doi:	https://doi.org/10.22331/q-2025-01-15-1594
Citation:	Quantum 9, 1594 (2025).

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Abstract

Quantum optimization methods use a continuous degree-of-freedom of quantum states to heuristically solve combinatorial problems, such as the MAX-CUT problem, which can be attributed to various NP-hard combinatorial problems. This paper shows that some existing quantum optimization methods can be unified into a solver to find the binary solution which is most likely measured from the optimal quantum state. Combining this finding with the concept of quantum random access codes (QRACs) for encoding bits into quantum states on fewer qubits, we propose an efficient recursive quantum relaxation method called recursive quantum random access optimization (RQRAO) for MAX-CUT. Experiments on standard benchmark graphs with several hundred nodes in the MAX-CUT problem, conducted in a fully classical manner using a tensor network technique, show that RQRAO not only outperforms the Goemans-Williamson and recursive QAOA methods, but also is comparable to state-of-the-art classical solvers. The code is available at https://github.com/ToyotaCRDL/rqrao.

Featured image: The concept image of recursive quantum random access optimization (RQRAO).

Popular summary

We propose a unified quantum-inspired method for solving combinatorial problems that integrates an existing technique for encoding the information of multiple classical bits into fewer qubits with a particular recursive protocol. To validate our framework, we proved several fundamental properties associated with the method. Notably, the encoding process from classical bits into qubits naturally coarsens the problem; this enables the application of ensemble heuristics, which thereby enhances the efficiency of the proposed method, leading to better solutions. Numerical experiments conducted on classical devices demonstrated that this method outperforms both existing classical approximation techniques and quantum heuristics.

► BibTeX data

@article{Kondo2025recursivequantum,
  doi = {10.22331/q-2025-01-15-1594},
  url = {https://doi.org/10.22331/q-2025-01-15-1594},
  title = {Recursive {Q}uantum {R}elaxation for {C}ombinatorial {O}ptimization {P}roblems},
  author = {Kondo, Ruho and Sato, Yuki and Raymond, Rudy and Yamamoto, Naoki},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1594},
  month = jan,
  year = {2025}
}

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