Abstract
The Maximum Boolean Satisfiability Problem (also known as the Max-SAT problem) is the problem of determining the maximum number of disjunctive clauses that can be satisfied (i.e., made true) by an assignment of truth values to the formula’s variables. This is a generalization of the well-known Boolean Satisfiability Problem (also known as the SAT problem), the first problem that was proven to be NP-complete. With the proliferation of quantum computing, a current approach to tackle this optimization problem is Quantum Annealing (QA). In this work, we compare several gadgets that translate 3-SAT problems into Quadratic Unconstrained Binary Optimization (QUBO) problems to be able to solve them in a quantum annealer. We show the performance superiority of the not-yet-considered gadgets in comparison to state-of-the-art approaches when solving random instances in D-Wave’s quantum annealer.
P. R. Farrés and R. Ballester—These authors contributed equally to this work.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
Notice that 3-SAT, Max-3-SAT, and even Max-2-SAT are NP-complete problems. Therefore, a bounded-error quantum polynomial-time algorithm for any of these problems would result in a proof of \(NP\subseteq BQP\), which, obviously, we do not have.
- 3.
References
Ansótegui, C., Levy, J.: SAT, Gadgets, Max2XOR, and Quantum Annealers (2024). http://arxiv.org/abs/2403.00182 [quant-ph]
Biamonte, J., Wittek, P., Pancotti, N., Rebentrost, P., Wiebe, N., Lloyd, S.: Quantum machine learning. Nature 549(7671), 195–202 (2017). https://doi.org/10.1038/nature23474, https://www.nature.com/articles/nature23474, number: 7671. Publisher: Nature Publishing Group
Bian, Z., Chudak, F., Israel, R., Lackey, B., Macready, W.G., Roy, A.: Discrete optimization using quantum annealing on sparse Ising models. Front. Phys. 2 (2014). https://www.frontiersin.org/articles/10.3389/fphy.2014.00056
Bian, Z., Chudak, F., Macready, W., Roy, A., Sebastiani, R., Varotti, S.: Solving SAT (and MaxSAT) with a Quantum Annealer: foundations, encodings, and preliminary results. Inf. Comput. 275, 104609 (2020). https://doi.org/10.1016/j.ic.2020.104609, https://www.sciencedirect.com/science/article/pii/S0890540120300973
Chancellor, N., Zohren, S., Warburton, P.A., Benjamin, S.C., Roberts, S.: A direct mapping of Max k-SAT and high order parity checks to a chimera graph. Sci. Rep. 6(1), 37107 (2016).https://doi.org/10.1038/srep37107, https://www.nature.com/articles/srep37107, number: 1. Publisher: Nature Publishing Group
Choi, V.: Adiabatic quantum algorithms for the NP-complete maximum-weight independent set, exact cover and 3SAT problems. ArXiv (2010). https://www.semanticscholar.org/paper/Adiabatic-Quantum-Algorithms-for-the-NP-Complete-Choi/1fcb4d5749074714ca2eb0b56ab986ca66fd0b95
Choi, V.: Minor-embedding in adiabatic quantum computation: I. The parameter setting problem. Quantum Inf. Process. 7(5), 193–209 (2008). https://doi.org/10.1007/s11128-008-0082-9
Cook, S.A.: The complexity of theorem-proving procedures. In: Proceedings of the Third Annual ACM Symposium on Theory of Computing, pp. 151–158. STOC ’71, Association for Computing Machinery, New York, NY, USA (1971). https://doi.org/10.1145/800157.805047
Farhi, E., Goldstone, J., Gutmann, S., Lapan, J., Lundgren, A., Preda, D.: A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292(5516), 472–475 (2001). https://doi.org/10.1126/science.1057726, http://arxiv.org/abs/quant-ph/0104129
Gabor, T., et al.: Assessing solution quality of 3SAT on a quantum annealing platform (2019). http://arxiv.org/abs/1902.04703 [quant-ph] version: 1
Glover, F., Kochenberger, G., Du, Y.: A tutorial on formulating and using QUBO models (2019). https://doi.org/10.48550/arXiv.1811.11538 [quant-ph]
Gonzalez Calaza, C.D., Willsch, D., Michielsen, K.: Garden optimization problems for benchmarking quantum annealers. Quantum Inf. Process. 20, 305 (2021). https://doi.org/10.1007/s11128-021-03226-6
Li, C.M.: Detecting disjoint inconsistent subformulas for computing lower bounds for Max- SAT (2006)
Lucas, A.: Ising formulations of many NP problems. Front. Phys. 2 (2014). https://doi.org/10.3389/fphy.2014.00005, http://arxiv.org/abs/1302.5843 [cond-mat, physics:quant-ph]
Nüßlein, J., Zielinski, S., Gabor, T., Linnhoff-Popien, C., Feld, S.: Solving (Max) 3-SAT via quadratic unconstrained binary optimization (2023). https://doi.org/10.48550/arXiv.2302.03536 [quant-ph]
Schuld, M., Petruccione, F.: Supervised Learning with Quantum Computers. QST, Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96424-9
Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997). https://doi.org/10.1137/S0097539795293172. Publisher: Society for Industrial and Applied Mathematics
Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Siekmann, J.H., Wrightson, G. (eds.) Automation of Reasoning: 2: Classical Papers on Computational Logic 1967–1970, pp. 466–483. Symbolic Computation, Springer, Berlin, Heidelberg (1983). https://doi.org/10.1007/978-3-642-81955-1_28
Acknowledgments
This work was founded partially by the Government of Catalonia Exp. Num. 2023 DI 00071, by Enzyme Advising Group under the development of an industrial PhD on Quantum Machine Learning, by CSIC’s JAE Intro JAEICU_23_00782 and by grant PID2022-136787NB-I00 funded by MCIN/AEI/10.13039/501100011033. Jesus Cerquides is funded by European Union KHealthInAir, GUARDEN, and Humane-AI projects with No. 101057693, 101060693, and 952026.
Author information
Authors and Affiliations
Contributions
Pol Farrés and Rocco Ballester: These authors contributed equally to this work.
Corresponding author
Editor information
Editors and Affiliations
Ethics declarations
Disclosure of Interests.
The authors have no competing interests to declare that are relevant to the content of this article.
A Number of Physical Qubits Required per Gadget
A Number of Physical Qubits Required per Gadget
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Rodríguez-Farrés, P., Ballester, R., Ansótegui, C., Levy, J., Cerquides, J. (2024). Implementing 3-SAT Gadgets for Quantum Annealers with Random Instances. In: Franco, L., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M.A. (eds) Computational Science – ICCS 2024. ICCS 2024. Lecture Notes in Computer Science, vol 14837. Springer, Cham. https://doi.org/10.1007/978-3-031-63778-0_20
Download citation
DOI: https://doi.org/10.1007/978-3-031-63778-0_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-63777-3
Online ISBN: 978-3-031-63778-0
eBook Packages: Computer ScienceComputer Science (R0)