Local classical MAX-CUT algorithm outperforms $p=2$ QAOA on high-girth regular graphs

Kunal Marwaha

Berkeley Center for Quantum Information and Computation, University of California, Berkeley, California 94720, USA

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The $p$-stage Quantum Approximate Optimization Algorithm (QAOA$_p$) is a promising approach for combinatorial optimization on noisy intermediate-scale quantum (NISQ) devices, but its theoretical behavior is not well understood beyond $p=1$. We analyze QAOA$_2$ for the $\textit{maximum cut problem}$ (MAX-CUT), deriving a graph-size-independent expression for the expected cut fraction on any $D$-regular graph of girth $> 5$ (i.e. without triangles, squares, or pentagons).

We show that for all degrees $D \ge 2$ and every $D$-regular graph $G$ of girth $> 5$, QAOA$_2$ has a larger expected cut fraction than QAOA$_1$ on $G$. However, we also show that there exists a $2$-local randomized $\textit{classical}$ algorithm $A$ such that $A$ has a larger expected cut fraction than QAOA$_2$ on all $G$. This supports our conjecture that for every constant $p$, there exists a local classical MAX-CUT algorithm that performs as well as QAOA$_p$ on all graphs.

For further reading go to our presentation Local Competitors to QAOA


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► References

[1] F. Arute, K. Arya, R. Babbush, et al. Quantum approximate optimization of non-planar graph problems on a planar superconducting processor. 2004.04197, doi:10.1038/​s41567-020-01105-y.

[2] B. Barak, A. Moitra, R. O'Donnell, et al. Beating the random assignment on constraint satisfaction problems of bounded degree. 1505.03424.

[3] E. Farhi, J. Goldstone, and S. Gutmann. A quantum approximate optimization algorithm. 1411.4028.

[4] E. Farhi, J. Goldstone, and S. Gutmann. A quantum approximate optimization algorithm applied to a bounded occurrence constraint problem. 1412.6062.

[5] M. X. Goemans and D. P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM (JACM), 42(6):1115–1145, 1995. doi:10.1145/​227683.227684, URL http:/​/​www-math.mit.edu/​ goemans/​PAPERS/​maxcut-jacm.pdf.

[6] M. B. Hastings. Classical and quantum bounded depth approximation algorithms. 1905.07047.

[7] J. Hirvonen, J. Rybicki, S. Schmid, and J. Suomela. Large cuts with local algorithms on triangle-free graphs. 1402.2543.

[8] S. Khot. On the power of unique 2-prover 1-round games. In Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, STOC '02, page 767–775, New York, NY, USA, 2002. Association for Computing Machinery. doi:10.1145/​509907.510017.

[9] G. B. Mbeng, R. Fazio, and G. Santoro. Quantum annealing: a journey through digitalization, control, and hybrid quantum variational schemes. 1906.08948.

[10] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18(2):023023, feb 2016. doi:10.1088/​1367-2630/​18/​2/​023023.

[11] A. Meurer, C. P. Smith, M. Paprocki, et al. Sympy: symbolic computing in python. PeerJ Computer Science, 3:e103, Jan. 2017. doi:10.7717/​peerj-cs.103.

[12] F. Pérez and B. E. Granger. IPython: a system for interactive scientific computing. Computing in Science and Engineering, 9(3):21–29, May 2007. doi:10.1109/​MCSE.2007.53, URL https:/​/​ipython.org.

[13] J. Preskill. Quantum computing in the nisq era and beyond. Quantum, 2:79, Aug 2018. doi:10.22331/​q-2018-08-06-79.

[14] J. B. Shearer. A note on bipartite subgraphs of triangle-free graphs. Random Structures & Algorithms, 3(2):223–226, 1992. doi:10.1002/​rsa.3240030211.

[15] M. Szegedy. What do qaoa energies reveal about graphs? 1912.12277.

[16] P. Virtanen, R. Gommers, T. E. Oliphant, et al. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods, 17:261–272, 2020. doi:10.1038/​s41592-019-0686-2.

[17] Z. Wang, S. Hadfield, Z. Jiang, and E. G. Rieffel. Quantum approximate optimization algorithm for maxcut: A fermionic view. Phys. Rev. A, 97:022304, Feb 2018. doi:10.1103/​PhysRevA.97.022304.

[18] J. Wurtz and P. J. Love. Bounds on maxcut qaoa performance for p$>$1. 2010.11209.

[19] L. Zhou, S.-T. Wang, S. Choi, et al. Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices. Physical Review X, 10(2), Jun 2020. doi:10.1103/​physrevx.10.021067.

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[2] Jordi R. Weggemans, Alexander Urech, Alexander Rausch, Robert Spreeuw, Richard Boucherie, Florian Schreck, Kareljan Schoutens, Jiří Minář, and Florian Speelman, "Solving correlation clustering with QAOA and a Rydberg qudit system: a full-stack approach", Quantum 6, 687 (2022).

[3] Kunal Marwaha and Stuart Hadfield, "Bounds on approximating Max kXOR with quantum and classical local algorithms", Quantum 6, 757 (2022).

[4] Karen Wintersperger, Hila Safi, and Wolfgang Mauerer, Lecture Notes in Computer Science 13642, 100 (2022) ISBN:978-3-031-21866-8.

[5] Zeqiao Zhou, Yuxuan Du, Xinmei Tian, and Dacheng Tao, "QAOA-in-QAOA: Solving Large-Scale MaxCut Problems on Small Quantum Machines", Physical Review Applied 19 2, 024027 (2023).

[6] Daniel J. Egger, Jakub Mareček, and Stefan Woerner, "Warm-starting quantum optimization", Quantum 5, 479 (2021).

[7] Phillip C. Lotshaw, Thien Nguyen, Anthony Santana, Alexander McCaskey, Rebekah Herrman, James Ostrowski, George Siopsis, and Travis S. Humble, "Scaling quantum approximate optimization on near-term hardware", Scientific Reports 12 1, 12388 (2022).

[8] Taylor L. Patti, Jean Kossaifi, Anima Anandkumar, and Susanne F. Yelin, "Variational quantum optimization with multibasis encodings", Physical Review Research 4 3, 033142 (2022).

[9] Xinwei Lee, Ningyi Xie, Dongsheng Cai, Yoshiyuki Saito, and Nobuyoshi Asai, "A Depth-Progressive Initialization Strategy for Quantum Approximate Optimization Algorithm", Mathematics 11 9, 2176 (2023).

[10] Ramin Fakhimi and Hamidreza Validi, Encyclopedia of Optimization 1 (2023) ISBN:978-3-030-54621-2.

[11] T. Miki, D. Tsukayama, R. Okita, M Shimada, and J. Shirakashi, 2022 IEEE 22nd International Conference on Nanotechnology (NANO) 303 (2022) ISBN:978-1-6654-5225-0.

[12] Stuart Hadfield, Tad Hogg, and Eleanor G Rieffel, "Analytical framework for quantum alternating operator ansätze", Quantum Science and Technology 8 1, 015017 (2023).

[13] Lis Arufe, Riccardo Rasconi, Angelo Oddi, Ramiro Varela, and Miguel A. González, "New coding scheme to compile circuits for Quantum Approximate Optimization Algorithm by genetic evolution", Applied Soft Computing 110456 (2023).

[14] Joao Basso, Edward Farhi, Kunal Marwaha, Benjamin Villalonga, and Leo Zhou, "The Quantum Approximate Optimization Algorithm at High Depth for MaxCut on Large-Girth Regular Graphs and the Sherrington-Kirkpatrick Model", arXiv:2110.14206, (2021).

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