Limitations of Local Quantum Algorithms on Random MAX-k-XOR and Beyond
We introduce a notion of generic local algorithm, which strictly generalizes existing frameworks of local algorithms such as factors of i.i.d. by capturing local quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA).
Motivated by a question of Farhi et al. [arXiv:1910.08187, 2019], we then show limitations of generic local algorithms including QAOA on random instances of constraint satisfaction problems (CSPs). Specifically, we show that any generic local algorithm whose assignment to a vertex depends only on a local neighborhood with o(n) other vertices (such as the QAOA at depth less than εlog(n)) cannot arbitrarily-well approximate boolean CSPs if the problem satisfies a geometric property from statistical physics called the coupled overlap-gap property (OGP) [Chen et al., Annals of Probability, 47(3), 2019]. We show that the random MAX-k-XOR problem has this property when k ≥ 4 is even by extending the corresponding result for diluted k-spin glasses.
Our concentration lemmas confirm a conjecture of Brandao et al. [arXiv:1812.04170, 2018] asserting that the landscape independence of QAOA extends to logarithmic depth - in other words, for every fixed choice of QAOA angle parameters, the algorithm at logarithmic depth performs almost equally well on almost all instances. One of these lemmas is a strengthening of McDiarmid’s inequality, applicable when the random variables have a highly biased distribution, and may be of independent interest.
Quantum Algorithms
Spin Glasses
Hardness of Approximation
Local Algorithms
Concentration Inequalities
Overlap Gap Property
Theory of computation~Randomness, geometry and discrete structures
Mathematics of computing~Probabilistic algorithms
Mathematics of computing~Combinatorics
Theory of computation~Quantum complexity theory
41:1-41:20
Track A: Algorithms, Complexity and Games
https://arxiv.org/abs/2108.06049
We thank Jonathan Wurtz for many insightful discussions about QAOA. We are grateful to Amartya Shankha Biswas for patiently explaining the factors of i.i.d. framework to us. We would also like to thank Antares Chen for many invigorating and profound discussions which culminated as the open problem proposed in Problem 5.1. Lastly, we would like to thank Boaz Barak for providing detailed and helpful feedback on a prior version of this manuscript, and David Gamarnik for his explanations on the state of the art results in the research area.
Chi-Ning
Chou
Chi-Ning Chou
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Supported by NSF awards CCF 1565264 and CNS 1618026.
Peter J.
Love
Peter J. Love
Department of Physics and Astronomy, Tufts University, Medford, MA, USA
Supported by NSF STAQ award PHY-1818914 and DARPA ONISQ program award HR001120C0068.
Juspreet Singh
Sandhu
Juspreet Singh Sandhu
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Supported by NSF STAQ award PHY-1818914 and DARPA ONISQ program award HR001120C0068.
Jonathan
Shi
Jonathan Shi
Department of Computing Sciences, Bocconi University, Milan, Italy
Supported by European Research Council (ERC) award No. 834861.
10.4230/LIPIcs.ICALP.2022.41
Dimitris Achlioptas and Federico Ricci-Tersenghi. On the solution-space geometry of random constraint satisfaction problems. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 130-139, 2006.
Ahmed El Alaoui, Andrea Montanari, and Mark Sellke. Local algorithms for maximum cut and minimum bisection on locally treelike regular graphs of large degree. arXiv preprint, 2021. URL: http://arxiv.org/abs/2111.06813.
http://arxiv.org/abs/2111.06813
Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, et al. Hartree-Fock on a superconducting qubit quantum computer. Science, 369(6507):1084-1089, 2020. URL: https://www.science.org/doi/abs/10.1126/science.abb9811.
https://www.science.org/doi/abs/10.1126/science.abb9811
Antonio Auffinger, Wei-Kuo Chen, and Qiang Zeng. The sk model is infinite step replica symmetry breaking at zero temperature. Communications on Pure and Applied Mathematics, 73(5), 2020.
Boaz Barak and Kunal Marwaha. Classical algorithms and quantum limitations for maximum cut on high-girth graphs. arXiv preprint, 2021. URL: http://arxiv.org/abs/2106.05900.
http://arxiv.org/abs/2106.05900
John S Bell. On the einstein-podolsky-rosen paradox. Physics Physique Fizika, 1(3):195, 1964.
Sally C Brailsford, Chris N Potts, and Barbara M Smith. Constraint satisfaction problems: Algorithms and applications. European journal of operational research, 119(3):557-581, 1999.
Fernando Brandao, Michael Broughton, Edward Farhi, Sam Gutmann, and Hartmut Neven. For fixed control parameters the quantum approximate optimization algorithm’s objective function value concentrates for typical instances. arXiv preprint, 2018. URL: http://arxiv.org/abs/1812.04170.
http://arxiv.org/abs/1812.04170
Alfredo Braunstein, Marc Mézard, and Riccardo Zecchina. Survey propagation: An algorithm for satisfiability. Random Structures & Algorithms, 27(2):201-226, 2005.
Wei-Kuo Chen, David Gamarnik, Dmitry Panchenko, et al. Suboptimality of local algorithms for a class of max-cut problems. Annals of Probability, 47(3):1587-1618, 2019.
Chi-Ning Chou, Peter J Love, Juspreet Singh Sandhu, and Jonathan Shi. Limitations of local quantum algorithms on random max-k-xor and beyond. arXiv preprint v3, 2021. URL: http://arxiv.org/abs/2108.06049.
http://arxiv.org/abs/2108.06049
Francesco Concetti. The full replica symmetry breaking in the ising spin glass on random regular graph. Journal of Statistical Physics, 173(5):1459-1483, 2018.
Giacomo De Palma, Milad Marvian, Dario Trevisan, and Seth Lloyd. The quantum wasserstein distance of order 1. IEEE Transactions on Information Theory, 67(10):6627-6643, 2021.
Amir Dembo, Andrea Montanari, and Subhabrata Sen. Extremal cuts of sparse random graphs. The Annals of Probability, 45(2):1190-1217, 2017.
Jian Ding, Allan Sly, and Nike Sun. Satisfiability threshold for random regular nae-sat. Communications in Mathematical Physics, 341(2):435-489, 2016.
Sepehr Ebadi, Tout T Wang, Harry Levine, et al. Quantum phases of matter on a 256-atom programmable quantum simulator. Nature, 595(7866):227-232, 2021.
Edward Farhi, David Gamarnik, and Sam Gutmann. The quantum approximate optimization algorithm needs to see the whole graph: A typical case. arXiv preprint, 2020. URL: http://arxiv.org/abs/2004.09002.
http://arxiv.org/abs/2004.09002
Edward Farhi, David Gamarnik, and Sam Gutmann. The quantum approximate optimization algorithm needs to see the whole graph: Worst case examples. arXiv preprint, 2020. URL: http://arxiv.org/abs/2005.08747.
http://arxiv.org/abs/2005.08747
Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm. arXiv preprint, 2014. URL: http://arxiv.org/abs/1411.4028.
http://arxiv.org/abs/1411.4028
Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Leo Zhou. The quantum approximate optimization algorithm and the sherrington-kirkpatrick model at infinite size. arXiv preprint, 2019. URL: http://arxiv.org/abs/1910.08187.
http://arxiv.org/abs/1910.08187
Silvio Franz and Michele Leone. Replica bounds for optimization problems and diluted spin systems. Journal of Statistical Physics, 111(3):535-564, 2003.
David Gamarnik and Aukosh Jagannath. The overlap gap property and approximate message passing algorithms for p-spin models. The Annals of Probability, 49(1):180-205, 2021.
David Gamarnik, Aukosh Jagannath, and Alexander S Wein. Low-degree hardness of random optimization problems. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 131-140. IEEE Computer Society, 2020.
David Gamarnik and Madhu Sudan. Limits of local algorithms over sparse random graphs. In Proceedings of the 5th conference on Innovations in theoretical computer science, pages 369-376, 2014.
David Gamarnik and Ilias Zadik. The landscape of the planted clique problem: Dense subgraphs and the overlap gap property. arXiv preprint, 2019. URL: http://arxiv.org/abs/1904.07174.
http://arxiv.org/abs/1904.07174
Ming Gong, Shiyu Wang, Chen Zha, Ming-Cheng Chen, He-Liang Huang, Yulin Wu, Qingling Zhu, Youwei Zhao, Shaowei Li, Shaojun Guo, et al. Quantum walks on a programmable two-dimensional 62-qubit superconducting processor. Science, 372(6545):948-952, 2021.
Francesco Guerra and Fabio Lucio Toninelli. The thermodynamic limit in mean field spin glass models. Communications in Mathematical Physics, 230(1):71-79, 2002.
Francesco Guerra and Fabio Lucio Toninelli. The high temperature region of the viana-bray diluted spin glass model. Journal of statistical physics, 115(1):531-555, 2004.
Matthew B Hastings. Classical and quantum bounded depth approximation algorithms. arXiv preprint, 2019. URL: http://arxiv.org/abs/1905.07047.
http://arxiv.org/abs/1905.07047
Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O’Donnell. Optimal inapproximability results for max-cut and other 2-variable csps? SIAM Journal on Computing, 37(1):319-357, 2007.
Subhash Khot and Nisheeth K Vishnoi. On the unique games conjecture. In FOCS, volume 5, page 3. Citeseer, 2005.
Vipin Kumar. Algorithms for constraint-satisfaction problems: A survey. AI magazine, 13(1):32-32, 1992.
Kunal Marwaha. Local classical max-cut algorithm outperforms p = 2 qaoa on high-girth regular graphs. Quantum, 5:437, 2021.
Marc Mézard and Giorgio Parisi. The bethe lattice spin glass revisited. The European Physical Journal B-Condensed Matter and Complex Systems, 20(2):217-233, 2001.
Dmitry Panchenko. Introduction to the sk model. arXiv preprint, 2014. URL: http://arxiv.org/abs/1412.0170.
http://arxiv.org/abs/1412.0170
Dmitry Panchenko. The parisi formula for mixed p-spin models. The Annals of Probability, 42(3):946-958, 2014.
Dmitry Panchenko and Michel Talagrand. Bounds for diluted mean-fields spin glass models. Probability Theory and Related Fields, 130(3):319-336, 2004.
Giorgio Parisi. A sequence of approximated solutions to the sk model for spin glasses. Journal of Physics A: Mathematical and General, 13(4):L115, 1980.
Giorgio Parisi, Federico Ricci-Tersenghi, and Tommaso Rizzo. Diluted mean-field spin-glass models at criticality. Journal of Statistical Mechanics: Theory and Experiment, 2014(4):P04013, 2014.
John Preskill. Quantum computing in the nisq era and beyond. Quantum, 2:79, 2018.
Prasad Raghavendra. Optimal algorithms and inapproximability results for every csp? In Proceedings of the 40th annual ACM symposium on Theory of computing, pages 245-254, 2008.
Subhabrata Sen. Optimization on sparse random hypergraphs and spin glasses. Random Structures & Algorithms, 53(3):504-536, 2018.
David Sherrington and Scott Kirkpatrick. Solvable model of a spin-glass. Physical review letters, 35(26):1792, 1975.
Michel Talagrand. The parisi formula. Annals of mathematics, pages 221-263, 2006.
Alexander S Wein. Optimal low-degree hardness of maximum independent set. Mathematical Statistics and Learning, 2022.
Jonathan Yedidia, William Freeman, Yair Weiss, et al. Understanding belief propagation and its generalizations. Exploring artificial intelligence in the new millennium, 8:236-239, 2003.
Chi-Ning Chou, Peter J. Love, Juspreet Singh Sandhu, and Jonathan Shi
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode