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Graph neural network initialisation of quantum approximate optimisation

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Nishant Jain1, Brian Coyle2, Elham Kashefi2,3, and Niraj Kumar2

1Indian Institute of Technology, Roorkee, India.
2School of Informatics, University of Edinburgh, EH8 9AB Edinburgh, United Kingdom.
3LIP6, CNRS, Sorbonne Université, 4 place Jussieu, 75005 Paris, France.

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Abstract

Approximate combinatorial optimisation has emerged as one of the most promising application areas for quantum computers, particularly those in the near term. In this work, we focus on the quantum approximate optimisation algorithm (QAOA) for solving the MaxCut problem. Specifically, we address two problems in the QAOA, how to initialise the algorithm, and how to subsequently train the parameters to find an optimal solution. For the former, we propose graph neural networks (GNNs) as a warm-starting technique for QAOA. We demonstrate that merging GNNs with QAOA can outperform both approaches individually. Furthermore, we demonstrate how graph neural networks enables warm-start generalisation across not only graph instances, but also to increasing graph sizes, a feature not straightforwardly available to other warm-starting methods. For training the QAOA, we test several optimisers for the MaxCut problem up to 16 qubits and benchmark against vanilla gradient descent. These include quantum aware/agnostic and machine learning based/neural optimisers. Examples of the latter include reinforcement and meta-learning. With the incorporation of these initialisation and optimisation toolkits, we demonstrate how the optimisation problems can be solved using QAOA in an end-to-end differentiable pipeline.

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► References

[1] John Preskill. Quantum Computing in the NISQ era and beyond. Quantum, 2:79, August 2018. URL: https:/​/​quantum-journal.org/​papers/​q-2018-08-06-79/​, doi:10.22331/​q-2018-08-06-79.
https:/​/​doi.org/​10.22331/​q-2018-08-06-79
https:/​/​quantum-journal.org/​papers/​q-2018-08-06-79/​

[2] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O’Brien. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5(1):1–7, July 2014. URL: https:/​/​www.nature.com/​articles/​ncomms5213, doi:10.1038/​ncomms5213.
https:/​/​doi.org/​10.1038/​ncomms5213
https:/​/​www.nature.com/​articles/​ncomms5213

[3] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A Quantum Approximate Optimization Algorithm. arXiv:1411.4028 [quant-ph], November 2014. URL: http:/​/​arxiv.org/​abs/​1411.4028, doi:10.48550/​arXiv.1411.4028.
https:/​/​doi.org/​10.48550/​arXiv.1411.4028
arXiv:1411.4028

[4] Jarrod R. McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18(2):023023, February 2016. URL:.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​2/​023023

[5] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles. Variational quantum algorithms. Nature Reviews Physics, 3(9):625–644, September 2021. URL: https:/​/​www.nature.com/​articles/​s42254-021-00348-9, doi:10.1038/​s42254-021-00348-9.
https:/​/​doi.org/​10.1038/​s42254-021-00348-9
https:/​/​www.nature.com/​articles/​s42254-021-00348-9

[6] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik. Noisy intermediate-scale quantum algorithms. Rev. Mod. Phys., 94(1):015004, February 2022. URL: https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.94.015004, doi:10.1103/​RevModPhys.94.015004.
https:/​/​doi.org/​10.1103/​RevModPhys.94.015004

[7] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii. Quantum circuit learning. Phys. Rev. A, 98(3):032309, September 2018. URL: https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.98.032309, doi:10.1103/​PhysRevA.98.032309.
https:/​/​doi.org/​10.1103/​PhysRevA.98.032309

[8] Edward Farhi and Hartmut Neven. Classification with Quantum Neural Networks on Near Term Processors. arXiv:1802.06002 [quant-ph], February 2018. URL: http:/​/​arxiv.org/​abs/​1802.06002, doi:10.48550/​arXiv.1802.06002.
https:/​/​doi.org/​10.48550/​arXiv.1802.06002
arXiv:1802.06002

[9] Marcello Benedetti, Erika Lloyd, Stefan Sack, and Mattia Fiorentini. Parameterized quantum circuits as machine learning models. Quantum Sci. Technol., 4(4):043001, November 2019. URL:.
https:/​/​doi.org/​10.1088/​2058-9565/​ab4eb5

[10] Francisco Barahona, Martin Grötschel, Michael Jünger, and Gerhard Reinelt. An application of combinatorial optimization to statistical physics and circuit layout design. Operations Research, 36(3):493–513, 1988. URL: http:/​/​jstor.org/​stable/​170992.
http:/​/​jstor.org/​stable/​170992

[11] Jan Poland and Thomas Zeugmann. Clustering Pairwise Distances with Missing Data: Maximum Cuts Versus Normalized Cuts. In Ljupco Todorovski, Nada Lavrac, and Klaus P. Jantke, editors, Discovery Science, 9th International Conference, DS 2006, Barcelona, Spain, October 7-10, 2006, Proceedings, volume 4265 of Lecture Notes in Computer Science, pages 197–208. Springer, 2006. URL: https:/​/​doi.org/​10.1007/​11893318_21, doi:10.1007/​11893318_21.
https:/​/​doi.org/​10.1007/​11893318_21

[12] Michael A. Nielsen and Isaac L. Chuang. Quantum computation and quantum information. Cambridge University Press, Cambridge ; New York, 10th anniversary ed edition, 2010. doi:10.1017/​CBO9780511976667.
https:/​/​doi.org/​10.1017/​CBO9780511976667

[13] Matthew B. Hastings. Classical and quantum bounded depth approximation algorithms. Quantum Inf. Comput., 19(13&14):1116–1140, 2019. doi:10.26421/​QIC19.13-14-3.
https:/​/​doi.org/​10.26421/​QIC19.13-14-3

[14] Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Leo Zhou. The Quantum Approximate Optimization Algorithm and the Sherrington-Kirkpatrick Model at Infinite Size. Quantum, 6:759, July 2022. URL: https:/​/​quantum-journal.org/​papers/​q-2022-07-07-759/​, doi:10.22331/​q-2022-07-07-759.
https:/​/​doi.org/​10.22331/​q-2022-07-07-759
https:/​/​quantum-journal.org/​papers/​q-2022-07-07-759/​

[15] Daniel Stilck França and Raul García-Patrón. Limitations of optimization algorithms on noisy quantum devices. Nature Physics, 17(11):1221–1227, November 2021. URL: https:/​/​www.nature.com/​articles/​s41567-021-01356-3, doi:10.1038/​s41567-021-01356-3.
https:/​/​doi.org/​10.1038/​s41567-021-01356-3
https:/​/​www.nature.com/​articles/​s41567-021-01356-3

[16] V. Akshay, H. Philathong, M. E. S. Morales, and J. D. Biamonte. Reachability Deficits in Quantum Approximate Optimization. Phys. Rev. Lett., 124(9):090504, March 2020. URL: https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.124.090504, doi:10.1103/​PhysRevLett.124.090504.
https:/​/​doi.org/​10.1103/​PhysRevLett.124.090504

[17] Sami Boulebnane. Improving the Quantum Approximate Optimization Algorithm with postselection. arXiv:2011.05425 [quant-ph], November 2020. URL: http:/​/​arxiv.org/​abs/​2011.05425, doi:10.48550/​arXiv.2011.05425.
https:/​/​doi.org/​10.48550/​arXiv.2011.05425
arXiv:2011.05425

[18] V. Akshay, D. Rabinovich, E. Campos, and J. Biamonte. Parameter Concentration in Quantum Approximate Optimization. Physical Review A, 104(1):L010401, July 2021. URL: http:/​/​arxiv.org/​abs/​2103.11976, doi:10.1103/​PhysRevA.104.L010401.
https:/​/​doi.org/​10.1103/​PhysRevA.104.L010401
arXiv:2103.11976

[19] D. Rabinovich, R. Sengupta, E. Campos, V. Akshay, and J. Biamonte. Progress towards analytically optimal angles in quantum approximate optimisation. arXiv:2109.11566 [math-ph, physics:quant-ph], September 2021. URL: http:/​/​arxiv.org/​abs/​2109.11566.
https:/​/​doi.org/​10.3390/​math10152601
arXiv:2109.11566

[20] 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. In François Le Gall and Tomoyuki Morimae, editors, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022), volume 232 of Leibniz International Proceedings in Informatics (LIPIcs), pages 7:1–7:21, Dagstuhl, Germany, 2022. Schloss Dagstuhl – Leibniz-Zentrum für Informatik. URL: https:/​/​drops.dagstuhl.de/​opus/​volltexte/​2022/​16514, doi:10.4230/​LIPIcs.TQC.2022.7.
https:/​/​doi.org/​10.4230/​LIPIcs.TQC.2022.7
https:/​/​drops.dagstuhl.de/​opus/​volltexte/​2022/​16514

[21] Stuart Hadfield, Zhihui Wang, Bryan O’Gorman, Eleanor G. Rieffel, Davide Venturelli, and Rupak Biswas. From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz. Algorithms, 12(2):34, February 2019. URL: https:/​/​www.mdpi.com/​1999-4893/​12/​2/​34, doi:10.3390/​a12020034.
https:/​/​doi.org/​10.3390/​a12020034
https:/​/​www.mdpi.com/​1999-4893/​12/​2/​34

[22] Ryan LaRose, Eleanor Rieffel, and Davide Venturelli. Mixer-Phaser Ansätze for Quantum Optimization with Hard Constraints. arXiv:2107.06651 [quant-ph], July 2021. URL: http:/​/​arxiv.org/​abs/​2107.06651, doi:10.48550/​arXiv.2107.06651.
https:/​/​doi.org/​10.48550/​arXiv.2107.06651
arXiv:2107.06651

[23] Linghua Zhu, Ho Lun Tang, George S. Barron, F. A. Calderon-Vargas, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou. Adaptive quantum approximate optimization algorithm for solving combinatorial problems on a quantum computer. Phys. Rev. Research, 4(3):033029, July 2022. URL: https:/​/​link.aps.org/​doi/​10.1103/​PhysRevResearch.4.033029, doi:10.1103/​PhysRevResearch.4.033029.
https:/​/​doi.org/​10.1103/​PhysRevResearch.4.033029

[24] Stuart Hadfield, Tad Hogg, and Eleanor G. Rieffel. Analytical Framework for Quantum Alternating Operator Ansätze. arXiv:2105.06996 [quant-ph], May 2021. URL: http:/​/​arxiv.org/​abs/​2105.06996, doi:10.48550/​arXiv.2105.06996.
https:/​/​doi.org/​10.48550/​arXiv.2105.06996
arXiv:2105.06996

[25] Guillaume Verdon, Juan Miguel Arrazola, Kamil Brádler, and Nathan Killoran. A Quantum Approximate Optimization Algorithm for continuous problems. arXiv:1902.00409 [quant-ph], February 2019. URL: http:/​/​arxiv.org/​abs/​1902.00409, doi:10.48550/​arXiv.1902.00409.
https:/​/​doi.org/​10.48550/​arXiv.1902.00409
arXiv:1902.00409

[26] Panagiotis Kl Barkoutsos, Giacomo Nannicini, Anton Robert, Ivano Tavernelli, and Stefan Woerner. Improving Variational Quantum Optimization using CVaR. Quantum, 4:256, April 2020. URL: https:/​/​quantum-journal.org/​papers/​q-2020-04-20-256/​, doi:10.22331/​q-2020-04-20-256.
https:/​/​doi.org/​10.22331/​q-2020-04-20-256
https:/​/​quantum-journal.org/​papers/​q-2020-04-20-256/​

[27] Ioannis Kolotouros and Petros Wallden. Evolving objective function for improved variational quantum optimization. Phys. Rev. Research, 4(2):023225, June 2022. URL: https:/​/​link.aps.org/​doi/​10.1103/​PhysRevResearch.4.023225, doi:10.1103/​PhysRevResearch.4.023225.
https:/​/​doi.org/​10.1103/​PhysRevResearch.4.023225

[28] David Amaro, Carlo Modica, Matthias Rosenkranz, Mattia Fiorentini, Marcello Benedetti, and Michael Lubasch. Filtering variational quantum algorithms for combinatorial optimization. Quantum Science and Technology, 7(1):015021, January 2022. doi:10.1088/​2058-9565/​ac3e54.
https:/​/​doi.org/​10.1088/​2058-9565/​ac3e54

[29] Daniel J. Egger, Jakub Mareček, and Stefan Woerner. Warm-starting quantum optimization. Quantum, 5:479, June 2021. URL: http:/​/​dx.doi.org/​10.22331/​q-2021-06-17-479, doi:10.22331/​q-2021-06-17-479.
https:/​/​doi.org/​10.22331/​q-2021-06-17-479

[30] Stefan H. Sack and Maksym Serbyn. Quantum annealing initialization of the quantum approximate optimization algorithm. Quantum, 5:491, July 2021. URL: http:/​/​dx.doi.org/​10.22331/​q-2021-07-01-491, doi:10.22331/​q-2021-07-01-491.
https:/​/​doi.org/​10.22331/​q-2021-07-01-491

[31] Gian Giacomo Guerreschi and Mikhail Smelyanskiy. Practical optimization for hybrid quantum-classical algorithms. arXiv:1701.01450 [quant-ph], January 2017. URL: http:/​/​arxiv.org/​abs/​1701.01450, doi:10.48550/​arXiv.1701.01450.
https:/​/​doi.org/​10.48550/​arXiv.1701.01450
arXiv:1701.01450

[32] Nikolaj Moll, Panagiotis Barkoutsos, Lev S Bishop, Jerry M Chow, Andrew Cross, Daniel J Egger, Stefan Filipp, Andreas Fuhrer, Jay M Gambetta, Marc Ganzhorn, and et al. Quantum optimization using variational algorithms on near-term quantum devices. Quantum Science and Technology, 3(3):030503, June 2018. URL: http:/​/​dx.doi.org/​10.1088/​2058-9565/​aab822, doi:10.1088/​2058-9565/​aab822.
https:/​/​doi.org/​10.1088/​2058-9565/​aab822

[33] Sami Khairy, Ruslan Shaydulin, Lukasz Cincio, Yuri Alexeev, and Prasanna Balaprakash. Reinforcement-Learning-Based Variational Quantum Circuits Optimization for Combinatorial Problems. arXiv:1911.04574 [quant-ph, stat], November 2019. URL: http:/​/​arxiv.org/​abs/​1911.04574, doi:10.48550/​arXiv.1911.04574.
https:/​/​doi.org/​10.48550/​arXiv.1911.04574
arXiv:1911.04574

[34] Michael Streif and Martin Leib. Training the quantum approximate optimization algorithm without access to a quantum processing unit. Quantum Science and Technology, 5(3):034008, May 2020. doi:10.1088/​2058-9565/​ab8c2b.
https:/​/​doi.org/​10.1088/​2058-9565/​ab8c2b

[35] Leo Zhou, Sheng-Tao Wang, Soonwon Choi, Hannes Pichler, and Mikhail D. Lukin. Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices. Phys. Rev. X, 10(2):021067, June 2020. URL: https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.10.021067, doi:10.1103/​PhysRevX.10.021067.
https:/​/​doi.org/​10.1103/​PhysRevX.10.021067

[36] David Amaro, Matthias Rosenkranz, Nathan Fitzpatrick, Koji Hirano, and Mattia Fiorentini. A case study of variational quantum algorithms for a job shop scheduling problem. EPJ Quantum Technology, 9(1):1–20, December 2022. URL: https:/​/​epjquantumtechnology.springeropen.com/​articles/​10.1140/​epjqt/​s40507-022-00123-4, doi:10.1140/​epjqt/​s40507-022-00123-4.
https:/​/​doi.org/​10.1140/​epjqt/​s40507-022-00123-4

[37] Matthew P. Harrigan, Kevin J. Sung, Matthew Neeley, Kevin J. Satzinger, Frank Arute, Kunal Arya, Juan Atalaya, Joseph C. Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B. Buckley, David A. Buell, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Sean Demura, Andrew Dunsworth, Daniel Eppens, Austin Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Rob Graff, Steve Habegger, Alan Ho, Sabrina Hong, Trent Huang, L. B. Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul V. Klimov, Alexander N. Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Mike Lindmark, Martin Leib, Orion Martin, John M. Martinis, Jarrod R. McClean, Matt McEwen, Anthony Megrant, Xiao Mi, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Charles Neill, Florian Neukart, Murphy Yuezhen Niu, Thomas E. O’Brien, Bryan O’Gorman, Eric Ostby, Andre Petukhov, Harald Putterman, Chris Quintana, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Andrea Skolik, Vadim Smelyanskiy, Doug Strain, Michael Streif, Marco Szalay, Amit Vainsencher, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Leo Zhou, Hartmut Neven, Dave Bacon, Erik Lucero, Edward Farhi, and Ryan Babbush. Quantum approximate optimization of non-planar graph problems on a planar superconducting processor. Nature Physics, 17(3):332–336, March 2021. URL: https:/​/​www.nature.com/​articles/​s41567-020-01105-y, doi:10.1038/​s41567-020-01105-y.
https:/​/​doi.org/​10.1038/​s41567-020-01105-y
https:/​/​www.nature.com/​articles/​s41567-020-01105-y

[38] Johannes Weidenfeller, Lucia C. Valor, Julien Gacon, Caroline Tornow, Luciano Bello, Stefan Woerner, and Daniel J. Egger. Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware, February 2022. URL: http:/​/​arxiv.org/​abs/​2202.03459, doi:10.48550/​arXiv.2202.03459.
https:/​/​doi.org/​10.48550/​arXiv.2202.03459
arXiv:2202.03459

[39] Cheng Xue, Zhao-Yun Chen, Yu-Chun Wu, and Guo-Ping Guo. Effects of Quantum Noise on Quantum Approximate Optimization Algorithm. Chinese Physics Letters, 38(3):030302, March 2021. URL: https:/​/​doi.org/​10.1088/​0256-307x/​38/​3/​030302, doi:10.1088/​0256-307X/​38/​3/​030302.
https:/​/​doi.org/​10.1088/​0256-307x/​38/​3/​030302

[40] Jeffrey Marshall, Filip Wudarski, Stuart Hadfield, and Tad Hogg. Characterizing local noise in QAOA circuits. IOP SciNotes, 1(2):025208, August 2020. doi:10.1088/​2633-1357/​abb0d7.
https:/​/​doi.org/​10.1088/​2633-1357/​abb0d7

[41] Ryan LaRose. Overview and Comparison of Gate Level Quantum Software Platforms. Quantum, 3:130, March 2019. URL: https:/​/​quantum-journal.org/​papers/​q-2019-03-25-130/​, doi:10.22331/​q-2019-03-25-130.
https:/​/​doi.org/​10.22331/​q-2019-03-25-130
https:/​/​quantum-journal.org/​papers/​q-2019-03-25-130/​

[42] Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes. Nature Communications, 9(1):4812, November 2018. URL: https:/​/​www.nature.com/​articles/​s41467-018-07090-4, doi:10.1038/​s41467-018-07090-4.
https:/​/​doi.org/​10.1038/​s41467-018-07090-4
https:/​/​www.nature.com/​articles/​s41467-018-07090-4

[43] Roeland Wiersema, Cunlu Zhou, Yvette de Sereville, Juan Felipe Carrasquilla, Yong Baek Kim, and Henry Yuen. Exploring Entanglement and Optimization within the Hamiltonian Variational Ansatz. PRX Quantum, 1(2):020319, December 2020. URL: https:/​/​link.aps.org/​doi/​10.1103/​PRXQuantum.1.020319, doi:10.1103/​PRXQuantum.1.020319.
https:/​/​doi.org/​10.1103/​PRXQuantum.1.020319

[44] M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nature Communications, 12(1):1791, March 2021. URL: https:/​/​www.nature.com/​articles/​s41467-021-21728-w, doi:10.1038/​s41467-021-21728-w.
https:/​/​doi.org/​10.1038/​s41467-021-21728-w
https:/​/​www.nature.com/​articles/​s41467-021-21728-w

[45] Martin Larocca, Piotr Czarnik, Kunal Sharma, Gopikrishnan Muraleedharan, Patrick J. Coles, and M. Cerezo. Diagnosing barren plateaus with tools from quantum optimal control, March 2022. URL: http:/​/​arxiv.org/​abs/​2105.14377, doi:10.48550/​arXiv.2105.14377.
https:/​/​doi.org/​10.48550/​arXiv.2105.14377
arXiv:2105.14377

[46] Xuchen You and Xiaodi Wu. Exponentially Many Local Minima in Quantum Neural Networks. In Marina Meila and Tong Zhang, editors, Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pages 12144–12155. PMLR, July 2021. URL: https:/​/​proceedings.mlr.press/​v139/​you21c.html, doi:10.48550/​arXiv.2110.02479.
https:/​/​doi.org/​10.48550/​arXiv.2110.02479
https:/​/​proceedings.mlr.press/​v139/​you21c.html

[47] Javier Rivera-Dean, Patrick Huembeli, Antonio Acín, and Joseph Bowles. Avoiding local minima in Variational Quantum Algorithms with Neural Networks. arXiv:2104.02955 [quant-ph], April 2021. URL: http:/​/​arxiv.org/​abs/​2104.02955, doi:10.48550/​arXiv.2104.02955.
https:/​/​doi.org/​10.48550/​arXiv.2104.02955
arXiv:2104.02955

[48] Andrew Arrasmith, Zoe Holmes, Marco Cerezo, and Patrick J Coles. Equivalence of quantum barren plateaus to cost concentration and narrow gorges. Quantum Science and Technology, 2022. URL: http:/​/​iopscience.iop.org/​article/​10.1088/​2058-9565/​ac7d06, doi:10.1088/​2058-9565/​ac7d06.
https:/​/​doi.org/​10.1088/​2058-9565/​ac7d06

[49] James Dborin, Fergus Barratt, Vinul Wimalaweera, Lewis Wright, and Andrew G. Green. Matrix product state pre-training for quantum machine learning. Quantum Science and Technology, 7(3):035014, May 2022. doi:10.1088/​2058-9565/​ac7073.
https:/​/​doi.org/​10.1088/​2058-9565/​ac7073

[50] Guillaume Verdon, Michael Broughton, Jarrod R. McClean, Kevin J. Sung, Ryan Babbush, Zhang Jiang, Hartmut Neven, and Masoud Mohseni. Learning to learn with quantum neural networks via classical neural networks. arXiv:1907.05415 [quant-ph], July 2019. URL: http:/​/​arxiv.org/​abs/​1907.05415, doi:10.48550/​arXiv.1907.05415.
https:/​/​doi.org/​10.48550/​arXiv.1907.05415
arXiv:1907.05415

[51] Frederic Sauvage, Sukin Sim, Alexander A. Kunitsa, William A. Simon, Marta Mauri, and Alejandro Perdomo-Ortiz. FLIP: A flexible initializer for arbitrarily-sized parametrized quantum circuits, May 2021. arXiv:2103.08572 [quant-ph]. URL: http:/​/​arxiv.org/​abs/​2103.08572, doi:10.48550/​arXiv.2103.08572.
https:/​/​doi.org/​10.48550/​arXiv.2103.08572
arXiv:2103.08572

[52] Alba Cervera-Lierta, Jakob S. Kottmann, and Alán Aspuru-Guzik. Meta-Variational Quantum Eigensolver: Learning Energy Profiles of Parameterized Hamiltonians for Quantum Simulation. PRX Quantum, 2(2):020329, May 2021. URL: https:/​/​link.aps.org/​doi/​10.1103/​PRXQuantum.2.020329, doi:10.1103/​PRXQuantum.2.020329.
https:/​/​doi.org/​10.1103/​PRXQuantum.2.020329

[53] Weichi Yao, Afonso S. Bandeira, and Soledad Villar. Experimental performance of graph neural networks on random instances of max-cut. In Wavelets and Sparsity XVIII, volume 11138, page 111380S. International Society for Optics and Photonics, September 2019. URL: https:/​/​www.spiedigitallibrary.org/​conference-proceedings-of-spie/​11138/​111380S/​Experimental-performance-of-graph-neural-networks-on-random-instances-of/​10.1117/​12.2529608.short, doi:10.1117/​12.2529608.
https:/​/​doi.org/​10.1117/​12.2529608

[54] Quentin Cappart, Didier Chételat, Elias B. Khalil, Andrea Lodi, Christopher Morris, and Petar Veličković. Combinatorial Optimization and Reasoning with Graph Neural Networks. In Zhi-Hua Zhou, editor, Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence, IJCAI-21, pages 4348–4355. International Joint Conferences on Artificial Intelligence Organization, August 2021. doi:10.24963/​ijcai.2021/​595.
https:/​/​doi.org/​10.24963/​ijcai.2021/​595

[55] James Kotary, Ferdinando Fioretto, Pascal Van Hentenryck, and Bryan Wilder. End-to-End Constrained Optimization Learning: A Survey. In Zhi-Hua Zhou, editor, Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence, IJCAI-21, pages 4475–4482. International Joint Conferences on Artificial Intelligence Organization, August 2021. doi:10.24963/​ijcai.2021/​610.
https:/​/​doi.org/​10.24963/​ijcai.2021/​610

[56] Martin J. A. Schuetz, J. Kyle Brubaker, and Helmut G. Katzgraber. Combinatorial optimization with physics-inspired graph neural networks. Nature Machine Intelligence, 4(4):367–377, April 2022. URL: https:/​/​www.nature.com/​articles/​s42256-022-00468-6, doi:10.1038/​s42256-022-00468-6.
https:/​/​doi.org/​10.1038/​s42256-022-00468-6
https:/​/​www.nature.com/​articles/​s42256-022-00468-6

[57] Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, Shahnawaz Ahmed, Vishnu Ajith, M. Sohaib Alam, Guillermo Alonso-Linaje, B. AkashNarayanan, Ali Asadi, Juan Miguel Arrazola, Utkarsh Azad, Sam Banning, Carsten Blank, Thomas R. Bromley, Benjamin A. Cordier, Jack Ceroni, Alain Delgado, Olivia Di Matteo, Amintor Dusko, Tanya Garg, Diego Guala, Anthony Hayes, Ryan Hill, Aroosa Ijaz, Theodor Isacsson, David Ittah, Soran Jahangiri, Prateek Jain, Edward Jiang, Ankit Khandelwal, Korbinian Kottmann, Robert A. Lang, Christina Lee, Thomas Loke, Angus Lowe, Keri McKiernan, Johannes Jakob Meyer, J. A. Montañez-Barrera, Romain Moyard, Zeyue Niu, Lee James O’Riordan, Steven Oud, Ashish Panigrahi, Chae-Yeun Park, Daniel Polatajko, Nicolás Quesada, Chase Roberts, Nahum Sá, Isidor Schoch, Borun Shi, Shuli Shu, Sukin Sim, Arshpreet Singh, Ingrid Strandberg, Jay Soni, Antal Száva, Slimane Thabet, Rodrigo A. Vargas-Hernández, Trevor Vincent, Nicola Vitucci, Maurice Weber, David Wierichs, Roeland Wiersema, Moritz Willmann, Vincent Wong, Shaoming Zhang, and Nathan Killoran. PennyLane: Automatic differentiation of hybrid quantum-classical computations, July 2022. arXiv:1811.04968 [physics, physics:quant-ph]. URL: http:/​/​arxiv.org/​abs/​1811.04968, doi:10.48550/​arXiv.1811.04968.
https:/​/​doi.org/​10.48550/​arXiv.1811.04968
arXiv:1811.04968

[58] Michael Broughton, Guillaume Verdon, Trevor McCourt, Antonio J. Martinez, Jae Hyeon Yoo, Sergei V. Isakov, Philip Massey, Ramin Halavati, Murphy Yuezhen Niu, Alexander Zlokapa, Evan Peters, Owen Lockwood, Andrea Skolik, Sofiene Jerbi, Vedran Dunjko, Martin Leib, Michael Streif, David Von Dollen, Hongxiang Chen, Shuxiang Cao, Roeland Wiersema, Hsin-Yuan Huang, Jarrod R. McClean, Ryan Babbush, Sergio Boixo, Dave Bacon, Alan K. Ho, Hartmut Neven, and Masoud Mohseni. TensorFlow Quantum: A Software Framework for Quantum Machine Learning, August 2021. arXiv:2003.02989 [cond-mat, physics:quant-ph]. URL: http:/​/​arxiv.org/​abs/​2003.02989, doi:10.48550/​arXiv.2003.02989.
https:/​/​doi.org/​10.48550/​arXiv.2003.02989
arXiv:2003.02989

[59] Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Yee Whye Teh and Mike Titterington, editors, Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, volume 9 of Proceedings of Machine Learning Research, pages 249–256, Chia Laguna Resort, Sardinia, Italy, May 2010. PMLR. URL: https:/​/​proceedings.mlr.press/​v9/​glorot10a.html.
https:/​/​proceedings.mlr.press/​v9/​glorot10a.html

[60] Michael R. Garey and David S. Johnson. Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., USA, 1990.

[61] Christos H. Papadimitriou and Mihalis Yannakakis. Optimization, approximation, and complexity classes. Journal of Computer and System Sciences, 43(3):425–440, December 1991. URL: https:/​/​www.sciencedirect.com/​science/​article/​pii/​002200009190023X, doi:10.1016/​0022-0000(91)90023-X.
https:/​/​doi.org/​10.1016/​0022-0000(91)90023-X
https:/​/​www.sciencedirect.com/​science/​article/​pii/​002200009190023X

[62] Subhash Khot. On the power of unique 2-prover 1-round games. In In Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 767–775. ACM Press, 2002. URL: https:/​/​doi.org/​10.1145/​509907.510017.
https:/​/​doi.org/​10.1145/​509907.510017

[63] 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, January 2007. URL: https:/​/​epubs.siam.org/​doi/​10.1137/​S0097539705447372, doi:10.1137/​S0097539705447372.
https:/​/​doi.org/​10.1137/​S0097539705447372

[64] Sergey Bravyi, Alexander Kliesch, Robert Koenig, and Eugene Tang. Hybrid quantum-classical algorithms for approximate graph coloring. Quantum, 6:678, March 2022. URL: https:/​/​quantum-journal.org/​papers/​q-2022-03-30-678/​, doi:10.22331/​q-2022-03-30-678.
https:/​/​doi.org/​10.22331/​q-2022-03-30-678
https:/​/​quantum-journal.org/​papers/​q-2022-03-30-678/​

[65] Sergey Bravyi, Alexander Kliesch, Robert Koenig, and Eugene Tang. Obstacles to Variational Quantum Optimization from Symmetry Protection. Phys. Rev. Lett., 125(26):260505, December 2020. URL: https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.125.260505, doi:10.1103/​PhysRevLett.125.260505.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.260505

[66] Michael Overton and Henry Wolkowicz. Semidefinite programming. Mathematical Programming, 77:105–109, April 1997. doi:10.1007/​BF02614431.
https:/​/​doi.org/​10.1007/​BF02614431

[67] Tadashi Kadowaki and Hidetoshi Nishimori. Quantum annealing in the transverse Ising model. Physical Review E, 58(5):5355–5363, November 1998. URL: http:/​/​dx.doi.org/​10.1103/​PhysRevE.58.5355, doi:10.1103/​physreve.58.5355.
https:/​/​doi.org/​10.1103/​PhysRevE.58.5355

[68] Philipp Hauke, Helmut G Katzgraber, Wolfgang Lechner, Hidetoshi Nishimori, and William D Oliver. Perspectives of quantum annealing: methods and implementations. Reports on Progress in Physics, 83(5):054401, May 2020. URL: http:/​/​dx.doi.org/​10.1088/​1361-6633/​ab85b8, doi:10.1088/​1361-6633/​ab85b8.
https:/​/​doi.org/​10.1088/​1361-6633/​ab85b8

[69] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. PyTorch: An Imperative Style, High-Performance Deep Learning Library. In Advances in Neural Information Processing Systems 32, pages 8024–8035. Curran Associates, Inc., 2019. URL: http:/​/​papers.neurips.cc/​paper/​9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf, doi:10.48550/​arXiv.1912.01703.
https:/​/​doi.org/​10.48550/​arXiv.1912.01703
http:/​/​papers.neurips.cc/​paper/​9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf

[70] Martín Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, Manjunath Kudlur, Josh Levenberg, Rajat Monga, Sherry Moore, Derek G. Murray, Benoit Steiner, Paul Tucker, Vijay Vasudevan, Pete Warden, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng. TensorFlow: A system for large-scale machine learning, May 2016. arXiv:1605.08695 [cs]. URL: http:/​/​arxiv.org/​abs/​1605.08695, doi:10.48550/​arXiv.1605.08695.
https:/​/​doi.org/​10.48550/​arXiv.1605.08695
arXiv:1605.08695

[71] Franco Scarselli, Marco Gori, Ah Chung Tsoi, Markus Hagenbuchner, and Gabriele Monfardini. The Graph Neural Network Model. IEEE Transactions on Neural Networks, 20(1):61–80, January 2009. doi:10.1109/​TNN.2008.2005605.
https:/​/​doi.org/​10.1109/​TNN.2008.2005605

[72] Michael M. Bronstein, Joan Bruna, Taco Cohen, and Petar Veličković. Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges, May 2021. URL: http:/​/​arxiv.org/​abs/​2104.13478, doi:10.48550/​arXiv.2104.13478.
https:/​/​doi.org/​10.48550/​arXiv.2104.13478
arXiv:2104.13478

[73] Guillaume Verdon, Trevor McCourt, Enxhell Luzhnica, Vikash Singh, Stefan Leichenauer, and Jack Hidary. Quantum Graph Neural Networks, September 2019. URL: http:/​/​arxiv.org/​abs/​1909.12264, doi:10.48550/​arXiv.1909.12264.
https:/​/​doi.org/​10.48550/​arXiv.1909.12264
arXiv:1909.12264

[74] Martín Larocca, Frédéric Sauvage, Faris M. Sbahi, Guillaume Verdon, Patrick J. Coles, and M. Cerezo. Group-Invariant Quantum Machine Learning. PRX Quantum, 3(3):030341, September 2022. Publisher: American Physical Society. URL: https:/​/​link.aps.org/​doi/​10.1103/​PRXQuantum.3.030341, doi:10.1103/​PRXQuantum.3.030341.
https:/​/​doi.org/​10.1103/​PRXQuantum.3.030341

[75] Andrea Skolik, Michele Cattelan, Sheir Yarkoni, Thomas Bäck, and Vedran Dunjko. Equivariant quantum circuits for learning on weighted graphs, May 2022. arXiv:2205.06109 [quant-ph]. URL: http:/​/​arxiv.org/​abs/​2205.06109, doi:10.48550/​arXiv.2205.06109.
https:/​/​doi.org/​10.48550/​arXiv.2205.06109
arXiv:2205.06109

[76] Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. Graph Attention Networks. In International Conference on Learning Representations, 2018. URL: https:/​/​openreview.net/​forum?id=rJXMpikCZ, doi:10.48550/​arXiv.1710.10903.
https:/​/​doi.org/​10.48550/​arXiv.1710.10903
https:/​/​openreview.net/​forum?id=rJXMpikCZ

[77] Si Zhang, Hanghang Tong, Jiejun Xu, and Ross Maciejewski. Graph convolutional networks: a comprehensive review. Computational Social Networks, 6(1):11, November 2019. doi:10.1186/​s40649-019-0069-y.
https:/​/​doi.org/​10.1186/​s40649-019-0069-y

[78] Jie Zhou, Ganqu Cui, Shengding Hu, Zhengyan Zhang, Cheng Yang, Zhiyuan Liu, Lifeng Wang, Changcheng Li, and Maosong Sun. Graph neural networks: A review of methods and applications. AI Open, 1:57–81, January 2020. URL: https:/​/​www.sciencedirect.com/​science/​article/​pii/​S2666651021000012, doi:10.1016/​j.aiopen.2021.01.001.
https:/​/​doi.org/​10.1016/​j.aiopen.2021.01.001
https:/​/​www.sciencedirect.com/​science/​article/​pii/​S2666651021000012

[79] Zhengdao Chen, Lisha Li, and Joan Bruna. Supervised Community Detection with Line Graph Neural Networks. In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. OpenReview.net, 2019. URL: https:/​/​openreview.net/​forum?id=H1g0Z3A9Fm, doi:10.48550/​arXiv.1705.08415.
https:/​/​doi.org/​10.48550/​arXiv.1705.08415
https:/​/​openreview.net/​forum?id=H1g0Z3A9Fm

[80] Elias Khalil, Hanjun Dai, Yuyu Zhang, Bistra Dilkina, and Le Song. Learning Combinatorial Optimization Algorithms over Graphs. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017. URL: https:/​/​proceedings.neurips.cc/​paper/​2017/​file/​d9896106ca98d3d05b8cbdf4fd8b13a1-Paper.pdf, doi:10.48550/​arXiv.1704.01665.
https:/​/​doi.org/​10.48550/​arXiv.1704.01665
https:/​/​proceedings.neurips.cc/​paper/​2017/​file/​d9896106ca98d3d05b8cbdf4fd8b13a1-Paper.pdf

[81] Michel Deudon, Pierre Cournut, Alexandre Lacoste, Yossiri Adulyasak, and Louis-Martin Rousseau. Learning Heuristics for the TSP by Policy Gradient. In Willem-Jan van Hoeve, editor, Integration of Constraint Programming, Artificial Intelligence, and Operations Research, Lecture Notes in Computer Science, pages 170–181, Cham, 2018. Springer International Publishing. doi:10.1007/​978-3-319-93031-2_12.
https:/​/​doi.org/​10.1007/​978-3-319-93031-2_12

[82] Wouter Kool, Herke van Hoof, and Max Welling. Attention, Learn to Solve Routing Problems! In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. OpenReview.net, 2019. URL: https:/​/​openreview.net/​forum?id=ByxBFsRqYm, doi:10.48550/​arXiv.1803.08475.
https:/​/​doi.org/​10.48550/​arXiv.1803.08475
https:/​/​openreview.net/​forum?id=ByxBFsRqYm

[83] Chaitanya K. Joshi, Quentin Cappart, Louis-Martin Rousseau, and Thomas Laurent. Learning TSP Requires Rethinking Generalization. In Laurent D. Michel, editor, 27th International Conference on Principles and Practice of Constraint Programming (CP 2021), volume 210 of Leibniz International Proceedings in Informatics (LIPIcs), pages 33:1–33:21, Dagstuhl, Germany, 2021. Schloss Dagstuhl – Leibniz-Zentrum für Informatik. URL: https:/​/​drops.dagstuhl.de/​opus/​volltexte/​2021/​15324, doi:10.4230/​LIPIcs.CP.2021.33.
https:/​/​doi.org/​10.4230/​LIPIcs.CP.2021.33
https:/​/​drops.dagstuhl.de/​opus/​volltexte/​2021/​15324

[84] Ryan Sweke, Frederik Wilde, Johannes Jakob Meyer, Maria Schuld, Paul K. Fährmann, Barthélémy Meynard-Piganeau, and Jens Eisert. Stochastic gradient descent for hybrid quantum-classical optimization. Quantum, 4:314, August 2020. URL: https:/​/​quantum-journal.org/​papers/​q-2020-08-31-314/​, doi:10.22331/​q-2020-08-31-314.
https:/​/​doi.org/​10.22331/​q-2020-08-31-314
https:/​/​quantum-journal.org/​papers/​q-2020-08-31-314/​

[85] Jonas M. Kübler, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles. An Adaptive Optimizer for Measurement-Frugal Variational Algorithms. Quantum, 4:263, May 2020. URL: https:/​/​quantum-journal.org/​papers/​q-2020-05-11-263/​, doi:10.22331/​q-2020-05-11-263.
https:/​/​doi.org/​10.22331/​q-2020-05-11-263
https:/​/​quantum-journal.org/​papers/​q-2020-05-11-263/​

[86] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo. Quantum Natural Gradient. Quantum, 4:269, May 2020. URL: https:/​/​quantum-journal.org/​papers/​q-2020-05-25-269/​, doi:10.22331/​q-2020-05-25-269.
https:/​/​doi.org/​10.22331/​q-2020-05-25-269
https:/​/​quantum-journal.org/​papers/​q-2020-05-25-269/​

[87] Diederik P. Kingma and Jimmy Ba. Adam: A Method for Stochastic Optimization. In Yoshua Bengio and Yann LeCun, editors, 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015. URL: http:/​/​arxiv.org/​abs/​1412.6980, doi:10.48550/​arXiv.1412.6980.
https:/​/​doi.org/​10.48550/​arXiv.1412.6980
arXiv:1412.6980

[88] Matthew D. Zeiler. ADADELTA: An Adaptive Learning Rate Method, December 2012. URL: http:/​/​arxiv.org/​abs/​1212.5701, doi:10.48550/​arXiv.1212.5701.
https:/​/​doi.org/​10.48550/​arXiv.1212.5701
arXiv:1212.5701

[89] M. J. D. Powell. A Direct Search Optimization Method That Models the Objective and Constraint Functions by Linear Interpolation. In Susana Gomez and Jean-Pierre Hennart, editors, Advances in Optimization and Numerical Analysis, pages 51–67. Springer Netherlands, Dordrecht, 1994. doi:10.1007/​978-94-015-8330-5_4.
https:/​/​doi.org/​10.1007/​978-94-015-8330-5_4

[90] Kevin J. Sung, Jiahao Yao, Matthew P. Harrigan, Nicholas C. Rubin, Zhang Jiang, Lin Lin, Ryan Babbush, and Jarrod R. McClean. Using models to improve optimizers for variational quantum algorithms. Quantum Science and Technology, 5(4):044008, October 2020. doi:10.1088/​2058-9565/​abb6d9.
https:/​/​doi.org/​10.1088/​2058-9565/​abb6d9

[91] Julien Gacon, Christa Zoufal, Giuseppe Carleo, and Stefan Woerner. Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information. Quantum, 5:567, October 2021. URL: https:/​/​quantum-journal.org/​papers/​q-2021-10-20-567/​, doi:10.22331/​q-2021-10-20-567.
https:/​/​doi.org/​10.22331/​q-2021-10-20-567
https:/​/​quantum-journal.org/​papers/​q-2021-10-20-567/​

[92] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware. Phys. Rev. A, 99(3):032331, March 2019. URL: https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.99.032331, doi:10.1103/​PhysRevA.99.032331.
https:/​/​doi.org/​10.1103/​PhysRevA.99.032331

[93] Ke Li and Jitendra Malik. Learning to Optimize, June 2016. arXiv:1606.01885 [cs, math, stat]. URL: http:/​/​arxiv.org/​abs/​1606.01885, doi:10.48550/​arXiv.1606.01885.
https:/​/​doi.org/​10.48550/​arXiv.1606.01885
arXiv:1606.01885

[94] John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal Policy Optimization Algorithms, August 2017. arXiv:1707.06347 [cs]. URL: http:/​/​arxiv.org/​abs/​1707.06347, doi:10.48550/​arXiv.1707.06347.
https:/​/​doi.org/​10.48550/​arXiv.1707.06347
arXiv:1707.06347

[95] Max Wilson, Rachel Stromswold, Filip Wudarski, Stuart Hadfield, Norm M. Tubman, and Eleanor G. Rieffel. Optimizing quantum heuristics with meta-learning. Quantum Machine Intelligence, 3(1):13, April 2021. doi:10.1007/​s42484-020-00022-w.
https:/​/​doi.org/​10.1007/​s42484-020-00022-w

[96] Amira Abbas, David Sutter, Christa Zoufal, Aurelien Lucchi, Alessio Figalli, and Stefan Woerner. The power of quantum neural networks. Nature Computational Science, 1(6):403–409, June 2021. URL: https:/​/​www.nature.com/​articles/​s43588-021-00084-1, doi:10.1038/​s43588-021-00084-1.
https:/​/​doi.org/​10.1038/​s43588-021-00084-1
https:/​/​www.nature.com/​articles/​s43588-021-00084-1

[97] Florent Krzakala, Cristopher Moore, Elchanan Mossel, Joe Neeman, Allan Sly, Lenka Zdeborová, and Pan Zhang. Spectral redemption in clustering sparse networks. Proceedings of the National Academy of Sciences, 110(52):20935–20940, 2013. URL: https:/​/​www.pnas.org/​content/​110/​52/​20935, doi:10.1073/​pnas.1312486110.
https:/​/​doi.org/​10.1073/​pnas.1312486110
https:/​/​www.pnas.org/​content/​110/​52/​20935

Cited by

[1] Stefan H. Sack, Raimel A. Medina, Richard Kueng, and Maksym Serbyn, “Transition states and greedy exploration of the QAOA optimization landscape”, arXiv:2209.01159.

[2] Samuel Duffield, Marcello Benedetti, and Matthias Rosenkranz, “Bayesian Learning of Parameterised Quantum Circuits”, arXiv:2206.07559.

[3] Brian Coyle, “Machine learning applications for noisy intermediate-scale quantum computers”, arXiv:2205.09414.

[4] Ohad Amosy, Tamuz Danzig, Ely Porat, Gal Chechik, and Adi Makmal, “Iterative-Free Quantum Approximate Optimization Algorithm Using Neural Networks”, arXiv:2208.09888.

[5] Ioannis Kolotouros, Ioannis Petrongonas, and Petros Wallden, “Adiabatic quantum computing with parameterized quantum circuits”, arXiv:2206.04373.

The above citations are from SAO/NASA ADS (last updated successfully 2022-11-18 02:57:39). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2022-11-18 02:57:38).

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