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Simulation of a feedback-based algorithm for quantum optimization for a realistic neutral atom system with an optimized small-angle controlled-phase gate

Published 16 May 2024 in quant-ph | (2405.10451v3)

Abstract: In contrast to the classical optimization process required by the quantum approximate optimization algorithm, FALQON, a feedback-based algorithm for quantum optimization [A. B. Magann {\it et al.,} {\color{blue}Phys. Rev. Lett. {\bf129}, 250502 (2022)}], enables one to obtain approximate solutions to combinatorial optimization problems without any classical optimization effort. In this study, we leverage the specifications of a recent experimental platform for the neutral atom system [Z. Fu {\it et al.,} {\color{blue}Phys. Rev. A {\bf105}, 042430 (2022)}] and present a scheme to implement an optimally tuned small-angle controlled-phase gate. By examining the 2- to 4-qubit FALQON algorithms in the Max-Cut problem and considering the spontaneous emission of the neutral atomic system, we have observed that the performance of FALQON implemented with small-angle controlled-phase gates exceeds that of FALQON utilizing CZ gates. This approach has the potential to significantly simplify the logic circuit required to simulate FALQON and effectively address the Max-Cut problem, which may pave a way for the experimental implementation of near-term noisy intermediate-scale quantum algorithms with neutral-atom systems.

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References (88)
  1. Yudong Cao, Jonathan Romero, Jonathan P Olson, Matthias Degroote, Peter D Johnson, Mária Kieferová, Ian D Kivlichan, Tim Menke, Borja Peropadre, Nicolas PD Sawaya, et al., “Quantum chemistry in the age of quantum computing,” Chemical reviews 119, 10856–10915 (2019).
  2. Kenneth Steiglitz and Christos H. Papadimitriou, Combinatorial optimization : algorithms and complexity, dover ed. (Dover Publications, 1998).
  3. Andrew Lucas, “Ising formulations of many np problems,” Frontiers in physics 2, 5 (2014).
  4. Alicia B. Magann, Kenneth M. Rudinger, Matthew D. Grace,  and Mohan Sarovar, “Feedback-based quantum optimization,” Phys. Rev. Lett. 129, 250502 (2022a).
  5. Alicia B. Magann, Kenneth M. Rudinger, Matthew D. Grace,  and Mohan Sarovar, “Lyapunov-control-inspired strategies for quantum combinatorial optimization,” Phys. Rev. A 106, 062414 (2022b).
  6. Edward Farhi, Jeffrey Goldstone,  and Sam Gutmann, “A quantum approximate optimization algorithm,” arXiv preprint arXiv:1411.4028  (2014).
  7. Michelle Chalupnik, Hans Melo, Yuri Alexeev,  and Alexey Galda, “Augmenting qaoa ansatz with multiparameter problem-independent layer,” in 2022 IEEE International Conference on Quantum Computing and Engineering (QCE) (2022) pp. 97–103.
  8. P. Chandarana, N. N. Hegade, K. Paul, F. Albarrán-Arriagada, E. Solano, A. del Campo,  and Xi Chen, “Digitized-counterdiabatic quantum approximate optimization algorithm,” Phys. Rev. Res. 4, 013141 (2022).
  9. Yunlong Yu, Chenfeng Cao, Carter Dewey, Xiang-Bin Wang, Nic Shannon,  and Robert Joynt, “Quantum approximate optimization algorithm with adaptive bias fields,” Phys. Rev. Res. 4, 023249 (2022).
  10. 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. Res. 4, 033029 (2022).
  11. Sergey Bravyi, Alexander Kliesch, Robert Koenig,  and Eugene Tang, “Obstacles to variational quantum optimization from symmetry protection,” Phys. Rev. Lett. 125, 260505 (2020).
  12. 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 (2019), 10.3390/a12020034.
  13. Daniel J. Egger, Jakub Mareček,  and Stefan Woerner, “Warm-starting quantum optimization,” Quantum 5, 479 (2021).
  14. Takuya Yoshioka, Keita Sasada, Yuichiro Nakano,  and Keisuke Fujii, “Fermionic quantum approximate optimization algorithm,” Phys. Rev. Res. 5, 023071 (2023).
  15. Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H. Booth,  and Jonathan Tennyson, “The variational quantum eigensolver: A review of methods and best practices,” Physics Reports 986, 1–128 (2022).
  16. Zoë Holmes, Kunal Sharma, M. Cerezo,  and Patrick J. Coles, “Connecting ansatz expressibility to gradient magnitudes and barren plateaus,” PRX Quantum 3, 010313 (2022).
  17. Gian Giacomo Guerreschi and Anne Y Matsuura, “Qaoa for max-cut requires hundreds of qubits for quantum speed-up,” Scientific reports 9, 6903 (2019).
  18. Andreas Bärtschi and Stephan Eidenbenz, “Grover mixers for qaoa: Shifting complexity from mixer design to state preparation,” in 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) (2020) pp. 72–82.
  19. Franz G Fuchs, Herman Øie Kolden, Niels Henrik Aase,  and Giorgio Sartor, “Efficient encoding of the weighted max k-cut on a quantum computer using qaoa,” SN Computer Science 2, 89 (2021).
  20. Nicholas Chancellor, “Domain wall encoding of discrete variables for quantum annealing and qaoa,” Quantum Science and Technology 4, 045004 (2019).
  21. Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, et al., “Variational quantum algorithms,” Nature Reviews Physics 3, 625–644 (2021a).
  22. Matthew P Harrigan, Kevin J Sung, Matthew Neeley, Kevin J Satzinger, Frank Arute, Kunal Arya, Juan Atalaya, Joseph C Bardin, Rami Barends, Sergio Boixo, et al., “Quantum approximate optimization of non-planar graph problems on a planar superconducting processor,” Nature Physics 17, 332–336 (2021).
  23. V. Akshay, H. Philathong, M. E. S. Morales,  and J. D. Biamonte, “Reachability deficits in quantum approximate optimization,” Phys. Rev. Lett. 124, 090504 (2020).
  24. 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 (2022).
  25. Seth Lloyd, “Quantum approximate optimization is computationally universal,” arXiv preprint arXiv:1812.11075  (2018).
  26. Edward Farhi and Aram W Harrow, “Quantum supremacy through the quantum approximate optimization algorithm,” arXiv preprint arXiv:1602.07674  (2016).
  27. Guido Pagano, Aniruddha Bapat, Patrick Becker, Katherine S Collins, Arinjoy De, Paul W Hess, Harvey B Kaplan, Antonis Kyprianidis, Wen Lin Tan, Christopher Baldwin, et al., “Quantum approximate optimization of the long-range ising model with a trapped-ion quantum simulator,” Proceedings of the National Academy of Sciences 117, 25396–25401 (2020).
  28. Pontus Vikstål, Mattias Grönkvist, Marika Svensson, Martin Andersson, Göran Johansson,  and Giulia Ferrini, “Applying the quantum approximate optimization algorithm to the tail-assignment problem,” Phys. Rev. Appl. 14, 034009 (2020).
  29. Ruslan Shaydulin, Ilya Safro,  and Jeffrey Larson, “Multistart methods for quantum approximate optimization,” in 2019 IEEE high performance extreme computing conference (HPEC) (IEEE, 2019) pp. 1–8.
  30. TM Graham, Y Song, J Scott, C Poole, L Phuttitarn, K Jooya, P Eichler, X Jiang, A Marra, B Grinkemeyer, et al., “Multi-qubit entanglement and algorithms on a neutral-atom quantum computer,” Nature 604, 457–462 (2022).
  31. Samson Wang, Enrico Fontana, Marco Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio,  and Patrick J Coles, “Noise-induced barren plateaus in variational quantum algorithms,” Nature communications 12, 6961 (2021).
  32. Marco Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio,  and Patrick J Coles, “Cost function dependent barren plateaus in shallow parametrized quantum circuits,” Nature communications 12, 1791 (2021b).
  33. Daniel Stilck França and Raul Garcia-Patron, “Limitations of optimization algorithms on noisy quantum devices,” Nature Physics 17, 1221–1227 (2021).
  34. Oscar Higgott, Daochen Wang,  and Stephen Brierley, “Variational quantum computation of excited states,” Quantum 3, 156 (2019).
  35. Samuel Yen-Chi Chen, Chao-Han Huck Yang, Jun Qi, Pin-Yu Chen, Xiaoli Ma,  and Hsi-Sheng Goan, “Variational quantum circuits for deep reinforcement learning,” IEEE Access 8, 141007–141024 (2020).
  36. Sumeet Khatri, Ryan LaRose, Alexander Poremba, Lukasz Cincio, Andrew T Sornborger,  and Patrick J Coles, “Quantum-assisted quantum compiling,” Quantum 3, 140 (2019).
  37. S. Ebadi, A. Keesling, M. Cain, T. T. Wang, H. Levine, D. Bluvstein, G. Semeghini, A. Omran, J.-G. Liu, R. Samajdar, X.-Z. Luo, B. Nash, X. Gao, B. Barak, E. Farhi, S. Sachdev, N. Gemelke, L. Zhou, S. Choi, H. Pichler, S.-T. Wang, M. Greiner, V. Vuletić,  and M. D. Lukin, “Quantum optimization of maximum independent set using rydberg atom arrays,” Science 376, 1209–1215 (2022).
  38. A. Galindo and M. A. Martín-Delgado, “Information and computation: Classical and quantum aspects,” Rev. Mod. Phys. 74, 347–423 (2002).
  39. Thaddeus D Ladd, Fedor Jelezko, Raymond Laflamme, Yasunobu Nakamura, Christopher Monroe,  and Jeremy Lloyd O’Brien, “Quantum computers,” nature 464, 45–53 (2010).
  40. Göran Wendin, “Quantum information processing with superconducting circuits: a review,” Reports on Progress in Physics 80, 106001 (2017).
  41. F. A. Calderon-Vargas, George S. Barron, Xiu-Hao Deng, A. J. Sigillito, Edwin Barnes,  and Sophia E. Economou, “Fast high-fidelity entangling gates for spin qubits in si double quantum dots,” Phys. Rev. B 100, 035304 (2019).
  42. David W. Kanaar, Sidney Wolin, Utkan Güngördü,  and J. P. Kestner, “Single-tone pulse sequences and robust two-tone shaped pulses for three silicon spin qubits with always-on exchange,” Phys. Rev. B 103, 235314 (2021).
  43. C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol,  and D. M. Lucas, “High-fidelity quantum logic gates using trapped-ion hyperfine qubits,” Phys. Rev. Lett. 117, 060504 (2016).
  44. J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried,  and D. J. Wineland, “High-fidelity universal gate set for Be+9superscriptsuperscriptBe9{{}^{9}\mathrm{Be}}^{+}start_FLOATSUPERSCRIPT 9 end_FLOATSUPERSCRIPT roman_Be start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ion qubits,” Phys. Rev. Lett. 117, 060505 (2016).
  45. Rami Barends, Julian Kelly, Anthony Megrant, Andrzej Veitia, Daniel Sank, Evan Jeffrey, Ted C White, Josh Mutus, Austin G Fowler, Brooks Campbell, et al., “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature 508, 500–503 (2014).
  46. Tenghui Wang, Zhenxing Zhang, Liang Xiang, Zhilong Jia, Peng Duan, Zhiwen Zong, Zhenhai Sun, Zhangjingzi Dong, Jianlan Wu, Yi Yin,  and Guoping Guo, “Experimental realization of a fast controlled-z gate via a shortcut to adiabaticity,” Phys. Rev. Appl. 11, 034030 (2019).
  47. M. A. Rol, F. Battistel, F. K. Malinowski, C. C. Bultink, B. M. Tarasinski, R. Vollmer, N. Haider, N. Muthusubramanian, A. Bruno, B. M. Terhal,  and L. DiCarlo, “Fast, high-fidelity conditional-phase gate exploiting leakage interference in weakly anharmonic superconducting qubits,” Phys. Rev. Lett. 123, 120502 (2019).
  48. Nathan Lacroix, Christoph Hellings, Christian Kraglund Andersen, Agustin Di Paolo, Ants Remm, Stefania Lazar, Sebastian Krinner, Graham J. Norris, Mihai Gabureac, Johannes Heinsoo, Alexandre Blais, Christopher Eichler,  and Andreas Wallraff, “Improving the performance of deep quantum optimization algorithms with continuous gate sets,” PRX Quantum 1, 020304 (2020).
  49. Immanuel Bloch, “Quantum coherence and entanglement with ultracold atoms in optical lattices,” Nature 453, 1016–1022 (2008).
  50. M. Saffman, T. G. Walker,  and K. Mølmer, “Quantum information with rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
  51. Antoine Browaeys, Daniel Barredo,  and Thierry Lahaye, “Experimental investigations of dipole–dipole interactions between a few rydberg atoms,” Journal of Physics B: Atomic, Molecular and Optical Physics 49, 152001 (2016).
  52. Xiao-Qiang Shao, Shi-Lei Su, Lin Li, Rejish Nath, Jin-Hui Wu,  and Weibin Li, “Rydberg superatoms: An artificial quantum system for quantum information processing and quantum optics,” arXiv preprint arXiv:2404.05330  (2024).
  53. E Urban, Todd A Johnson, T Henage, L Isenhower, DD Yavuz, TG Walker,  and M Saffman, “Observation of rydberg blockade between two atoms,” Nature Physics 5, 110–114 (2009).
  54. D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler,  and P. L. Gould, “Local blockade of rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93, 063001 (2004).
  55. Alpha Gaëtan, Yevhen Miroshnychenko, Tatjana Wilk, Amodsen Chotia, Matthieu Viteau, Daniel Comparat, Pierre Pillet, Antoine Browaeys,  and Philippe Grangier, “Observation of collective excitation of two individual atoms in the rydberg blockade regime,” Nature Physics 5, 115–118 (2009).
  56. D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté,  and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. 85, 2208–2211 (2000).
  57. M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac,  and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
  58. E Brion, LH Pedersen,  and K Mølmer, “Implementing a neutral atom rydberg gate without populating the rydberg state,” Journal of Physics B: Atomic, Molecular and Optical Physics 40, S159 (2007).
  59. Huai-Zhi Wu, Zhen-Biao Yang,  and Shi-Biao Zheng, “Implementation of a multiqubit quantum phase gate in a neutral atomic ensemble via the asymmetric rydberg blockade,” Phys. Rev. A 82, 034307 (2010).
  60. M. M. Müller, D. M. Reich, M. Murphy, H. Yuan, J. Vala, K. B. Whaley, T. Calarco,  and C. P. Koch, “Optimizing entangling quantum gates for physical systems,” Phys. Rev. A 84, 042315 (2011).
  61. T. Xia, X. L. Zhang,  and M. Saffman, “Analysis of a controlled phase gate using circular rydberg states,” Phys. Rev. A 88, 062337 (2013).
  62. David Petrosyan and Klaus Mølmer, “Binding potentials and interaction gates between microwave-dressed rydberg atoms,” Phys. Rev. Lett. 113, 123003 (2014).
  63. Lörinc Sárkány, József Fortágh,  and David Petrosyan, “Long-range quantum gate via rydberg states of atoms in a thermal microwave cavity,” Phys. Rev. A 92, 030303 (2015).
  64. Mark Saffman, “Quantum computing with atomic qubits and rydberg interactions: progress and challenges,” Journal of Physics B: Atomic, Molecular and Optical Physics 49, 202001 (2016).
  65. Shi-Lei Su, Erjun Liang, Shou Zhang, Jing-Ji Wen, Li-Li Sun, Zhao Jin,  and Ai-Dong Zhu, “One-step implementation of the rydberg-rydberg-interaction gate,” Phys. Rev. A 93, 012306 (2016).
  66. Shi-Lei Su, Ya Gao, Erjun Liang,  and Shou Zhang, “Fast rydberg antiblockade regime and its applications in quantum logic gates,” Phys. Rev. A 95, 022319 (2017).
  67. Xiao-Feng Shi, “Deutsch, toffoli, and cnot gates via rydberg blockade of neutral atoms,” Phys. Rev. Appl. 9, 051001 (2018).
  68. Xiao-Feng Shi and T. A. B. Kennedy, “Annulled van der waals interaction and fast rydberg quantum gates,” Phys. Rev. A 95, 043429 (2017).
  69. Xiao-Feng Shi, “Rydberg quantum gates free from blockade error,” Phys. Rev. Appl. 7, 064017 (2017).
  70. Xi-Rong Huang, Zong-Xing Ding, Chang-Sheng Hu, Li-Tuo Shen, Weibin Li, Huaizhi Wu,  and Shi-Biao Zheng, “Robust rydberg gate via landau-zener control of förster resonance,” Phys. Rev. A 98, 052324 (2018).
  71. S. L. Su, H. Z. Shen, Erjun Liang,  and Shou Zhang, “One-step construction of the multiple-qubit rydberg controlled-phase gate,” Phys. Rev. A 98, 032306 (2018).
  72. D. X. Li and X. Q. Shao, “Unconventional rydberg pumping and applications in quantum information processing,” Phys. Rev. A 98, 062338 (2018).
  73. Xiao-Feng Shi, “Fast, accurate, and realizable two-qubit entangling gates by quantum interference in detuned rabi cycles of rydberg atoms,” Phys. Rev. Appl. 11, 044035 (2019).
  74. Hong-Da Yin, Xiao-Xuan Li, Gang-Cheng Wang,  and Xiao-Qiang Shao, “One-step implementation of toffoli gate for neutral atoms based on unconventional rydberg pumping,” Opt. Express 28, 35576–35587 (2020).
  75. Rui Li, Shurui Li, Dongmin Yu, Jing Qian,  and Weiping Zhang, “Optimal model for fewer-qubit cnot gates with rydberg atoms,” Phys. Rev. Appl. 17, 024014 (2022a).
  76. X. X. Li, J. B. You, X. Q. Shao,  and Weibin Li, “Coherent ground-state transport of neutral atoms,” Phys. Rev. A 105, 032417 (2022b).
  77. Harry Levine, Alexander Keesling, Giulia Semeghini, Ahmed Omran, Tout T. Wang, Sepehr Ebadi, Hannes Bernien, Markus Greiner, Vladan Vuletić, Hannes Pichler,  and Mikhail D. Lukin, “Parallel implementation of high-fidelity multiqubit gates with neutral atoms,” Phys. Rev. Lett. 123, 170503 (2019).
  78. Michael J Martin, Yuan-Yu Jau, Jongmin Lee, Anupam Mitra, Ivan H Deutsch,  and Grant W Biedermann, “A mølmer-sørensen gate with rydberg-dressed atoms,” arXiv e-prints , arXiv–2111 (2021).
  79. Zhuo Fu, Peng Xu, Yuan Sun, Yang-Yang Liu, Xiao-Dong He, Xiao Li, Min Liu, Run-Bing Li, Jin Wang, Liang Liu,  and Ming-Sheng Zhan, “High-fidelity entanglement of neutral atoms via a rydberg-mediated single-modulated-pulse controlled-phase gate,” Phys. Rev. A 105, 042430 (2022).
  80. X. X. Li, X. Q. Shao,  and Weibin Li, “Single temporal-pulse-modulated parameterized controlled-phase gate for rydberg atoms,” Phys. Rev. Appl. 18, 044042 (2022c).
  81. J. S. Otterbach, R. Manenti, N. Alidoust, A. Bestwick, M. Block, B. Bloom, S. Caldwell, N. Didier, E. Schuyler Fried, S. Hong, P. Karalekas, C. B. Osborn, A. Papageorge, E. C. Peterson, G. Prawiroatmodjo, N. Rubin, Colm A. Ryan, D. Scarabelli, M. Scheer, E. A. Sete, P. Sivarajah, Robert S. Smith, A. Staley, N. Tezak, W. J. Zeng, A. Hudson, Blake R. Johnson, M. Reagor, M. P. da Silva,  and C. Rigetti, “Unsupervised machine learning on a hybrid quantum computer,”  (2017), arXiv:1712.05771 [quant-ph] .
  82. Andreas Bengtsson, Pontus Vikstål, Christopher Warren, Marika Svensson, Xiu Gu, Anton Frisk Kockum, Philip Krantz, Christian Krizan, Daryoush Shiri, Ida-Maria Svensson, et al., “Quantum approximate optimization of the exact-cover problem on a superconducting quantum processor,” arXiv preprint arXiv:1912.10495  (2019).
  83. Nikola Šibalić, Jonathan D Pritchard, Charles S Adams,  and Kevin J Weatherill, “Arc: An open-source library for calculating properties of alkali rydberg atoms,” Computer Physics Communications 220, 319–331 (2017).
  84. Sylvain de Léséleuc, Daniel Barredo, Vincent Lienhard, Antoine Browaeys,  and Thierry Lahaye, “Analysis of imperfections in the coherent optical excitation of single atoms to rydberg states,” Phys. Rev. A 97, 053803 (2018).
  85. Woojun Lee, Minhyuk Kim, Hanlae Jo, Yunheung Song,  and Jaewook Ahn, “Coherent and dissipative dynamics of entangled few-body systems of rydberg atoms,” Phys. Rev. A 99, 043404 (2019).
  86. Hikaru Tamura, Tomotake Yamakoshi,  and Ken’ichi Nakagawa, “Analysis of coherent dynamics of a rydberg-atom quantum simulator,” Phys. Rev. A 101, 043421 (2020).
  87. Ivaylo S Madjarov, Jacob P Covey, Adam L Shaw, Joonhee Choi, Anant Kale, Alexandre Cooper, Hannes Pichler, Vladimir Schkolnik, Jason R Williams,  and Manuel Endres, “High-fidelity entanglement and detection of alkaline-earth rydberg atoms,” Nature Physics 16, 857–861 (2020).
  88. Svetoslav S. Ivanov and Nikolay V. Vitanov, “Composite two-qubit gates,” Phys. Rev. A 92, 022333 (2015).
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Authors (4)

  1. S. X. Li 
  2. W. L. Mu 
  3. J. B. You 
  4. X. Q. Shao 

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