Counterdiabaticity and the quantum approximate optimization algorithm

Jonathan Wurtz; Peter J. Love

doi:10.22331/q-2022-01-27-635

Counterdiabaticity and the quantum approximate optimization algorithm

Jonathan Wurtz and Peter J. Love

Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA

Published:	2022-01-27, volume 6, page 635
Eprint:	arXiv:2106.15645v3
Doi:	https://doi.org/10.22331/q-2022-01-27-635
Citation:	Quantum 6, 635 (2022).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

The quantum approximate optimization algorithm (QAOA) is a near-term hybrid algorithm intended to solve combinatorial optimization problems, such as MaxCut. QAOA can be made to mimic an adiabatic schedule, and in the $p\to\infty$ limit the final state is an exact maximal eigenstate in accordance with the adiabatic theorem. In this work, the connection between QAOA and adiabaticity is made explicit by inspecting the regime of $p$ large but finite. By connecting QAOA to counterdiabatic (CD) evolution, we construct CD-QAOA angles which mimic a counterdiabatic schedule by matching Trotter "error" terms to approximate adiabatic gauge potentials which suppress diabatic excitations arising from finite ramp speed. In our construction, these "error" terms are helpful, not detrimental, to QAOA. Using this matching to link QAOA with quantum adiabatic algorithms (QAA), we show that the approximation ratio converges to one at least as $1-C(p)\sim 1/p^{\mu}$. We show that transfer of parameters between graphs, and interpolating angles for $p+1$ given $p$ are both natural byproducts of CD-QAOA matching. Optimization of CD-QAOA angles is equivalent to optimizing a continuous adiabatic schedule. Finally, we show that, using a property of variational adiabatic gauge potentials, QAOA is at least counterdiabatic, not just adiabatic, and has better performance than finite time adiabatic evolution. We demonstrate the method on three examples: a 2 level system, an Ising chain, and the MaxCut problem.

Featured image: Schematic diagram of the CD-QAOA angle matching procedure. As input, the procedure receives the counterdiabatic annealing protocol (Black line), effective Hamiltonian H, and number of QAOA steps p. For each step of the CD-QAOA evolution (pink), the effective unitaries over the interval are matched by minimizing the difference of their generators (right). This procedure sets the value of each angle γ, β in the QAOA (blue). By best matching along each step, the procedure fixes the angles {γ, β} and total adiabatic annealing time T.

► BibTeX data

@article{Wurtz2022counterdiabaticity,
  doi = {10.22331/q-2022-01-27-635},
  url = {https://doi.org/10.22331/q-2022-01-27-635},
  title = {Counterdiabaticity and the quantum approximate optimization algorithm},
  author = {Wurtz, Jonathan and Love, Peter J.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {635},
  month = jan,
  year = {2022}
}

► References

[1] E. Farhi, J. Goldstone, and S. Gutmann, (2014), arXiv:1411.4028 [quant-ph].
arXiv:1411.4028

[2] J. Preskill, Quantum 2, 79 (2018).
https://doi.org/10.22331/q-2018-08-06-79

[3] K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, (2021), arXiv:2101.08448 [quant-ph].
arXiv:2101.08448

[4] A. Lucas, Frontiers in Physics 2 (2014), 10.3389/fphy.2014.00005.
https://doi.org/10.3389/fphy.2014.00005

[5] T. Albash and D. A. Lidar, Rev. Mod. Phys. 90, 015002 (2018).
https://doi.org/10.1103/RevModPhys.90.015002

[6] M. H. S. Amin, Phys. Rev. Lett. 102, 220401 (2009).
https://doi.org/10.1103/PhysRevLett.102.220401

[7] S. H. Sack and M. Serbyn, (2021), arXiv:2101.05742 [quant-ph].
https://doi.org/10.22331/q-2021-07-01-491
arXiv:2101.05742

[8] D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, Rev. Mod. Phys. 91, 045001 (2019).
https://doi.org/10.1103/RevModPhys.91.045001

[9] G. Rigolin, G. Ortiz, and V. H. Ponce, Phys. Rev. A 78, 052508 (2008).
https://doi.org/10.1103/PhysRevA.78.052508

[10] S. Bachmann, W. De Roeck, and M. Fraas, Phys. Rev. Lett. 119, 060201 (2017).
https://doi.org/10.1103/PhysRevLett.119.060201

[11] A. del Campo and W. H. Zurek, International Journal of Modern Physics A 29, 1430018 (2014).
https://doi.org/10.1142/s0217751x1430018x

[12] J. I. Latorre and R. Orús, Phys. Rev. A 69, 062302 (2004).
https://doi.org/10.1103/PhysRevA.69.062302

[13] R. Barankov and A. Polkovnikov, Physical Review Letters 101 (2008), 10.1103/physrevlett.101.076801.
https://doi.org/10.1103/physrevlett.101.076801

[14] B. F. Schiffer, J. Tura, and J. I. Cirac, (2021), arXiv:2103.01226 [quant-ph].
arXiv:2103.01226

[15] L. Zhou, S.-T. Wang, S. Choi, H. Pichler, and M. D. Lukin, Physical Review X 10 (2020), 10.1103/physrevx.10.021067.
https://doi.org/10.1103/physrevx.10.021067

[16] F. G. S. L. Brandao, M. Broughton, E. Farhi, S. Gutmann, and H. Neven, (2018), arXiv:1812.04170 [quant-ph].
arXiv:1812.04170

[17] Z.-C. Yang, A. Rahmani, A. Shabani, H. Neven, and C. Chamon, Physical Review X 7 (2017), 10.1103/physrevx.7.021027.
https://doi.org/10.1103/physrevx.7.021027

[18] L. T. Brady, C. L. Baldwin, A. Bapat, Y. Kharkov, and A. V. Gorshkov, Phys. Rev. Lett. 126, 070505 (2021a).
https://doi.org/10.1103/PhysRevLett.126.070505

[19] M. Streif and M. Leib, Quantum Science and Technology 5, 034008 (2020).
https://doi.org/10.1088/2058-9565/ab8c2b

[20] M. Born and V. Fock, Zeitschrift für Physik 51, 165 (1928).
https://doi.org/10.1007/BF01343193

[21] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, (2000), arXiv:quant-ph/0001106 [quant-ph].
arXiv:quant-ph/0001106

[22] A. T. Rezakhani, W.-J. Kuo, A. Hamma, D. A. Lidar, and P. Zanardi, Phys. Rev. Lett. 103, 080502 (2009).
https://doi.org/10.1103/PhysRevLett.103.080502

[23] C. Brif, M. D. Grace, M. Sarovar, and K. C. Young, New Journal of Physics 16, 065013 (2014).
https://doi.org/10.1088/1367-2630/16/6/065013

[24] L. T. Brady, L. Kocia, P. Bienias, A. Bapat, Y. Kharkov, and A. V. Gorshkov, (2021b), arXiv:2107.01218 [quant-ph].
arXiv:2107.01218

[25] S. Bao, S. Kleer, R. Wang, and A. Rahmani, Phys. Rev. A 97, 062343 (2018).
https://doi.org/10.1103/PhysRevA.97.062343

[26] T. Chasseur, L. S. Theis, Y. R. Sanders, D. J. Egger, and F. K. Wilhelm, Phys. Rev. A 91, 043421 (2015).
https://doi.org/10.1103/PhysRevA.91.043421

[27] P. W. Claeys, M. Pandey, D. Sels, and A. Polkovnikov, Phys. Rev. Lett. 123, 090602 (2019).
https://doi.org/10.1103/PhysRevLett.123.090602

[28] S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, 2011).

[29] H. Nishimori and K. Takada, Frontiers in ICT 4, 2 (2017).
https://doi.org/10.3389/fict.2017.00002

[30] M. W. Johnson, M. H. S. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk, E. M. Chapple, C. Enderud, J. P. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkacheva, C. J. S. Truncik, S. Uchaikin, J. Wang, B. Wilson, and G. Rose, Nature 473, 194 (2011).
https://doi.org/10.1038/nature10012

[31] S. Sugiura, P. W. Claeys, A. Dymarsky, and A. Polkovnikov, Phys. Rev. Research 3, 013102 (2021).
https://doi.org/10.1103/PhysRevResearch.3.013102

[32] M. V. Berry, Journal of Physics A: Mathematical and Theoretical 42, 365303 (2009).
https://doi.org/10.1088/1751-8113/42/36/365303

[33] M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, Physics Reports 697, 1 (2017).
https://doi.org/10.1016/j.physrep.2017.07.001

[34] M. Pandey, P. W. Claeys, D. K. Campbell, A. Polkovnikov, and D. Sels, Phys. Rev. X 10, 041017 (2020).
https://doi.org/10.1103/PhysRevX.10.041017

[35] D. Sels and A. Polkovnikov, Proceedings of the National Academy of Sciences 114, E3909 (2017).
https://doi.org/10.1073/pnas.1619826114

[36] M. B. Hastings and X.-G. Wen, Phys. Rev. B 72, 045141 (2005).
https://doi.org/10.1103/PhysRevB.72.045141

[37] J. Wurtz and A. Polkovnikov, Phys. Rev. B 101, 195138 (2020).
https://doi.org/10.1103/PhysRevB.101.195138

[38] N. Hatano and M. Suzuki, Lecture Notes in Physics , 37–68 (2005).
https://doi.org/10.1007/11526216_2

[39] S. Blanes, F. Casas, J. Oteo, and J. Ros, Physics Reports 470, 151–238 (2009).
https://doi.org/10.1016/j.physrep.2008.11.001

[40] X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, Phys. Rev. Lett. 105, 123003 (2010).
https://doi.org/10.1103/PhysRevLett.105.123003

[41] J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005).
https://doi.org/10.1103/PhysRevLett.95.245701

[42] M. Kolodrubetz, B. K. Clark, and D. A. Huse, Physical Review Letters 109 (2012), 10.1103/physrevlett.109.015701.
https://doi.org/10.1103/physrevlett.109.015701

[43] A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosenbaum, and D. Sen, (2015), arXiv:1012.0653 [cond-mat.stat-mech].
arXiv:1012.0653

[44] D. J. Egger, J. Mareček, and S. Woerner, Quantum 5, 479 (2021).
https://doi.org/10.22331/q-2021-06-17-479

[45] A. G. R. Day, M. Bukov, P. Weinberg, P. Mehta, and D. Sels, Phys. Rev. Lett. 122, 020601 (2019).
https://doi.org/10.1103/PhysRevLett.122.020601

[46] G. B. Mbeng, R. Fazio, and G. Santoro, (2019), arXiv:1906.08948 [quant-ph].
arXiv:1906.08948

[47] V. Viswanath and G. Müller, The Recursion Method (Springer US, 2008).

[48] J. Wurtz and P. Love, Phys. Rev. A 103, 042612 (2021).
https://doi.org/10.1103/PhysRevA.103.042612

Cited by

[1] Maxime Dupont, Bhuvanesh Sundar, Bram Evert, David E. Bernal Neira, Zedong Peng, Stephen Jeffrey, and Mark J. Hodson, "Benchmarking quantum optimization for the maximum-cut problem on a superconducting quantum computer", Physical Review Applied 23 1, 014045 (2025).

[2] Zekun He, A. F. Kemper, and J. K. Freericks, "Hidden local adiabatic ramp in the modulated time evolution and the quantum approximate optimization algorithm", APL Quantum 3 1, 016107 (2026).

[3] Naruo Ohga and Takuya Hatomura, "Improving Variational Counterdiabatic Driving with Weighted Actions and Computer Algebra", PRX Quantum 7 2, 020347 (2026).

[4] Akel Hashim, Rich Rines, Victory Omole, Ravi K. Naik, John Mark Kreikebaum, David I. Santiago, Frederic T. Chong, Irfan Siddiqi, and Pranav Gokhale, "Optimized SWAP networks with equivalent circuit averaging for QAOA", Physical Review Research 4 3, 033028 (2022).

[5] Zhiqiang Fan, Jinchen Xu, Guoqiang Shu, Xiaodong Ding, Hang Lian, and Zheng Shan, "Solving the Shortest Path Problem with QAOA", SPIN 13 01, 2350002 (2023).

[6] Danylo Lykov, Jonathan Wurtz, Cody Poole, Mark Saffman, Tom Noel, and Yuri Alexeev, "Sampling frequency thresholds for the quantum advantage of the quantum approximate optimization algorithm", npj Quantum Information 9 1, 73 (2023).

[7] Lucas K. Kovalsky, Fernando A. Calderon-Vargas, Matthew D. Grace, Alicia B. Magann, James B. Larsen, Andrew D. Baczewski, and Mohan Sarovar, "Self-Healing of Trotter Error in Digital Adiabatic State Preparation", Physical Review Letters 131 6, 060602 (2023).

[8] Stefan H. Sack, Raimel A. Medina, Richard Kueng, and Maksym Serbyn, "Recursive greedy initialization of the quantum approximate optimization algorithm with guaranteed improvement", Physical Review A 107 6, 062404 (2023).

[9] Takuya Hatomura, "Shortcuts to adiabaticity: theoretical framework, relations between different methods, and versatile approximations", Journal of Physics B: Atomic, Molecular and Optical Physics 57 10, 102001 (2024).

[10] Zewen Zhang, Roger Paredes, Bhuvanesh Sundar, David Quiroga, Anastasios Kyrillidis, Leonardo Duenas-Osorio, Guido Pagano, and Kaden R A Hazzard, "Grover-QAOA for 3-SAT: quadratic speedup, fair-sampling, and parameter clustering", Quantum Science and Technology 10 1, 015022 (2025).

[11] Gregory Quiroz, Paraj Titum, Phillip Lotshaw, Pavel Lougovski, Kevin Schultz, Eugene Dumitrescu, and Itay Hen, "Quantifying the impact of precision errors on quantum approximate optimization algorithms", Physical Review Research 7 2, 023240 (2025).

[12] Jonathan Wurtz, Stefan H. Sack, and Sheng-Tao Wang, "Solving Nonnative Combinatorial Optimization Problems Using Hybrid Quantum–Classical Algorithms", IEEE Transactions on Quantum Engineering 5, 1 (2024).

[13] Michelle Chalupnik, Hans Melo, Yuri Alexeev, and Alexey Galda, 2022 IEEE International Conference on Quantum Computing and Engineering (QCE) 97 (2022) ISBN:978-1-6654-9113-6.

[14] Zheng-Hang Sun, Yong-Yi Wang, Jian Cui, and Heng Fan, "Improving the performance of quantum approximate optimization for preparing non-trivial quantum states without translational symmetry", New Journal of Physics 25 1, 013015 (2023).

[15] Yintai Zhang and Jacek Dziarmaga, "Bang-bang preparation of quantum many-body ground states in two dimensions: Optimization of the algorithm with a two-dimensional tensor network", Physical Review B 109 23, 235124 (2024).

[16] Choy Boy and David J. Wales, "Energy landscapes for the quantum approximate optimization algorithm", Physical Review A 109 6, 062602 (2024).

[17] Zichang He, Ruslan Shaydulin, Dylan Herman, Changhao Li, Rudy Raymond, Shree Hari Sureshbabu, and Marco Pistoia, Proceedings of the 43rd IEEE/ACM International Conference on Computer-Aided Design 1 (2024) ISBN:9798400710773.

[18] Stewart Morawetz and Anatoli Polkovnikov, "Efficient paths for local counterdiabatic driving", Physical Review B 110 2, 024304 (2024).

[19] Antonio A. Mele, Glen B. Mbeng, Giuseppe E. Santoro, Mario Collura, and Pietro Torta, "Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz", Physical Review A 106 6, L060401 (2022).

[20] Alejandro Gomez Cadavid, Iraitz Montalban, Archismita Dalal, Enrique Solano, and Narendra N. Hegade, "Efficient digitized counterdiabatic quantum optimization algorithm within the impulse regime for portfolio optimization", Physical Review Applied 22 5, 054037 (2024).

[21] P. Chandarana, N. N. Hegade, K. Paul, F. Albarrán-Arriagada, E. Solano, A. del Campo, and Xi Chen, "Digitized-counterdiabatic quantum approximate optimization algorithm", Physical Review Research 4 1, 013141 (2022).

[22] Callum W. Duncan, Pablo M. Poggi, Marin Bukov, Nikolaj Thomas Zinner, and Steve Campbell, "Taming Quantum Systems: A Tutorial for Using Shortcuts-To-Adiabaticity, Quantum Optimal Control, and Reinforcement Learning", PRX Quantum 6 4, 040201 (2025).

[23] Takuya Hatomura, "Classical algorithm inspired by the feedback-based algorithm for quantum optimization and local counterdiabatic driving", Physical Review E 112 5, 055303 (2025).

[24] Ihor Sokolov and Jacek Dziarmaga, "Bang-bang preparation of a quantum many-body ground state in a finite lattice: Optimization of the algorithm with a tensor network", Physical Review B 111 24, 245144 (2025).

[25] Lucas T. Brady and Stuart Hadfield, "Feedback-based optimally controlled quantum states", Physical Review A 111 6, 062406 (2025).

[26] N. N. Hegade, P. Chandarana, K. Paul, Xi Chen, F. Albarrán-Arriagada, and E. Solano, "Portfolio optimization with digitized counterdiabatic quantum algorithms", Physical Review Research 4 4, 043204 (2022).

[27] Wenyang Qian, Robert A. M. Basili, Mary Mehrnoosh Eshaghian-Wilner, Ashfaq Khokhar, Glenn Luecke, and James P. Vary, "Comparative Study of Variations in Quantum Approximate Optimization Algorithms for the Traveling Salesman Problem", Entropy 25 8, 1238 (2023).

[28] Qing Xie, Kazuhiro Seki, and Seiji Yunoki, "Variational counterdiabatic driving of the Hubbard model for ground-state preparation", Physical Review B 106 15, 155153 (2022).

[29] Ruiyi Wang, Vincenzo Roberto Arezzo, Kiran Thengil, Giovanni Pecci, and Giuseppe E Santoro, "From exponential to quadratic: optimal control for a frustrated Ising ring model", Quantum Science and Technology 10 3, 035052 (2025).

[30] Vincenzo Roberto Arezzo, Ruiyi Wang, Kiran Thengil, Giovanni Pecci, and Giuseppe E. Santoro, "Digital controllability of transverse-field Ising chains", Physical Review A 113 1, 012610 (2026).

[31] Kazutaka Takahashi and Adolfo del Campo, "Shortcuts to Adiabaticity in Krylov Space", Physical Review X 14 1, 011032 (2024).

[32] Yajie Hao, Qiming Ding, Xiao Yuan, and Xiaoting Wang, "Constraint-aware quantum optimization via Hamming weight operators", Science China Physics, Mechanics & Astronomy 69 5, 250314 (2026).

[33] Laith Abualigah, Saif AlNajdawi, Abiodun M. Ikotun, Agostino Forestiero, Faiza Gul, Absalom E. Ezugwu, Heming Jia, Mohsen Zare, Shubham Mahajan, and Mohammad Alshinwan, Metaheuristic Optimization Algorithms 147 (2024) ISBN:9780443139253.

[34] Phillip C. Lotshaw, Kevin D. Battles, Bryan Gard, Gilles Buchs, Travis S. Humble, and Creston D. Herold, "Modeling noise in global Mølmer-Sørensen interactions applied to quantum approximate optimization", Physical Review A 107 6, 062406 (2023).

[35] Kostas Blekos, Dean Brand, Andrea Ceschini, Chiao-Hui Chou, Rui-Hao Li, Komal Pandya, and Alessandro Summer, "A review on Quantum Approximate Optimization Algorithm and its variants", Physics Reports 1068, 1 (2024).

[36] James Sud, Stuart Hadfield, Eleanor Rieffel, Norm Tubman, and Tad Hogg, "Parameter-setting heuristic for the quantum alternating operator ansatz", Physical Review Research 6 2, 023171 (2024).

[37] Yu-Cheng Lin, Chuan-Chi Wang, Chia-Heng Tu, and Shih-Hao Hung, Proceedings of the 39th ACM/SIGAPP Symposium on Applied Computing 1487 (2024) ISBN:9798400702433.

[38] Yahui Chai, Yong-Jian Han, Yu-Chun Wu, Ye Li, Menghan Dou, and Guo-Ping Guo, "Shortcuts to the quantum approximate optimization algorithm", Physical Review A 105 4, 042415 (2022).

[39] 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).

[40] Giulia Salatino, Maximilian Matzler, Annarita Scocco, Procolo Lucignano, and Gianluca Passarelli, "Noise effects on diabatic quantum annealing protocols", Physical Review A 112 2, 022433 (2025).

[41] Narendra N. Hegade, Xi Chen, and Enrique Solano, "Digitized counterdiabatic quantum optimization", Physical Review Research 4 4, L042030 (2022).

[42] Camille Grange, Michael Poss, and Eric Bourreau, "An introduction to variational quantum algorithms for combinatorial optimization problems", 4OR 21 3, 363 (2023).

[43] Gopal Chandra Santra, Fred Jendrzejewski, Philipp Hauke, and Daniel J. Egger, "Squeezing and quantum approximate optimization", Physical Review A 109 1, 012413 (2024).

[44] C.-C. Joseph Wang, Phillip C. Lotshaw, Titus Morris, Vicente Leyton-Ortega, Daniel Claudino, and Travis S. Humble, "Ancilla-entangling Floquet kicks for accelerating quantum algorithms", Physical Review A 111 1, L010401 (2025).

[45] Yue Ruan, Pengyue Chen, Qi Li, Ling Yang, Zhiqiang Yuan, Xiling Xue, Xi Li, and Zhihao Liu, "XY-mixer ansatz assisted by counterdiabatic driving for combinational optimization", Physical Review Research 7 1, 013243 (2025).

[46] Filip B. Maciejewski, Jacob Biamonte, Stuart Hadfield, and Davide Venturelli, "Improving Quantum Approximate Optimization by Noise-Directed Adaptive Remapping", Quantum 9, 1906 (2025).

[47] Giovanni Pecci, Ruiyi Wang, Pietro Torta, Glen Bigan Mbeng, and Giuseppe Santoro, "Beyond quantum annealing: optimal control solutions to maxcut problems", Quantum Science and Technology 9 4, 045013 (2024).

[48] Nongmeikapam Brajabidhu Singh, Arnab Roy, and Anish Kumar Saha, "Max-flow min-cut theorem in quantum computing", Physica A: Statistical Mechanics and its Applications 649, 129990 (2024).

[49] Camille Grange, Michael Poss, and Eric Bourreau, "An introduction to variational quantum algorithms for combinatorial optimization problems", Annals of Operations Research 343 2, 847 (2024).

[50] Ewen Lawrence, Sebastian F J Schmid, Ieva Čepaitė, Peter Kirton, and Callum W Duncan, "A numerical approach for calculating exact non-adiabatic terms in quantum dynamics", SciPost Physics 18 1, 014 (2025).

[51] Mara Vizzuso, Gianluca Passarelli, Giovanni Cantele, and Procolo Lucignano, "Role of gaps in the digitized counterdiabatic quantum approximate optimization algorithm for fully connected spin models", Physical Review A 110 6, 062606 (2024).

[52] Yunlong Yu, Chenfeng Cao, Xiang-Bin Wang, Nic Shannon, and Robert Joynt, "Solution of SAT problems with the adaptive-bias quantum approximate optimization algorithm", Physical Review Research 5 2, 023147 (2023).

[53] James Walkling and Marin Bukov, "Walsh-Floquet theory of periodic kick drives", Physical Review Research 7 4, L042063 (2025).

[54] Shaojun Wu, Shan Jin, Abolfazl Bayat, and Xiaoting Wang, "Enhancing the reachability of variational quantum algorithms via input-state design", Communications Physics 9 1, 194 (2026).

[55] Daniil Rabinovich, Andrey Kardashin, and Soumik Adhikary, "Role of overparametrization in quantum approximate optimization", Physical Review A 113 6, 062617 (2026).

[56] Edric Matwiejew, Jonathan Wurtz, Jing Chen, Pascal Jahan Elahi, Tommaso Macri, and Ugo Varetto, "Continuous-time quantum-walk-based ansätze on neutral-atom hardware", Physical Review A 114 1, 012425 (2026).

[57] Lukas Schmidbauer, Carlos A. Riofrío, Florian Heinrich, Vanessa Junk, Ulrich Schwenk, Thomas Husslein, and Wolfgang Mauerer, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 2101 (2025) ISBN:979-8-3315-5736-2.

[58] Luis S. Yagüe Bosch, Tim Ehret, Francesco Petiziol, Ennio Arimondo, and Sandro Wimberger, "Shortcut‐to‐Adiabatic Controlled‐Phase Gate in Rydberg Atoms", Annalen der Physik 535 12, 2300275 (2023).

[59] Xiaoyang Wang, Yahui Chai, Xu Feng, Yibin Guo, Karl Jansen, and Cenk Tüysüz, "Imaginary Hamiltonian variational Ansatz for combinatorial optimization problems", Physical Review A 111 3, 032612 (2025).

[60] Ouyang Lei, "Infidelity Analysis of Digital Counter-Diabatic Driving in Simple Two-Qubit System", Entropy 26 10, 877 (2024).

[61] John Golden, Andreas Bärtschi, Daniel O'Malley, and Stephan Eidenbenz, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 307 (2023) ISBN:979-8-3503-4323-6.

[62] Naphan Benchasattabuse, Andreas Bärtschi, Luis Pedro García-Pintos, John Golden, Nathan Lemons, and Stephan Eidenbenz, "Lower bounds on the number of rounds of the quantum approximate optimization algorithm required for guaranteed approximation ratios", Physical Review A 111 6, 062411 (2025).

[63] Phillip C. Lotshaw, George Siopsis, James Ostrowski, Rebekah Herrman, Rizwanul Alam, Sarah Powers, and Travis S. Humble, "Approximate Boltzmann distributions in quantum approximate optimization", Physical Review A 108 4, 042411 (2023).

[64] Maxime Dupont and Bhuvanesh Sundar, "Extending relax-and-round combinatorial optimization solvers with quantum correlations", Physical Review A 109 1, 012429 (2024).

[65] Yuchen Luo, Xiaopeng Li, and Jian Lin, "Adaptive quantum optimization algorithms for programmable atom-cavity systems", Physical Review A 112 5, 052453 (2025).

[66] Gilles Buchs, Thomas L. Beck, Ryan S. Bennink, Daniel Claudino, Andrea Delgado, Nur Aiman Fadel, Peter Groszkowski, Kathleen E. Hamilton, Travis S. Humble, Neeraj Kumar, Ang Li, Phillip C. Lotshaw, Olli Mukkula, Ryousei Takano, Amit Saxena, In-Saeng Suh, Miwako Tsuji, Roel Van Beeumen, Ugo Varetto, Yan Wang, Kazuya Yamazaki, and Mikael P. Johansson, "The role of quantum computing in advancing scientific high-performance computing: A perspective from the ADAC institute", Future Generation Computer Systems 182, 108487 (2026).

[67] Ieva Čepaitė, Anatoli Polkovnikov, Andrew J. Daley, and Callum W. Duncan, "Counterdiabatic Optimized Local Driving", PRX Quantum 4 1, 010312 (2023).

[68] Yanjun Ji, Kathrin F. Koenig, and Ilia Polian, "Improving the performance of digitized counterdiabatic quantum optimization via algorithm-oriented qubit mapping", Physical Review A 110 3, 032421 (2024).

[69] Nicolas PD Sawaya, Albert T Schmitz, and Stuart Hadfield, "Encoding trade-offs and design toolkits in quantum algorithms for discrete optimization: coloring, routing, scheduling, and other problems", Quantum 7, 1111 (2023).

[70] Dan Sun, Pranav Chandarana, Zi-Hua Xin, and Xi Chen, "Optimizing counterdiabaticity by variational quantum circuits", Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 2239, 20210282 (2022).

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

[72] 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).

[73] András Grabarits, Federico Balducci, and Adolfo del Campo, "Fighting Exponentially Small Gaps by Counterdiabatic Driving", PRX Quantum 7 1, 010322 (2026).

[74] Pranav Chandarana, Pablo Suárez Vieites, Narendra N Hegade, Enrique Solano, Yue Ban, and Xi Chen, "Meta-learning digitized-counterdiabatic quantum optimization", Quantum Science and Technology 8 4, 045007 (2023).

[75] Elijah Pelofske, Andreas Bärtschi, and Stephan Eidenbenz, "Short-depth QAOA circuits and quantum annealing on higher-order ising models", npj Quantum Information 10 1, 30 (2024).

[76] Sebastiano Corli, Daniele Dragoni, Massimiliano Proietti, Massimiliano Dispenza, Carlo Cavazzoni, and Enrico Prati, "Benchmarking the emulation of measurement-based quantum computing through the Max K-Cut algorithm", Quantum Information Processing 24 12, 396 (2025).

[77] Pranav Chandarana, Narendra N. Hegade, Iraitz Montalban, Enrique Solano, and Xi Chen, "Digitized Counterdiabatic Quantum Algorithm for Protein Folding", Physical Review Applied 20 1, 014024 (2023).

[78] Diego Tancara and Francisco Albarrán-Arriagada, "High dimensional counterdiabatic quantum computing", npj Quantum Information 11 1, 116 (2025).

[79] András Grabarits, Federico Balducci, Barry C. Sanders, and Adolfo del Campo, "Nonadiabatic quantum optimization for crossing quantum phase transitions", Physical Review A 111 1, 012215 (2025).

[80] Mara Vizzuso, Gianluca Passarelli, Giovanni Cantele, and Procolo Lucignano, "Convergence of digitized-counterdiabatic QAOA: circuit depth versus free parameters", New Journal of Physics 26 1, 013002 (2024).

[81] Pietro Torta, Glen B. Mbeng, Carlo Baldassi, Riccardo Zecchina, and Giuseppe E. Santoro, "Quantum approximate optimization algorithm applied to the binary perceptron", Physical Review B 107 9, 094202 (2023).

[82] 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", Quantum 6, 870 (2022).

[83] Abhinav Suresh, Vishal Varma, Priya Batra, and T. S. Mahesh, "Counterdiabatic driving for long-lived singlet state preparation", The Journal of Chemical Physics 159 2, 024202 (2023).

[84] G. Wendin, Encyclopedia of Condensed Matter Physics 246 (2024) ISBN:9780323914086.

[85] Kuan–Cheng Chen, Hiromichi Matsuyama, and Wei–hao Huang, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 01 (2025) ISBN:979-8-3315-5736-2.

[86] Alberto Di Meglio, Karl Jansen, Ivano Tavernelli, Constantia Alexandrou, Srinivasan Arunachalam, Christian W. Bauer, Kerstin Borras, Stefano Carrazza, Arianna Crippa, Vincent Croft, Roland de Putter, Andrea Delgado, Vedran Dunjko, Daniel J. Egger, Elias Fernández-Combarro, Elina Fuchs, Lena Funcke, Daniel González-Cuadra, Michele Grossi, Jad C. Halimeh, Zoë Holmes, Stefan Kühn, Denis Lacroix, Randy Lewis, Donatella Lucchesi, Miriam Lucio Martinez, Federico Meloni, Antonio Mezzacapo, Simone Montangero, Lento Nagano, Vincent R. Pascuzzi, Voica Radescu, Enrique Rico Ortega, Alessandro Roggero, Julian Schuhmacher, Joao Seixas, Pietro Silvi, Panagiotis Spentzouris, Francesco Tacchino, Kristan Temme, Koji Terashi, Jordi Tura, Cenk Tüysüz, Sofia Vallecorsa, Uwe-Jens Wiese, Shinjae Yoo, and Jinglei Zhang, "Quantum Computing for High-Energy Physics: State of the Art and Challenges", PRX Quantum 5 3, 037001 (2024).

[87] Francesco Pio Barone, Oriel Kiss, Michele Grossi, Sofia Vallecorsa, and Antonio Mandarino, "Counterdiabatic optimized driving in quantum phase sensitive models", New Journal of Physics 26 3, 033031 (2024).

[88] Lucas T. Brady and Stuart Hadfield, "Quantum Davis-Putnam-Logemann-Loveland and generalized constraints in iterative quantum algorithms", Physical Review A 114 2, 022404 (2026).

[89] V Vijendran, Aritra Das, Dax Enshan Koh, Syed M Assad, and Ping Koy Lam, "An expressive ansatz for low-depth quantum approximate optimisation", Quantum Science and Technology 9 2, 025010 (2024).

[90] Elijah Pelofske, Andreas Bärtschi, Lukasz Cincio, John Golden, and Stephan Eidenbenz, "Scaling whole-chip QAOA for higher-order ising spin glass models on heavy-hex graphs", npj Quantum Information 10 1, 109 (2024).

[91] Haoqian Pan and Changhong Lu, "Solving the Independent Domination Problem by the Quantum Approximate Optimization Algorithm", Entropy 26 12, 1057 (2024).

[92] Vladimir Kremenetski, Tad Hogg, Stuart Hadfield, Stephen J. Cotton, and Norm M. Tubman, "Quantum Alternating Operator Ansatz (QAOA) Phase Diagrams and Applications for Quantum Chemistry", arXiv:2108.13056, (2021).

[93] Jiahao Yao, Lin Lin, and Marin Bukov, "Reinforcement Learning for Many-Body Ground-State Preparation Inspired by Counterdiabatic Driving", Physical Review X 11 3, 031070 (2021).

[94] Jonathan Wurtz and Danylo Lykov, "Fixed-angle conjectures for the quantum approximate optimization algorithm on regular MaxCut graphs", Physical Review A 104 5, 052419 (2021).

[95] Lucas T. Brady, Lucas Kocia, Przemyslaw Bienias, Aniruddha Bapat, Yaroslav Kharkov, and Alexey V. Gorshkov, "Behavior of Analog Quantum Algorithms", arXiv:2107.01218, (2021).

[96] Jonathan Wurtz and Danylo Lykov, "The fixed angle conjecture for QAOA on regular MaxCut graphs", arXiv:2107.00677, (2021).

[97] Ewen D C Lawrence, Sebastian F J Schmid, Ieva Čepaitė, Peter Kirton, and Callum W Duncan, "A numerical approach for calculating exact non-adiabatic terms in quantum dynamics", arXiv:2401.10985, (2024).

[98] Tobias Hartung and Karl Jansen, "Convergence and efficiency proof of quantum imaginary time evolution for bounded order systems", arXiv:2506.03014, (2025).

[99] Changhao Yi, "Success of digital adiabatic simulation with large Trotter step", Physical Review A 104 5, 052603 (2021).

[100] Ruslan Shaydulin, Phillip C. Lotshaw, Jeffrey Larson, James Ostrowski, and Travis S. Humble, "Parameter Transfer for Quantum Approximate Optimization of Weighted MaxCut", arXiv:2201.11785, (2022).

[101] Xiaoyang Wang, Yuexin Su, and Tongyang Li, "Performance guarantees of light-cone variational quantum algorithms for the maximum cut problem", arXiv:2504.12896, (2025).

[102] Julien Gacon, "Scalable Quantum Algorithms for Noisy Quantum Computers", arXiv:2403.00940, (2024).

[103] Caleb Rotello, Peter Graf, Matthew Reynolds, Eric B. Jones, Cody James Winkleblack, and Wesley Jones, "Calculating the expected value function of a two-stage stochastic optimization program with a quantum algorithm", arXiv:2402.15029, (2024).

[104] Jonathan Wurtz and Peter J. Love, "Classically Optimal Variational Quantum Algorithms", IEEE Transactions on Quantum Engineering 2, TQE.2021 (2021).

[105] Jiahao Yao, Haoya Li, Marin Bukov, Lin Lin, and Lexing Ying, "Monte Carlo Tree Search based Hybrid Optimization of Variational Quantum Circuits", arXiv:2203.16707, (2022).

[106] Lukas Schmidbauer, Carlos A. Riofrío, Florian Heinrich, Vanessa Junk, Ulrich Schwenk, Thomas Husslein, and Wolfgang Mauerer, "Path Matters: Industrial Data Meet Quantum Optimization", arXiv:2504.16607, (2025).

[107] Stefano Cipolla and Fabio Durastante, "Pauli-Sparse regularised Counterdiabatic Shortcuts for Linear-Ramp QAOA", arXiv:2606.28536, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-07 08:25:57) and SAO/NASA ADS (last updated successfully 2026-08-06 19:46:43). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch ADS cited-by data during last attempt 2026-08-07 08:25:57: Cannot retrieve data from ADS due to rate limitations.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.