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Analyzing the Quantum Approximate Optimization Algorithm: Ansätze, Symmetries, and Lie Algebras

Sujay Kazi1,2,3, Martín Larocca2,4,*, Marco Farinati5, Patrick J. Coles6,2, M. Cerezo7,†, and Robert Zeier8,‡

  • *Contact author: larocca@lanl.gov
  • †Contact author: cerezo@lanl.gov
  • ‡Contact author: r.zeier@fz-juelich.de
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PRX Quantum 6, 040345 – Published 25 November, 2025

DOI: https://doi.org/10.1103/yfwq-yqmk

Abstract

The quantum approximate optimization algorithm (QAOA) has been proposed as a method to obtain approximate solutions for combinatorial optimization tasks. In this work, we study the underlying algebraic properties of three QAOA ansätze for the maximum-cut problem on connected graphs, while focusing on the generated Lie algebras as well as their invariant subspaces. Specifically, we analyze the standard QAOA ansatz as well as the orbit and multiangle ansätze. We are able to fully characterize the Lie algebras of the multiangle ansatz across arbitrary connected graphs, finding that they only fall into one of just six families. Aside from the cycle and path graphs, the Lie dimensions for every graph are exponentially large in the system size, meaning that multiangle ansätze are extremely prone to exhibiting barren plateaus. Then, a similar quasi-graph-independent Lie-algebraic characterization beyond the multiangle ansatz is impeded as the circuit exhibits additional “hidden” symmetries besides those naturally arising from a certain parity-superselection operator and all automorphisms of the considered graph. Disregarding the “hidden” symmetries, we can upper bound the dimensions of the orbit and the standard Lie algebras, and the dimensions of the associated invariant subspaces are determined via explicit character formulas. To finish, we conjecture that (for most graphs) the standard Lie algebras have only components that are either exponential or that grow, at most, polynomially with the system size. This would imply that the QAOA is either prone to barren plateaus or classically simulable. More generally, our work provides a symmetry framework and tools to analyze any desired variational quantum algorithm.

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Popular Summary

The quantum approximate optimization algorithm (QAOA) is a leading proposal for tackling hard optimization problems on near-term quantum computers. However, its performance is not guaranteed and depends strongly on both the specific problem instance and the circuit design, known as the ansatz. We have developed a symmetry framework to characterize the dynamics of the QAOA, focusing on the maximum-cut problem on graphs. This framework highlights trainability limitations in sufficiently deep circuits and helps explain practical challenges of optimizing QAOA parameters.

By considering independent QAOA parameters in the so-called multiangle ansatz, we classified problem instances into six families on the basis of their symmetries. For relevant cases, this implies trainability limitations caused by optimization landscapes with gradients that vanish on average across exponentially large spaces. In contrast, the original standard ansatz has dependent parameters and additional symmetries, for which a similar classification is not readily attainable. Our symmetry analysis and initial data suggest that most problem instances either suffer from similar trainability limitations or result in dynamics restricted to small, structured subspaces, which are often classically tractable.

These structural insights offer practical guidance for choosing and designing QAOA variants: additional expressiveness must be well directed to maintain stable training and support the possibility of quantum advantage. More broadly, our symmetry framework can be applied to analyze a wide range of variational quantum algorithms.

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References (206)

  1. E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv:1411.4028.
  2. E. Farhi and A. W. Harrow, Quantum supremacy through the quantum approximate optimization algorithm, arXiv:1602.07674.
  3. M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
  4. K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke et al., Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
  5. S. Endo, Z. Cai, S. C. Benjamin, and X. Yuan, Hybrid quantum-classical algorithms and quantum error mitigation, J. Phys. Soc. Jpn. 90, 032001 (2021).
  6. M. Schuld, I. Sinayskiy, and F. Petruccione, An introduction to quantum machine learning, Contemp. Phys. 56, 172 (2015).
  7. J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Nature 549, 195 (2017).
  8. M. Cerezo, G. Verdon, H.-Y. Huang, L. Cincio, and P. J. Coles, Challenges and opportunities in quantum machine learning, Nat. Comput. Sci. 2, 567 (2022).
  9. M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Biamonte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, Barren plateaus in variational quantum computing, Nat. Rev. Phys. 7, 174 (2025).
  10. J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nat. Commun. 9, 1 (2018).
  11. M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, Cost function dependent barren plateaus in shallow parametrized quantum circuits, Nat. Commun. 12, 1 (2021).
  12. E. R. Anschuetz and B. T. Kiani, Quantum variational algorithms are swamped with traps, Nat. Commun. 13, 7760 (2022).
  13. E. R. Anschuetz, in International Conference on Learning Representations (2022), https://openreview.net/forum?id=2f1z55GVQN.
  14. L. Bittel and M. Kliesch, Training variational quantum algorithms is NP-hard, Phys. Rev. Lett. 127, 120502 (2021).
  15. X. You and X. Wu, in Proceedings of the 38th International Conference on Machine Learning, edited by M. Meila and T. Zhang (PMLR, 2021), pp. 12144–12155, https://proceedings.mlr.press/v139/you21c/you21c.pdf.
  16. E. Fontana, M. Cerezo, A. Arrasmith, I. Rungger, and P. J. Coles, Non-trivial symmetries in quantum landscapes and their resilience to quantum noise, Quantum 6, 804 (2022).
  17. G. Koßmann, L. Binkowski, L. van Luijk, T. Ziegler, and R. Schwonnek, Deep-circuit QAOA, Quantum 9, 1882 (2025).
  18. J. Rajakumar, J. Golden, A. Bärtschi, and S. Eidenbenz, in Proceedings of the 21st ACM International Conference on Computing Frontiers (ACM, New York, NY, 2024), pp. 199–206.
  19. P. Bermejo, B. Aizpurua, and R. Orús, Improving gradient methods via coordinate transformations: Applications to quantum machine learning, Phys. Rev. Res. 6, 023069 (2024).
  20. R. Zeier and T. Schulte-Herbrüggen, Symmetry principles in quantum systems theory, J. Math. Phys. 52, 113510 (2011).
  21. M. Larocca, P. Czarnik, K. Sharma, G. Muraleedharan, P. J. Coles, and M. Cerezo, Diagnosing barren plateaus with tools from quantum optimal control, Quantum 6, 824 (2022).
  22. M. Ragone, B. N. Bakalov, F. Sauvage, A. F. Kemper, C. Ortiz Marrero, M. Larocca, and M. Cerezo, A Lie algebraic theory of barren plateaus for deep parameterized quantum circuits, Nat. Commun. 15, 7172 (2024).
  23. E. Fontana, D. Herman, S. Chakrabarti, N. Kumar, R. Yalovetzky, J. Heredge, S. H. Sureshbabu, and M. Pistoia, Characterizing barren plateaus in quantum ansätze with the adjoint representation, Nat. Commun. 15, 7171 (2024).
  24. N. L. Diaz, D. García-Martín, S. Kazi, M. Larocca, and M. Cerezo, Showcasing a barren plateau theory beyond the dynamical Lie algebra, arXiv:2310.11505.
  25. M. Larocca, N. Ju, D. García-Martín, P. J. Coles, and M. Cerezo, Theory of overparametrization in quantum neural networks, Nat. Comput. Sci. 3, 542 (2023).
  26. L. Schatzki, M. Larocca, Q. T. Nguyen, F. Sauvage, and M. Cerezo, Theoretical guarantees for permutation-equivariant quantum neural networks, npj Quantum Inf. 10, 12 (2024).
  27. M. L. Goh, M. Larocca, L. Cincio, M. Cerezo, and F. Sauvage, Lie-algebraic classical simulations for variational quantum computing, Phys. Rev. Res. 7, 033266 (2025).
  28. Z. Zimborás, R. Zeier, T. Schulte-Herbrüggen, and D. Burgarth, Symmetry criteria for quantum simulability of effective interactions, Phys. Rev. A 92, 042309 (2015).
  29. F. Albertini and D. D’Alessandro, Controllability of symmetric spin networks, J. Math. Phys. 59, 052102 (2018).
  30. S. Lloyd, Quantum approximate optimization is computationally universal, arXiv:1812.11075.
  31. M. E. Morales, J. Biamonte, and Z. Zimborás, On the universality of the quantum approximate optimization algorithm, Quantum Inf. Process. 19, 1 (2020).
  32. S. Kazi, M. Larocca, and M. Cerezo, On the universality of Sn-equivariant k-body gates, New J. Phys. 26, 053030 (2024).
  33. R. Wiersema, E. Kökcü, A. F. Kemper, and B. N. Bakalov, Classification of dynamical Lie algebras of 2-local spin systems on linear, circular and fully connected topologies, npj Quantum Inf. 10, 110 (2024).
  34. D. D’Alessandro and Y. Isik, Controllability of the periodic quantum Ising spin chain and the Onsager algebra, J. Phys. A 58, 115202 (2025).
  35. G. Aguilar, S. Cichy, J. Eisert, and L. Bittel, Full classification of Pauli Lie algebras, arXiv:2408.00081.
  36. J. Allcock, M. Santha, P. Yuan, and S. Zhang, On the dynamical Lie algebras of quantum approximate optimization algorithms, arXiv:2407.12587.
  37. E. Kökcü, R. Wiersema, A. F. Kemper, and B. N. Bakalov, Classification of dynamical Lie algebras generated by spin interactions on undirected graphs, arXiv:2409.19797.
  38. M. B. Mansky, S. L. Castillo, V. R. Puigvert, and C. Linnhoff-Popien, Permutation-invariant quantum circuits, arXiv:2312.14909.
  39. M. B. Mansky, M. A. Martinez, A. B. de la Serna, S. L. Castillo, D. Nikoladou, G. Sathish, Z. Wang, S. Wölckert, and C. Linnhoff-Popien, Scaling of symmetry-restricted Lie groups, Phys. Scr. 100, 085222 (2025).
  40. E. Kökcü, T. Steckmann, Y. Wang, J. K. Freericks, E. F. Dumitrescu, and A. F. Kemper, Fixed depth Hamiltonian simulation via Cartan decomposition, Phys. Rev. Lett. 129, 070501 (2022).
  41. F. Sauvage, M. Larocca, P. J. Coles, and M. Cerezo, Building spatial symmetries into parameterized quantum circuits for faster training, Quantum Sci. Technol. 9, 015029 (2024).
  42. R. Herrman, P. C. Lotshaw, J. Ostrowski, T. S. Humble, and G. Siopsis, Multi-angle quantum approximate optimization algorithm, Sci. Rep. 12, 6781 (2022).
  43. K. Shi, R. Herrman, R. Shaydulin, S. Chakrabarti, M. Pistoia, and J. Larson, in 2022 IEEE/ACM 7th Symposium on Edge Computing (SEC) (IEEE, Piscataway, NJ, 2022), pp. 414–419.
  44. L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65, 117 (1944).
  45. I. Marvian, Theory of quantum circuits with Abelian symmetries, Phys. Rev. Res. 6, 043292 (2024).
  46. S. Boulebnane and A. Montanaro, Predicting parameters for the Quantum Approximate Optimization Algorithm for MAX-CUT from the infinite-size limit, arXiv:2110.10685.
  47. J. Basso, E. Farhi, K. Marwaha, B. Villalonga, and L. Zhou, The quantum approximate optimization algorithm at high depth for MaxCut on large-girth regular graphs and the Sherrington-Kirkpatrick model, arXiv:2110.14206v3 (2022).
  48. J. Basso, E. Farhi, K. Marwaha, B. Villalonga, and L. Zhou, in Proceedings of the 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC ‘22) (2022), pp. 7:1–7:21.
  49. S. Boulebnane, On the power of the Quantum Approximate Optimization Algorithm, Ph.D. thesis, University College, London, 2024, https://discovery.ucl.ac.uk/id/eprint/10200871.
  50. L. Zhou, S.-T. Wang, S. Choi, H. Pichler, and M. D. Lukin, Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices, Phys. Rev. X 10, 021067 (2020).
  51. E. Farhi, J. Goldstone, S. Gutmann, and L. Zhou, The quantum approximate optimization algorithm and the Sherrington-Kirkpatrick model at infinite size, Quantum 6, 759 (2022).
  52. C. Moore and S. Mertens, The Nature of Computation (Oxford University Press, New York, 2011).
  53. C. W. Commander, in Encyclopedia of Optimization, edited by C. A. Floudas and P. M. Pardalos (Springer, Boston, 2008), pp. 1991–1999.
  54. A. Newman, in Encyclopedia of Algorithms, edited by M.-Y. Kao (Springer, Boston, MA, 2008), pp. 489–492.
  55. M. R. Garey and D. S. Johnson, Computers and Intractability (W. H. Freeman and Company, San Francisco, 1979).
  56. R. Diestel, Graph Theory (Springer, Berlin, 2017, 5th ed.
  57. R. M. Karp, in Complexity of Computer Computations, edited by R. E. Miller and J. W. Thatcher (Plenum Press, New York, 1972), pp. 85–103.
  58. M. R. Garey, D. S. Johnson, and L. Stockmeyer, Some simplified NP-complete graph problems, Theor. Comput. Sci. 1, 237 (1976).
  59. M. M. Deza and M. Laurent, Geometry of Cuts and Metrics (Springer, Berlin, 1997).
  60. Lectures on Proof Verification and Approximation Algorithms, edited by E. W. Mayr, H. J. Prömel, and A. Steger (Springer, Berlin, 1998).
  61. V. V. Vazirani, Approximation Algorithms (Springer, Berlin, 2001).
  62. D. P. Williamson and D. B. Shmoys, The Design of Approximation Algorithms (Cambridge University Press, Cambridge, 2011).
  63. M. X. Goemans and D. P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM 42, 1115 (1995).
  64. H. Karloff, How good is the Goemans-Williamson MAX CUT algorithm? SIAM J. Comput. 29, 336 (1999).
  65. U. Feige and G. Schechtman, On the optimality of random hyperplane rounding technique for MAX CUT, Random Struct. Alg. 20, 403 (2002).
  66. J. Håstad, Some optimal inapproximability results, J. ACM 48, 798 (2001).
  67. S. Hadfield, Z. Wang, B. O’Gorman, E. G. Rieffel, D. Venturelli, and R. Biswas, From the quantum approximate optimization algorithm to a quantum alternating operator ansatz, Algorithms 12, 34 (2019).
  68. M. P. Harrigan, K. J. Sung, M. Neeley, K. J. Satzinger, F. Arute, K. Arya, J. Atalaya, J. C. Bardin, R. Barends, S. Boixo et al., Quantum approximate optimization of non-planar graph problems on a planar superconducting processor, Nat. Phys. 17, 1 (2021).
  69. I. Gaidai and R. Herrman, Performance analysis of multi-angle QAOA for p>1, Sci. Rep. 14, 18911 (2024).
  70. A. Wilkie, J. Ostrowski, and R. Herrman, An angle rounding parameter initialization technique for ma-QAOA, arXiv:2404.10743.
  71. Z. Holmes, K. Sharma, M. Cerezo, and P. J. Coles, Connecting ansatz expressibility to gradient magnitudes and barren plateaus, PRX Quantum 3, 010313 (2022).
  72. B. C. Hall, Lie Groups, Lie Algebras, and Representations (Springer, New York, 2015), 2nd ed.
  73. N. Jacobson, Lie Algebras (Dover Publications, New York, 1979).
  74. W. A. de Graaf, Lie Algebras: Theory and Algorithms (Elsevier, Amsterdam, 2000).
  75. D. D’Alessandro, Introduction to Quantum Control and Dynamics (CRC Press, Boca Raton, 2022), 2nd ed.
  76. D. Elliott, Bilinear Control Systems (Springer, London, 2009).
  77. N. Bourbaki, Elements of Mathematics, Lie Groups and Lie Algebras, Chapters 1–3 (Springer, Berlin, 1989).
  78. N. Bourbaki, Elements of Mathematics, Lie Groups and Lie Algebras, Chapters 7–9 (Springer, Berlin, 2008).
  79. D. A. Roozemond, Algorithms for Lie algebras of algebraic groups, Ph.D. thesis, Technische Universiteit Eindhoven, Netherlands, 2010.
  80. W. Bosma, J. Cannon, and C. Playoust, The MAGMA algebra system I: The user language, J. Symb. Comput. 24, 235 (1997).
  81. M. Grötschel and W. R. Pulleyblank, Weakly bipartite graphs and the Max-cut problem, Oper. Res. Lett. 1, 23 (1981).
  82. F. Lowenthal, Uniform finite generation of the rotation group, Rocky Mt. J. Math. 1, 575 (1971).
  83. F. Silva Leite, Bounds on the order of generation of SO(n,R) by one-parameter subgroups, Rocky Mt. J. Math. 21, 879 (1991).
  84. D. D’Alessandro, Uniform finite generation of compact Lie groups, Syst. Control Lett. 47, 87 (2002).
  85. B. T. Kiani, S. Lloyd, and R. Maity, Learning unitaries by gradient descent, arXiv:2001.11897.
  86. V. Akshay, H. Philathong, M. E. S. Morales, and J. D. Biamonte, Reachability deficits in quantum approximate optimization, Phys. Rev. Lett. 124, 090504 (2020).
  87. A. W. Harrow and R. A. Low, Random quantum circuits are approximate 2-designs, Commun. Math. Phys. 291, 257 (2009).
  88. R. A. Low, Pseudo-randomness and Learning in Quantum Computation, Ph.D. thesis, University of Bristol, UK, 2010.
  89. W. G. Brown and L. Viola, Convergence rates for arbitrary statistical moments of random quantum circuits, Phys. Rev. Lett. 104, 250501 (2010).
  90. F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, Commun. Math. Phys. 346, 397 (2016).
  91. F. G. S. L. Brandão, W. Chemissany, N. Hunter-Jones, R. Kueng, and J. Preskill, Models of quantum complexity growth, PRX Quantum 2, 030316 (2021).
  92. J. Haferkamp and N. Hunter-Jones, Improved spectral gaps for random quantum circuits: Large local dimensions and all-to-all interactions, Phys. Rev. A 104, 022417 (2021).
  93. M. Oszmaniec, A. Sawicki, and M. Horodecki, Epsilon-nets, unitary designs, and random quantum circuits, IEEE Trans. Inf. Theory 68, 989 (2022).
  94. J. Haferkamp, Random quantum circuits are approximate unitary t-designs in depth O(nt5+o(1)), Quantum 6, 795 (2022).
  95. A. W. Harrow and S. Mehraban, Approximate unitary t-designs by short random quantum circuits using nearest-neighbor and long-range gates, Commun. Math. Phys. 401, 1531 (2023).
  96. T. Schuster, J. Haferkamp, and H.-Y. Huang, Random unitaries in extremely low depth, Science 389, 92 (2025).
  97. N. Laracuente and F. Leditzky, in 16th Innovations in Theoretical Computer Science Conference (ITCS 2025), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 325, edited by R. Meka (Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2025), pp. 69:1–69:2.
  98. A. E. Deneris, P. Bermejo, P. Braccia, L. Cincio, and M. Cerezo, Exact spectral gaps of random one-dimensional quantum circuits, arXiv:2408.11201.
  99. R. O’Donnell, R. A. Servedio, and P. Paredes, in 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, Piscataway, NJ, 2023).
  100. J. Haah, Y. Liu, and X. Tan, in 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, Piscataway, NJ, 2024), pp. 463–475.
  101. D. Belkin, J. Allen, S. Ghosh, C. Kang, S. Lin, J. Sud, F. T. Chong, B. Fefferman, and B. K. Clark, Approximate t-designs in generic circuit architectures, PRX Quantum 5, 040344 (2024).
  102. A. Pesah, M. Cerezo, S. Wang, T. Volkoff, A. T. Sornborger, and P. J. Coles, Absence of barren plateaus in quantum convolutional neural networks, Phys. Rev. X 11, 041011 (2021).
  103. V. Vijendran, A. Das, D. E. Koh, S. M. Assad, and P. K. Lam, An expressive ansatz for low-depth quantum approximate optimisation, Quantum Sci. Technol. 9, 025010 (2024).
  104. A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets, Nature 549, 242 (2017).
  105. M. Larocca, F. Sauvage, F. M. Sbahi, G. Verdon, P. J. Coles, and M. Cerezo, Group-invariant quantum machine learning, PRX Quantum 3, 030341 (2022).
  106. B. E. Sagan, The Symmetric Group (Springer, New York, 2001), 2nd ed.
  107. D. Shemesh, Common eigenvectors of two matrices, Lin. Alg. Appl. 62, 11 (1984).
  108. A. Jamiołkowski and G. Pastuszak, Generalized Shemesh criterion, common invariant subspaces and irreducible completely positive superoperators, Lin. Multilin. Alg. 63, 314 (2015).
  109. E. Malvetti, Computing common eigenvectors and simultaneous triangulation, arXiv:2309.14344.
  110. P.-G. Rozon and K. Agarwal, Broken unitary picture of dynamics in quantum many-body scars, Phys. Rev. Res. 6, 023041 (2024).
  111. S. Moudgalya, N. Regnault, and B. A. Bernevig, η-pairing in Hubbard models: From spectrum generating algebras to quantum many-body scars, Phys. Rev. B 102, 085140 (2020).
  112. S. Moudgalya and O. I. Motrunich, Hilbert space fragmentation and commutant algebras, Phys. Rev. X 12, 011050 (2022).
  113. S. Moudgalya and O. I. Motrunich, Exhaustive characterization of quantum many-body scars using commutant algebras, Phys. Rev. X 14, 041069 (2024).
  114. C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papié, Weak ergodicity breaking from quantum many-body scars, Nat. Phys. 14, 745 (2018).
  115. M. Serbyn, D. A. Abanin, and Z. Papié, Quantum many-body scars and weak breaking of ergodicity, Nat. Phys. 17, 675 (2021).
  116. K. Pakrouski, P. N. Pallegar, F. K. Popov, and I. R. Klebanov, Many-body scars as a group invariant sector of Hilbert space, Phys. Rev. Lett. 125, 230602 (2020).
  117. K. Pakrouski, P. N. Pallegar, F. K. Popov, and I. R. Klebanov, Group theoretic approach to many-body scar states in fermionic lattice models, Phys. Rev. Res. 3, 043156 (2021).
  118. P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Pollmann, Ergodicity breaking arising from Hilbert space fragmentation in dipole-conserving Hamiltonians, Phys. Rev. X 10, 011047 (2020).
  119. K. Bull, J.-Y. Desaules, and Z. Papić, Quantum scars as embeddings of weakly broken Lie algebra representations, Phys. Rev. B 101, 165139 (2020).
  120. S. Moudgalya and O. I. Motrunich, Numerical methods for detecting symmetries and commutant algebras, Phys. Rev. B 107, 224312 (2023).
  121. I. Marvian, Restrictions on realizable unitary operations imposed by symmetry and locality, Nat. Phys. 18, 283 (2022).
  122. D. P. DiVincenzo, Two-bit gates are universal for quantum computation, Phys. Rev. A 51, 1015 (1995).
  123. S. Lloyd, Almost any quantum logic gate is universal, Phys. Rev. Lett. 75, 346 (1995).
  124. Z. Zimborás, R. Zeier, M. Keyl, and T. Schulte-Herbrüggen, A dynamic systems approach to fermions and their relation to spins, EPJ Quantum Technol. 1, 11 (2014).
  125. D. S. Dummit and R. M. Foote, Abstract Algebra (John Wiley & Sons, Hoboken, 2004), 3rd ed.
  126. W. A. Adkins and S. H. Weintraub, Algebra: An Approach via Module Theory (Springer, New York, 1992).
  127. J.-P. Serre, Linear Representations of Finite Groups (Springer, New York, 1977).
  128. W. Ledermann, Introduction to Group Characters (Cambridge University Press, Cambridge, 1987), 2nd ed.
  129. A. N. Sengupta, Representing Finite Groups: A Semisimple Introduction (Springer, New York, 2012).
  130. P. Erdős and A. Rényi, Asymmetric graphs, Acta Math. Acad. Sci. Hung. 14, 295 (1963).
  131. C. Godsil and G. Royle, Algebraic Graph Theory (Springer, New York, 2001).
  132. M. Cerezo, M. Larocca, D. García-Martín, N. L. Diaz, P. Braccia, E. Fontana, M. S. Rudolph, P. Bermejo, A. Ijaz, S. Thanasilp, E. R. Anschuetz, and Z. Holmes, Does provable absence of barren plateaus imply classical simulability? Nat. Commun. 16, 7907 (2025).
  133. A. Chapman and S. T. Flammia, Characterization of solvable spin models via graph invariants, Quantum 4, 278 (2020).
  134. G. E. Crooks, Performance of the quantum approximate optimization algorithm on the maximum cut problem, arXiv:1811.08419.
  135. F. G. Brandão, M. Broughton, E. Farhi, S. Gutmann, and H. Neven, For fixed control parameters the quantum approximate optimization algorithm’s objective function value concentrates for typical instances, arXiv:1812.04170.
  136. J. Basso, D. Gamarnik, S. Mei, and L. Zhou, in 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, Piscataway, NJ, 2022).
  137. J. Basso, D. Gamarnik, S. Mei, and L. Zhou, Performance and limitations of the QAOA at constant levels on large sparse hypergraphs and spin glass models (2024), arXiv:2204.10306.
  138. A. Dembo, A. Montanari, and S. Sen, Extremal cuts of sparse random graphs, Ann. Probab. 45, 1190 (2017).
  139. K. Blekos, D. Brand, A. Ceschini, C.-H. Chou, R.-H. Li, K. Pandey, and A. Summer, A review on Quantum Approximate Optimization Algorithm and its variants, Phys. Rep. 1068, 1 (2024).
  140. S. Bravyi, A. Kliesch, R. Koenig, and E. Tang, Obstacles to variational quantum optimization from symmetry protection, Phys. Rev. Lett. 125, 260505 (2020).
  141. E. Farhi, D. Gamarnik, and S. Gutmann, The quantum approximate optimization algorithm needs to see the whole graph: A typical case, arXiv:2004.09002.
  142. E. Farhi, D. Gamarnik, and S. Gutmann, The quantum approximate optimization algorithm needs to see the whole graph: Worst case examples, arXiv:2005.08747.
  143. C.-N. Chou, P. J. Love, J. S. Sandhu, and J. Shi, in 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 229, edited by M. Bojańczyk, E. Merelli, and D. P. Woodruff (Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2022), pp. 41:1–41:20.
  144. A. Misra-Spieldenner, T. Bode, P. K. Schuhmacher, T. Stollenwerk, D. Bagrets, and F. K. Wilhelm, Mean-field approximate optimization algorithm, PRX Quantum 4, 030335 (2023).
  145. G. Scriva, N. Astrakhantsev, S. Pilati, and G. Mazzola, Challenges of variational quantum optimization with measurement shot noise, Phys. Rev. A 109, 032408 (2024).
  146. M. H. Muñoz-Arias, S. Kourtis, and A. Blais, Low-depth Clifford circuits approximately solve MaxCut, Phys. Rev. Res. 6, 023294 (2024).
  147. L. Gerblich, T. Dasanjh, H. Wong, D. Ross, L. Novo, N. Chancellor, and V. Kendon, Advantages of multistage quantum walks over QAOA, arXiv:2407.06663.
  148. F. Rendl, G. Rinaldi, and A. Wiegele, Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations, Math. Program., Ser. A 121, 307 (2010).
  149. I. Dunning, S. Gupta, and J. Silberholz, What works best when? A systematic evaluation of heuristics for Max-Cut and QUBO, INFORMS J. Comput. 30, 608 (2018).
  150. M. Jünger, E. Lobe, P. Mutzel, G. Reinelt, F. Rendl, G. Rinaldi, and T. Stollenwerk, Quantum annealing versus digital computing: An experimental comparison, ACM J. Exp. Algorithmics 26, 1 (2021).
  151. V. H. Nguyen and M. Minoux, Linear size MIP formulation of Max-Cut: New properties, links with cycle inequalities and computational results, Optim. Lett. 15, 1041 (2021).
  152. D. Rehfeldt, T. Koch, and Y. Shinano, Faster exact solution of sparse MaxCut and QUBO problems, Math. Program. Comput. 15, 445 (2023).
  153. J. Charfreitag, M. Jünger, S. Mallach, and P. Mutzel, in Proceedings of the Symposium on Algorithm Engineering and Experiments, ALENEX, edited by C. A. Phillips and B. Speckmann (SIAM, Philadelphia, PA, 2022) pp. 54–66.
  154. C. H. Papadimitriou and M. Yannakakis, Optimization, approximation, and complexity classes, J. Comput. Syst. Sci. 43, 425 (1991).
  155. S. Arora, D. Karger, and M. Karpinski, Polynomial time approximation schemes for dense instances of NP-hard problems, J. Comput. Syst. Sci. 58, 193 (1999).
  156. K. Jansen, M. Karpinski, and A. L. E. Seidel, Polynomial time approximation schemes for max-bisection on planar and geometric graphs, SIAM J. Comput. 35, 110 (2005).
  157. S. Gharibian and O. Parekh, in Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 145, edited by D. Achlioptas and L. A. Végh (Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2019), pp. 31:1–31:17.
  158. E. Gaar and F. Rendl, A computational study of exact subgraph based SDP bounds for Max-Cut, stable set and coloring, Math. Program., Ser. B 183, 283 (2020).
  159. J. K. Fichte, D. L. Berre, M. Hecher, and S. Szeider, The silent (r)evolution of SAT, SIAM J. Comput. 66, 64 (2023).
  160. D. E. Knuth, The Art of Computer Programming, Volume 4B: Combinatorial Algorithms, Part 2 (Addison-Wesley, Upper Saddle River, New Jersey, 2023).
  161. R. Zeier and Z. Zimborás, On squares of representations of compact Lie algebras, J. Math. Phys. 56, 081702 (2015).
  162. T. Schulte-Herbrüggen, G. Dirr, and R. Zeier, Quantum systems theory viewed from Kossakowski-Lindblad Lie semigroups — and vice versa, Open Sys. Inf. Dyn. 24, 1740019 (2017).
  163. T. Schulte-Herbrüggen, R. Zeier, M. Keyl, and G. Dirr, in Quantum Information: From Foundations to Quantum Technology Applications, Vol. 2, edited by D. Bruss and G. Leuchs (Wiley, Weinheim, 2019), 2nd ed., pp. 607–641.
  164. R. Shaydulin and S. M. Wild, Exploiting symmetry reduces the cost of training QAOA, IEEE Trans. Quantum Eng. 2, 3101409 (2021).
  165. R. Shaydulin, S. Hadfield, T. Hogg, and I. Safro, Classical symmetries and the quantum approximate optimization algorithm, Quantum Inf. Process. 20, 359 (2021).
  166. I. Marvian, H. Liu, and A. Hulse, Rotationally invariant circuits: Universality with the exchange interaction and two ancilla qubits, Phys. Rev. Lett. 132, 130201 (2024).
  167. I. Marvian, H. Liu, and A. Hulse, Qudit circuits with SU(d) symmetry: Locality imposes additional conservation laws, arXiv:2105.12877.
  168. C. V. Kraus, M. M. Wolf, and J. I. Cirac, Quantum simulations under translational symmetry, Phys. Rev. A. 75, 022303 (2007).
  169. R. Gargiulo, M. Rizzi, and R. Zeier, in ReAQCT ‘24: Proceedings of Recent Advances in Quantum Computing and Technology, June 19–20, 2024, Budapest, Hungary (ACM, New York, NY, 2024), pp. 20–50.
  170. S. Kazi, M. Larocca, M. Farinati, P. J. Coles, M. Cerezo, and R. Zeier, Data for “Analyzing the Quantum Approximate Optimization Algorithm: Ansätze, Symmetries, and Lie Algebras,” [Data set], 2025, https://doi.org/10.5281/zenodo.17497740.
  171. M. Brešar, Introduction to Noncommutative Algebra (Springer, Cham, 2014).
  172. F. Lorenz, Algebra, Volume II (Springer, New York, 2008).
  173. N. Jacobson, Basic Algebra I (Freeman, San Francisco, 1985), 2nd ed.
  174. N. Jacobson, Basic Algebra II (Freeman, San Francisco, 1989), 2nd ed.
  175. K. Lux and H. Pahlings, Representations of Groups: A Computational Approach (Cambridge University Press, Cambridge, 2010).
  176. C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras (Wiley, New York, 1962).
  177. C. W. Curtis and I. Reiner, Methods of Representation Theory, Volume I (Wiley, New York, 1981).
  178. E. Noether, Hyperkomplexe Größen und Darstellungstheorie, Math. Z. 30, 641 (1929).
  179. E. Noether, Nichtkommutative Algebren, Math. Z. 37, 514 (1933).
  180. E. Noether, Gesammelte Abhandlungen – Collected Papers (Springer, Berlin, 1983).
  181. B. L. van der Waerden, Algebra, Volume II (Springer, New York, 1991), 3rd ed.
  182. H. Weyl, Generalized Riemann matrices and factor sets, Ann. Math. 37, 709 (1936).
  183. H. Weyl, The Classical Groups (Princeton University Press, Princeton, 1953).
  184. M.-C. Bañuls, J. I. Cirac, and M. M. Wolf, Entanglement in fermionic systems, Phys. Rev. A 76, 022311 (2007).
  185. G. C. Wick, A. S. Wightman, and E. P. Wigner, The intrinsic parity of elementary particles, Phys. Rev. 88, 101 (1952).
  186. R. Brauer and H. Weyl, Spinors in n dimensions, Am. J. Math. 57, 425 (1935).
  187. F. D. Murnaghan, The Theory of Group Representations (Dover Publications, New York, 1963).
  188. B. Kaufman, Crystal statistics. II. Partition function evaluated by spinor analysis, Phys. Rev. 76, 1232 (1949).
  189. H. Boerner, Representation of Groups (North-Holland, Amsterdam, 1969), 2nd ed.
  190. W. Miller, Jr., Symmetry Groups and Their Applications (Academic Press, New York, 1972).
  191. D. H. Sattinger and O. L. Weaver, Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics (Springer, New York, 1986).
  192. W. Fulton and J. Harris, Representation Theory: A First Course (Springer, New York, 1991).
  193. H. Georgi, Lie Algebras in Particle Physics (CRC Press, Boca Raton, 1999), 2nd ed.
  194. A. Zee, Group Theory in a Nutshell for Physicists (Princeton University Press, Princeton, 2016).
  195. H. Samelson, Notes on Lie Algebras (Springer, New York, 1999), 2nd ed.
  196. M. Obata, On subgroups of the orthogonal group, Trans. Am. Math. Soc. 87, 347 (1958).
  197. E. B. Dynkin, Maximal subgroups of the classical groups, Amer. Math. Soc. Transl. Ser. 2 6, 245 (1957).
  198. E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl. Ser. 2 6, 111 (1957).
  199. V. V. Gorbatsevich, A. L. Onishchik, and E. B. Vinberg, in Lie Groups and Lie Algebras III (Springer, Berlin, 1994), pp. 1–244.
  200. F. Antoneli, M. Forger, and P. Gaviria, Maximal subgroups of compact Lie groups, J. Lie Theory 22, 949 (2012).
  201. A. Borel, Semisimple Groups and Riemannian Symmetric Spaces (Hindustan Book Agency, New Delhi, 1998).
  202. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic Press, New York, 1978).
  203. N. Bourbaki, Elements of Mathematics, Lie Groups and Lie Algebras, Chapters 4–6 (Springer, Berlin, 2008).
  204. A. Borel and J. de Siebenthal, Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helv. 23, 200 (1949).
  205. M. Goto and F. D. Grosshans, Semisimple Lie Algebras (Marcel Dekker, New York, 1978).
  206. R. P. Stanley, Enumerative Combinatorics, Vol. I (Cambridge University Press, Cambridge, 2012), 2nd ed.

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PRX Quantum 6, 040345– Published 25 November, 2025
  • Received 14 July 2025
  • Revised 3 October 2025
  • Accepted 14 October 2025
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