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Ising machines as hardware solvers of combinatorial optimization problems

Abstract

Ising machines are hardware solvers that aim to find the absolute or approximate ground states of the Ising model. The Ising model is of fundamental computational interest because any problem in the complexity class NP can be formulated as an Ising problem with only polynomial overhead, and thus a scalable Ising machine that outperforms existing standard digital computers could have a huge impact for practical applications. We survey the status of various approaches to constructing Ising machines and explain their underlying operational principles. The types of Ising machines considered here include classical thermal annealers based on technologies such as spintronics, optics, memristors and digital hardware accelerators; dynamical systems solvers implemented with optics and electronics; and superconducting-circuit quantum annealers. We compare and contrast their performance using standard metrics such as the ground-state success probability and time-to-solution, give their scaling relations with problem size, and discuss their strengths and weaknesses.

Key points

  • Dedicated hardware solvers for the Ising model are of great interest, owing to their many potential practical applications and the end of Moore’s law, which motivate alternative computational approaches.

  • Three main computing methods that Ising machines use are classical annealing, quantum annealing and dynamical system evolution. A single machine can operate on the basis of multiple computing approaches.

  • Today, Ising hardware based on classical digital technologies is the best performing for common benchmark problems. However, the performance is problem-dependent, and alternative methods can perform well for particular classes of problems.

  • For particular crafted problem instances, quantum approaches have been observed to have superior performance over classical algorithms, motivating quantum hardware approaches and quantum-inspired classical algorithms.

  • Hybrid quantum–classical and digital–analogue algorithms are promising for future development; they may harness the complementary advantages of both.

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Fig. 1: Combinatorial problems, the Ising model, and its energy landscape.
Fig. 2: Example technologies used to realize various types of Ising machines.
Fig. 3: Success probability comparison of Ising machines.
Fig. 4: Time-to-solution (TTS) comparison of Ising machines.

References

  1. Lucas, A. Ising formulations of many NP problems. Front. Phys. 2, 5 (2014).

    Article  Google Scholar 

  2. Tanahashi, K., Takayanagi, S., Motohashi, T. & Tanaka, S. Application of Ising machines and a software development for Ising machines. J. Phys. Soc. Japan 88, 061010 (2019).

    ADS  Article  Google Scholar 

  3. Smelyanskiy, V. N. et al. A near-term quantum computing approach for hard computational problems in space exploration. Preprint at https://arxiv.org/abs/1204.2821 (2012).

  4. Hauke, P., Katzgraber, H. G., Lechner, W., Nishimori, H. & Oliver, W. D. Perspectives of quantum annealing: methods and implementations. Rep. Prog. Phys. 83, 054401 (2020).

    ADS  Article  Google Scholar 

  5. Karp, R. M. in Complexity of Computer Computations, 85–103 (Springer, 1972).

  6. Mézard, M., Parisi, G. & Virasoro, M. A. Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications vol. 9 (World Scientific, 1987).

  7. Barahona, F., Grötschel, M., Jünger, M. & Reinelt, G. An application of combinatorial optimization to statistical physics and circuit layout design. Oper. Res. 36, 493–513 (1988).

    MATH  Article  Google Scholar 

  8. Chang, K. & Du, D. C. Efficient algorithms for layer assignment problem. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 6, 67–78 (1987).

    Article  Google Scholar 

  9. Wang, J., Jebara, T. & Chang, S.-F. Semi-supervised learning using greedy Max-cut. J. Mach. Learn. Res. 14, 771–800 (2013).

    MathSciNet  MATH  Google Scholar 

  10. Collins, T. Graph Cut Matching in Computer Vision (Univ. Edinburgh, 2004).

  11. Arora, C., Banerjee, S., Kalra, P. & Maheshwari, S. An efficient graph cut algorithm for computer vision problems. In European Conf. Computer Vision, 552–565 (Springer, 2010).

  12. Turing, A. M. On computable numbers, with an application to the entscheidungsproblem. Proc. Lond. Math. Soc. 2, 230–265 (1937).

    MathSciNet  MATH  Article  Google Scholar 

  13. Bournez, O. & Pouly, A. in Handbook of Computability and Complexity in Analysis 2018 (eds Brattka, V. & Hertling, P.) 173–226 (Springer, 2018).

  14. Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982).

    MathSciNet  Article  Google Scholar 

  15. Buluta, I. & Nori, F. Quantum simulators. Science 326, 108–111 (2009).

    ADS  Article  Google Scholar 

  16. Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153 (2014).

    ADS  Article  Google Scholar 

  17. Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  18. Greiner, M., Mandel, O., Esslinger, T., Hänsch, T. W. & Bloch, I. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415, 39–44 (2002).

    ADS  Article  Google Scholar 

  19. Labuhn, H. et al. Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models. Nature 534, 667–670 (2016).

    ADS  Article  Google Scholar 

  20. Bernien, H. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579–584 (2017).

    ADS  Article  Google Scholar 

  21. Keesling, A. et al. Quantum Kibble–Zurek mechanism and critical dynamics on a programmable Rydberg simulator. Nature 568, 207–211 (2019).

    ADS  Article  Google Scholar 

  22. Scholl, P. et al. Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms. Nature 595, 233–238 (2021).

    ADS  Article  Google Scholar 

  23. Farhi, E. et al. A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292, 472–475 (2001).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  24. Byrnes, T., Yan, K. & Yamamoto, Y. Accelerated optimization problem search using Bose–Einstein condensation. New J. Phys. 13, 113025 (2011).

    ADS  Article  Google Scholar 

  25. King, A. D. et al. Observation of topological phenomena in a programmable lattice of 1,800 qubits. Nature 560, 456–460 (2018).

    ADS  Article  Google Scholar 

  26. Harris, R. et al. Phase transitions in a programmable quantum spin glass simulator. Science 361, 162–165 (2018).

    ADS  MathSciNet  Article  Google Scholar 

  27. Vadlamani, S. K., Xiao, T. P. & Yablonovitch, E. Physics successfully implements Lagrange multiplier optimization. Proc. Natl Acad. Sci. USA 117, 26639–26650 (2020).

    ADS  MathSciNet  Article  Google Scholar 

  28. Geman, S. & Geman, D. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).

    MATH  Article  Google Scholar 

  29. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. & Teller, E. Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953).

    ADS  MATH  Article  Google Scholar 

  30. Hastings, W. K. Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970).

    MathSciNet  MATH  Article  Google Scholar 

  31. Swendsen, R. H. & Wang, J.-S. Replica Monte Carlo simulation of spin-glasses. Phys. Rev. Lett. 57, 2607–2609 (1986).

    ADS  MathSciNet  Article  Google Scholar 

  32. Earl, D. J. & Deem, M. W. Parallel tempering: theory, applications, and new perspectives. Phys. Chem. Chem. Phys. 7, 3910–3916 (2005).

    Article  Google Scholar 

  33. Wang, W., Machta, J. & Katzgraber, H. G. Population annealing: theory and application in spin glasses. Phys. Rev. E 92, 063307 (2015).

    ADS  Article  Google Scholar 

  34. Zhu, Z., Ochoa, A. J. & Katzgraber, H. G. Efficient cluster algorithm for spin glasses in any space dimension. Phys. Rev. Lett. 115, 077201 (2015).

    ADS  Article  Google Scholar 

  35. Kirkpatrick, S., Gelatt, C. D. & Vecchi, M. P. Optimization by simulated annealing. Science 220, 671–680 (1983).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  36. Černý, V. Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. J. Optim. Theory Appl. 45, 41–51 (1985).

    MathSciNet  MATH  Article  Google Scholar 

  37. Camsari, K. Y., Faria, R., Sutton, B. M. & Datta, S. Stochastic p-bits for invertible logic. Phys. Rev. X 7, 031014 (2017).

    Google Scholar 

  38. Camsari, K. Y., Sutton, B. M. & Datta, S. p-bits for probabilistic spin logic. Appl. Phys. Rev. 6, 011305 (2019).

    ADS  Article  Google Scholar 

  39. Borders, W. A. et al. Integer factorization using stochastic magnetic tunnel junctions. Nature 573, 390–393 (2019).

    ADS  Article  Google Scholar 

  40. Sutton, B., Camsari, K. Y., Behin-Aein, B. & Datta, S. Intrinsic optimization using stochastic nanomagnets. Sci. Rep. 7, 1–9 (2017).

    Article  Google Scholar 

  41. Shim, Y., Jaiswal, A. & Roy, K. Ising computation based combinatorial optimization using spin-Hall effect induced stochastic magnetization reversal. J. Appl. Phys. 121, 193902 (2017).

    ADS  Article  Google Scholar 

  42. Arnalds, U. B. et al. A new look on the two-dimensional Ising model: thermal artificial spins. New J. Phys. 18, 023008 (2016).

    ADS  Article  Google Scholar 

  43. Bhanja, S., Karunaratne, D., Panchumarthy, R., Rajaram, S. & Sarkar, S. Non-Boolean computing with nanomagnets for computer vision applications. Nature Nanotechnol. 11, 177–183 (2016).

    ADS  Article  Google Scholar 

  44. Lee, A. et al. A thermodynamic core using voltage-controlled spin-orbit-torque magnetic tunnel junctions. Nanotechnology https://doi.org/10.1088/1361-6528/abeb9b (2021).

  45. Mizushima, K., Goto, H. & Sato, R. Large-scale Ising-machines composed of magnetic neurons. Appl. Phys. Lett. 111, 172406 (2017).

    ADS  Article  Google Scholar 

  46. Pierangeli, D., Marcucci, G. & Conti, C. Large-scale photonic Ising machine by spatial light modulation. Phys. Rev. Lett. 122, 213902 (2019).

    ADS  Article  Google Scholar 

  47. Pierangeli, D., Marcucci, G. & Conti, C. Adiabatic evolution on a spatial-photonic Ising machine. Optica 7, 1535–1543 (2020).

    ADS  Article  Google Scholar 

  48. Roques-Carmes, C. et al. Heuristic recurrent algorithms for photonic Ising machines. Nat. Commun. 11, 1–8 (2020).

    Article  Google Scholar 

  49. Cai, F. et al. Power-efficient combinatorial optimization using intrinsic noise in memristor Hopfield neural networks. Nat. Electron. 3, 409–418 (2020).

    Article  Google Scholar 

  50. Bojnordi, M. N. & Ipek, E. Memristive Boltzmann machine: a hardware accelerator for combinatorial optimization and deep learning. In 2016 IEEE Int. Symp. High-Performance Computer Architecture (HPCA), https://doi.org/10.1109/HPCA.2016.7446049 (IEEE, 2016).

  51. Behin-Aein, B., Diep, V. & Datta, S. A building block for hardware belief networks. Sci. Rep. 6, 29893 (2016).

    ADS  Article  Google Scholar 

  52. Sarkar, S. & Bhanja, S. Synthesizing energy minimizing quantum-dot cellular automata circuits for vision computing. In 5th IEEE Conf. Nanotechnol. 2005, 541–544 (IEEE, 2005).

  53. Kiraly, B., Knol, E. J., van Weerdenburg, W. M., Kappen, H. J. & Khajetoorians, A. A. An atomic Boltzmann machine capable of self-adaption. Nat. Nanotechnol. 16, 414–420 (2021).

    ADS  Article  Google Scholar 

  54. Guo, S. Y. et al. A molecular computing approach to solving optimization problems via programmable microdroplet arrays. Matter 4, 1107–1124 (2021).

    ADS  Article  Google Scholar 

  55. Byrnes, T., Koyama, S., Yan, K. & Yamamoto, Y. Neural networks using two-component Bose–Einstein condensates. Sci. Rep. 3, 2531 (2013).

    ADS  Article  Google Scholar 

  56. Yamaoka, M. et al. A 20k-spin Ising chip to solve combinatorial optimization problems with CMOS annealing. IEEE J. Solid-State Circuits 51, 303–309 (2015).

    Google Scholar 

  57. Matsubara, S. et al. Digital annealer for high-speed solving of combinatorial optimization problems and its applications. In 2020 25th Asia and South Pacific Design Automation Conf. (ASP-DAC), 667–672 (IEEE, 2020).

  58. Aramon, M. et al. Physics-inspired optimization for quadratic unconstrained problems using a digital annealer. Front. Phys. 7, 48 (2019).

    Article  Google Scholar 

  59. Su, Y., Kim, H. & Kim, B. Cim-spin: A 0.5-to-1.2 V scalable annealing processor using digital compute-in-memory spin operators and register-based spins for combinatorial optimization problems. In 2020 IEEE Int. Solid-State Circuits Conf. (ISSCC), 480–482 (IEEE, 2020).

  60. Yamamoto, K. et al. Statica: a 512-spin 0.25 m-weight full-digital annealing processor with a near-memory all-spin-updates-at-once architecture for combinatorial optimization with complete spin–spin interactions. In 2020 IEEE Int. Solid-State Circuits Conf. (ISSCC), 138–140 (IEEE, 2020).

  61. Tsukamoto, S., Takatsu, M., Matsubara, S. & Tamura, H. An accelerator architecture for combinatorial optimization problems. Fujitsu Sci. Tech. J. 53, 8–13 (2017).

    Google Scholar 

  62. Matsubara, S. et al. Ising-model optimizer with parallel-trial bit-sieve engine. In Conf. Complex, Intelligent, and Software Intensive Systems, 432–438 (Springer, 2017).

  63. Yamamoto, K. et al. A time-division multiplexing Ising machine on FPGAs. In Proc. 8th Int. Symp. Highly Efficient Accelerators and Reconfigurable Technologies (HEART), https://doi.org/10.1145/3120895.3120905 (2017).

  64. Patel, S., Chen, L., Canoza, P. & Salahuddin, S. Ising model optimization problems on a FPGA accelerated restricted Boltzmann machine. Preprint at https://arxiv.org/abs/2008.04436 (2020).

  65. Aadit, N. A. et al. Massively parallel probabilistic computing with sparse Ising machines. Preprint at https://arxiv.org/abs/2110.02481 (2021).

  66. Reuther, A. et al. Survey and benchmarking of machine learning accelerators. In 2019 IEEE High Perform. Extreme Comput. Conf. (HPEC), 1–9 (IEEE, 2019).

  67. Arima, Y. et al. A 336-neuron, 28 K-synapse, self-learning neural network chip with branch-neuron-unit architecture. IEEE J. Solid-state Circuits 26, 1637–1644 (1991).

    ADS  Article  Google Scholar 

  68. Alspector, J., Allen, R. B., Jayakumar, A., Zeppenfeld, T. & Meir, R. Relaxation networks for large supervised learning problems. In Adv. Neural Inf. Process. Syst., 1015–1021 (Citeseer, 1991).

  69. Merolla, P. A. et al. A million spiking-neuron integrated circuit with a scalable communication network and interface. Science 345, 668–673 (2014).

    ADS  Article  Google Scholar 

  70. Skubiszewski, M. An exact hardware implementation of the Boltzmann machine. In Proc. 4th IEEE Symp. Parallel and Distributed Processing, 107–110 (IEEE, 1992).

  71. Zhu, J. & Sutton, P. FPGA implementations of neural networks — a survey of a decade of progress. In Int. Conf. Field Programmable Logic and Applications, 1062–1066 (Springer, 2003).

  72. Kim, S. K., McAfee, L. C., McMahon, P. L. & Olukotun, K. A highly scalable restricted Boltzmann machine FPGA implementation. In 2009 Int. Conf. Field Programmable Logic and Applications, 367–372 (IEEE, 2009).

  73. Kim, S. K., McMahon, P. L. & Olukotun, K. A large-scale architecture for restricted Boltzmann machines. In 18th IEEE Annu. Int. Symp. Field-Programmable Custom Computing Machines, 201–208 (IEEE, 2010).

  74. Le Ly, D. & Chow, P. High-performance reconfigurable hardware architecture for restricted Boltzmann machines. IEEE Trans. Neural Networks 21, 1780–1792 (2010).

    Article  Google Scholar 

  75. Kim, L.-W., Asaad, S. & Linsker, R. A fully pipelined FPGA architecture of a factored restricted boltzmann machine artificial neural network. ACM Transactions on Reconfigurable Technology and Systems (TRETS) 7, 1–23 (2014).

    Google Scholar 

  76. Ly, D. L., Paprotski, V. & Yen, D. Neural Networks on GPUs: Restricted Boltzmann Machines (2008); https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.431.8720&rep=rep1&type=pdf

  77. Zhu, Y., Zhang, Y. & Pan, Y. Large-scale restricted Boltzmann machines on single GPU. In 2013 IEEE Int. Conf. Big Data, 169–174 (IEEE, 2013).

  78. Okuyama, T., Sonobe, T., Kawarabayashi, K.-i & Yamaoka, M. Binary optimization by momentum annealing. Phys. Rev. E 100, 012111 (2019).

    ADS  Article  Google Scholar 

  79. Hopfield, J. J. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl Acad. Sci. USA 79, 2554–2558 (1982).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  80. Hopfield, J. J. & Tank, D. W. ‘Neural’ computation of decisions in optimization problems. Biol. Cybern. 52, 141–152 (1985).

    MATH  Article  Google Scholar 

  81. Von Neumann, J. Non-linear capacitance or inductance switching, amplifying, and memory organs. US Patent 2,815,488 (1957).

  82. Wigington, R. A new concept in computing. Proc. IRE 47, 516–523 (1959).

    MATH  Article  Google Scholar 

  83. Goto, E. The parametron, a digital computing element which utilizes parametric oscillation. Proc. IRE 47, 1304–1316 (1959).

    Article  Google Scholar 

  84. Csaba, G. & Porod, W. Coupled oscillators for computing: a review and perspective. Appl. Phys. Rev. 7, 011302 (2020).

    Article  Google Scholar 

  85. Raychowdhury, A. et al. Computing with networks of oscillatory dynamical systems. Proc. IEEE 107, 73–89 (2018).

    Article  Google Scholar 

  86. Kuramoto, Y. in International Symposium on Mathematical Problems in Theoretical Physics, 420 (Lecture Notes in Physics vol. 30, Springer, 1975).

  87. Acebrón, J. A., Bonilla, L. L., Vicente, C. J. P., Ritort, F. & Spigler, R. The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137 (2005).

    ADS  Article  Google Scholar 

  88. Breakspear, M., Heitmann, S. & Daffertshofer, A. Generative models of cortical oscillations: neurobiological implications of the Kuramoto model. Front. Human Neurosci. 4, 190 (2010).

    Article  Google Scholar 

  89. Wu, C. W. & Chua, L. O. Application of graph theory to the synchronization in an array of coupled nonlinear oscillators. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 42, 494–497 (1995).

    Article  Google Scholar 

  90. Wu, C. W. Graph coloring via synchronization of coupled oscillators. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 45, 974–978 (1998).

    Article  Google Scholar 

  91. Wu, J., Jiao, L., Li, R. & Chen, W. Clustering dynamics of nonlinear oscillator network: application to graph coloring problem. Physica D 240, 1972–1978 (2011).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  92. Kalinin, K. P. & Berloff, N. G. Global optimization of spin Hamiltonians with gain-dissipative systems. Sci. Rep. 8, 1–9 (2018).

    Google Scholar 

  93. Wang, T. & Roychowdhury, J. OIM: oscillator-based Ising machines for solving combinatorial optimisation problems. In Int. Conf. Unconventional Computation and Natural Computation, 232–256 (Springer, 2019).

  94. Afoakwa, R., Zhang, Y., Vengalam, U. K. R., Ignjatovic, Z. & Huang, M. BRIM: bistable resistively-coupled Ising machine. In 2021 IEEE Int. Symp. High-Performance Computer Architecture (HPCA), 749–760 (IEEE, 2021).

  95. McGoldrick, B. C., Sun, J. Z. & Liu, L. Ising machine based on electrically coupled spin Hall nano-oscillators. Phys. Rev. Appl. 17, 014006 (2021).

    ADS  Article  Google Scholar 

  96. Albertsson, D. I. et al. Ultrafast Ising machines using spin torque nano-oscillators. Appl. Phys. Lett. 118, 112404 (2021).

    ADS  Article  Google Scholar 

  97. Chou, J., Bramhavar, S., Ghosh, S. & Herzog, W. Analog coupled oscillator based weighted Ising machine. Sci. Rep. 9, 14786 (2019).

    ADS  Article  Google Scholar 

  98. Xiao, T. P. Optoelectronics for Refrigeration and Analog Circuits for Combinatorial Optimization. PhD thesis, Univ. California Berkeley (2019).

  99. Saito, K., Aono, M. & Kasai, S. Amoeba-inspired analog electronic computing system integrating resistance crossbar for solving the travelling salesman problem. Sci. Rep. 10, 20772 (2020).

    Article  Google Scholar 

  100. Shukla, N. et al. Synchronized charge oscillations in correlated electron systems. Sci. Rep. 4, 4964 (2014).

    Article  Google Scholar 

  101. Parihar, A., Shukla, N., Jerry, M., Datta, S. & Raychowdhury, A. Vertex coloring of graphs via phase dynamics of coupled oscillatory networks. Sci. Rep. 7, 911 (2017).

    ADS  Article  Google Scholar 

  102. Dutta, S. et al. An Ising Hamiltonian solver using stochastic phase-transition nano-oscillators. Preprint at https://arxiv.org/abs/2007.12331 (2020).

  103. Zahedinejad, M. et al. Two-dimensional mutually synchronized spin Hall nano-oscillator arrays for neuromorphic computing. Nat. Nanotechnol. 15, 47–52 (2020).

    ADS  Article  Google Scholar 

  104. Houshang, A. et al. A spin Hall Ising machine. Preprint at https://arxiv.org/abs/2006.02236 (2020).

  105. Mallick, A. et al. Using synchronized oscillators to compute the maximum independent set. Nat. Commun. 11, 4689 (2020).

    ADS  Article  Google Scholar 

  106. Bashar, M. K. et al. Experimental demonstration of a reconfigurable coupled oscillator platform to solve the Max-Cut problem. IEEE J. Exploratory Solid-State Computational Devices and Circuits 6, 116–121 (2020).

    ADS  Article  Google Scholar 

  107. Ahmed, I., Chiu, P.-W., Moy, W. & Kim, C. H. A probabilistic compute fabric based on coupled ring oscillators for solving combinatorial optimization problems. IEEE J. Solid-State Circuits 56, 2870–2880 (2021).

  108. Traversa, F. L. & Di Ventra, M. Universal memcomputing machines. IEEE Trans. Neural Netw. Learn. Syst. 26, 2702–2715 (2015).

    MathSciNet  Article  Google Scholar 

  109. Di Ventra, M. & Traversa, F. L. Perspective: Memcomputing: leveraging memory and physics to compute efficiently. J. Appl. Phys. 123, 180901 (2018).

    Article  Google Scholar 

  110. Sheldon, F., Traversa, F. L. & Di Ventra, M. Taming a nonconvex landscape with dynamical long-range order: memcomputing Ising benchmarks. Phys. Rev. E 100, 053311 (2019).

    ADS  Article  Google Scholar 

  111. Aiken, J. & Traversa, F. L. Memcomputing for accelerated optimization. Preprint at https://arxiv.org/abs/2003.10644 (2020).

  112. Utsunomiya, S., Takata, K. & Yamamoto, Y. Mapping of Ising models onto injection-locked laser systems. Opt. Express 19, 18091–18108 (2011).

    ADS  Article  Google Scholar 

  113. Wang, Z., Marandi, A., Wen, K., Byer, R. L. & Yamamoto, Y. Coherent Ising machine based on degenerate optical parametric oscillators. Phys. Rev. A 88, 063853 (2013).

    ADS  Article  Google Scholar 

  114. Marandi, A., Wang, Z., Takata, K., Byer, R. L. & Yamamoto, Y. Network of time-multiplexed optical parametric oscillators as a coherent Ising machine. Nat. Photonics 8, 937–942 (2014).

    ADS  Article  Google Scholar 

  115. McMahon, P. L. et al. A fully programmable 100-spin coherent Ising machine with all-to-all connections. Science 354, 614–617 (2016).

    ADS  MathSciNet  Article  Google Scholar 

  116. Inagaki, T. et al. A coherent Ising machine for 2000-node optimization problems. Science 354, 603–606 (2016).

    ADS  Article  Google Scholar 

  117. Yamamoto, Y. et al. Coherent Ising machines — optical neural networks operating at the quantum limit. npj Quantum Inf. 3, 49 (2017).

    ADS  Article  Google Scholar 

  118. Leleu, T., Yamamoto, Y., McMahon, P. L. & Aihara, K. Destabilization of local minima in analog spin systems by correction of amplitude heterogeneity. Phys. Rev. Letters 122, 040607 (2019).

    ADS  Article  Google Scholar 

  119. Hamerly, R. et al. Experimental investigation of performance differences between coherent Ising machines and a quantum annealer. Sci. Adv. 5, eaau0823 (2019).

    ADS  Article  Google Scholar 

  120. Yamamoto, Y., Leleu, T., Ganguli, S. & Mabuchi, H. Coherent Ising machines — quantum optics and neural network perspectives. Appl. Phys. Lett. 117, 160501 (2020).

    ADS  Article  Google Scholar 

  121. Honjo, T. et al. 100,000-spin coherent Ising machine. Science Advances 7, eabh0952 (2021).

    ADS  Article  Google Scholar 

  122. Marandi, A., Leindecker, N. C., Vodopyanov, K. L. & Byer, R. L. All-optical quantum random bit generation from intrinsically binary phase of parametric oscillators. Opt. Express 20, 19322–19330 (2012).

    ADS  Article  Google Scholar 

  123. Kalinin, K. P. & Berloff, N. G. Networks of non-equilibrium condensates for global optimization. New J. Phys. 20, 113023 (2018).

    ADS  Article  Google Scholar 

  124. Takata, K. et al. A 16-bit coherent Ising machine for one-dimensional ring and cubic graph problems. Sci. Rep. 6, 34089 (2016).

    ADS  Article  Google Scholar 

  125. Inagaki, T. et al. Large-scale Ising spin network based on degenerate optical parametric oscillators. Nat. Photonics 10, 415–419 (2016).

    ADS  Article  Google Scholar 

  126. Okawachi, Y. et al. Demonstration of chip-based coupled degenerate optical parametric oscillators for realizing a nanophotonic spin-glass. Nat. Commun. 11, 4119 (2020).

    ADS  Article  Google Scholar 

  127. Goto, H. Bifurcation-based adiabatic quantum computation with a nonlinear oscillator network. Sci. Rep. 6, 21686 (2016).

    ADS  Article  Google Scholar 

  128. Tamate, S., Yamamoto, Y., Marandi, A., McMahon, P. & Utsunomiya, S. Simulating the classical XY model with a laser network. Preprint at https://arxiv.org/abs/1608.00358 (2016).

  129. Babaeian, M. et al. A single shot coherent Ising machine based on a network of injection-locked multicore fiber lasers. Nat. Commun. 10, 3516 (2019).

    ADS  Article  Google Scholar 

  130. Parto, M., Hayenga, W., Marandi, A., Christodoulides, D. N. & Khajavikhan, M. Realizing spin Hamiltonians in nanoscale active photonic lattices. Nat. Mater. 19, 725–731 (2020).

    ADS  Article  Google Scholar 

  131. Nixon, M., Ronen, E., Friesem, A. A. & Davidson, N. Observing geometric frustration with thousands of coupled lasers. Phys. Rev. Lett. 110, 184102 (2013).

    ADS  Article  Google Scholar 

  132. Böhm, F., Verschaffelt, G. & Van der Sande, G. A poor man’s coherent Ising machine based on opto-electronic feedback systems for solving optimization problems. Nat. Commun. 10, 3538 (2019).

    ADS  Article  Google Scholar 

  133. Lagoudakis, P. G. & Berloff, N. G. A polariton graph simulator. New J. Phys. 19, 125008 (2017).

    ADS  Article  Google Scholar 

  134. Berloff, N. G. et al. Realizing the classical XY Hamiltonian in polariton simulators. Nat. Mater. 16, 1120–1126 (2017).

    ADS  Article  Google Scholar 

  135. Kalinin, K. P. & Berloff, N. G. Simulating Ising and n-state planar Potts models and external fields with nonequilibrium condensates. Phys. Rev. Lett. 121, 235302 (2018).

    ADS  Article  Google Scholar 

  136. Kyriienko, O., Sigurdsson, H. & Liew, T. C. H. Probabilistic solving of NP-hard problems with bistable nonlinear optical networks. Phys. Rev. B 99, 195301 (2019).

    ADS  Article  Google Scholar 

  137. Mahboob, I., Okamoto, H. & Yamaguchi, H. An electromechanical Ising Hamiltonian. Sci. Adv. 2, e1600236 (2016).

    ADS  Article  Google Scholar 

  138. Tezak, N. et al. Integrated coherent Ising machines based on self-phase modulation in microring resonators. IEEE J. Sel. Top. Quant. Electron. 26, 1–15 (2019).

    Article  Google Scholar 

  139. Bernaschi, M., Billoire, A., Maiorano, A., Parisi, G. & Ricci-Tersenghi, F. Strong ergodicity breaking in aging of mean-field spin glasses. Proc. Natl Acad. Sci. USA 117, 17522–17527 (2020).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  140. Ercsey-Ravasz, M. & Toroczkai, Z. Optimization hardness as transient chaos in an analog approach to constraint satisfaction. Nat. Phys. 7, 966–970 (2011).

    Article  Google Scholar 

  141. Molnár, B., Molnár, F., Varga, M., Toroczkai, Z. & Ercsey-Ravasz, M. A continuous-time MaxSAT solver with high analog performance. Nat. Commun. 9, 4864 (2018).

    ADS  Article  Google Scholar 

  142. Leleu, T. et al. Chaotic amplitude control for neuromorphic Ising machine in silico. Preprint at https://arxiv.org/abs/2009.04084 (2020).

  143. Yin, X. et al. Efficient analog circuits for Boolean satisfiability. IEEE Trans. Very Large Scale Integration (VLSI) Systems 26, 155–167 (2017).

    Article  Google Scholar 

  144. Elser, V., Rankenburg, I. & Thibault, P. Searching with iterated maps. Proc. Natl Acad. Sci. USA 104, 418–423 (2007).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  145. Albash, T. & Lidar, D. A. Adiabatic quantum computation. Rev. Mod. Phys. 90, 015002 (2018).

    ADS  MathSciNet  Article  Google Scholar 

  146. Das, A. & Chakrabarti, B. K. Colloquium: Quantum annealing and analog quantum computation. Rev. Mod. Phys. 80, 1061–1081 (2008).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  147. Crosson, E. J. & Lidar, D. A. Prospects for quantum enhancement with diabatic quantum annealing. Nat. Rev. Phys. 3, 466 (2021).

    Article  Google Scholar 

  148. Apolloni, B., Carvalho, C. & De Falco, D. Quantum stochastic optimization. Stoch. Process. Their Appl. 33, 233–244 (1989).

    MathSciNet  MATH  Article  Google Scholar 

  149. Kadowaki, T. & Nishimori, H. Quantum annealing in the transverse Ising model. Phys. Rev. E 58, 5355–5363 (1998).

    ADS  Article  Google Scholar 

  150. Farhi, E., Goldstone, J., Gutmann, S. & Sipser, M. Quantum computation by adiabatic evolution. Preprint at https://arxiv.org/abs/quant-ph/0001106 (2000).

  151. Roland, J. & Cerf, N. J. Quantum search by local adiabatic evolution. Phys. Rev. A 65, 042308 (2002).

    ADS  Article  Google Scholar 

  152. Amin, M. Effect of local minima on adiabatic quantum optimization. Phys. Rev. Lett. 100, 130503 (2008).

    ADS  Article  Google Scholar 

  153. Schaller, G., Mostame, S. & Schützhold, R. General error estimate for adiabatic quantum computing. Phys. Rev. A 73, 062307 (2006).

    ADS  Article  Google Scholar 

  154. Lidar, D. A., Rezakhani, A. T. & Hamma, A. Adiabatic approximation with exponential accuracy for many-body systems and quantum computation. J. Math Phys. 50, 102106 (2009).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  155. Katzgraber, H. G., Hamze, F., Zhu, Z., Ochoa, A. J. & Munoz-Bauza, H. Seeking quantum speedup through spin glasses: the good, the bad, and the ugly. Phys. Rev. X 5, 031026 (2015).

    Google Scholar 

  156. Muthukrishnan, S., Albash, T. & Lidar, D. A. Tunneling and speedup in quantum optimization for permutation-symmetric problems. Phys. Rev. X 6, 031010 (2016).

    Google Scholar 

  157. Albash, T. & Lidar, D. A. Demonstration of a scaling advantage for a quantum annealer over simulated annealing. Phys. Rev. X 8, 031016 (2018).

    Google Scholar 

  158. Denchev, V. S. et al. What is the computational value of finite-range tunneling? Phys. Rev. X 6, 031015 (2016).

    Google Scholar 

  159. Boixo, S. et al. Computational multiqubit tunnelling in programmable quantum annealers. Nat. Commun. 7, 10327 (2016).

    ADS  Article  Google Scholar 

  160. Cerezo, M. et al. Variational quantum algorithms. Nat. Rev. Phys. 3, 625–644 (2021).

    Article  Google Scholar 

  161. Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).

    Article  Google Scholar 

  162. Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm. Preprint at https://arxiv.org/abs/1411.4028 (2014).

  163. Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm applied to a bounded occurrence constraint problem. Preprint at https://arxiv.org/abs/1412.6062 (2014).

  164. Zhou, L., Wang, S.-T., Choi, S., Pichler, H. & Lukin, M. D. Quantum approximate optimization algorithm: performance, mechanism, and implementation on near-term devices. Phys. Rev. X 10, 021067 (2020).

    Google Scholar 

  165. Guerreschi, G. G. & Smelyanskiy, M. Practical optimization for hybrid quantum-classical algorithms. Preprint at https://arxiv.org/abs/1701.01450 (2017).

  166. Khairy, S., Shaydulin, R., Cincio, L., Alexeev, Y. & Balaprakash, P. Learning to optimize variational quantum circuits to solve combinatorial problems. In Proc. AAAI Conf. Artificial Intelligence vol. 34, 2367–2375 (2020).

  167. Pagano, G. et al. Quantum approximate optimization of the long-range Ising model with a trapped-ion quantum simulator. Proc. Natl Acad. Sci. USA 117, 25396–25401 (2020).

    ADS  MathSciNet  Article  Google Scholar 

  168. Farhi, E. & Harrow, A. W. Quantum supremacy through the quantum approximate optimization algorithm. Preprint at https://arxiv.org/abs/1602.07674 (2016).

  169. Harrigan, M. P. et al. Quantum approximate optimization of non-planar graph problems on a planar superconducting processor. Nat. Phys. 17, 332–336 (2021).

    Article  Google Scholar 

  170. Qiang, X. et al. Large-scale silicon quantum photonics implementing arbitrary two-qubit processing. Nat. Photonics 12, 534–539 (2018).

    ADS  Article  Google Scholar 

  171. Ozaeta, A., van Dam, W. & McMahon, P. L. Expectation values from the single-layer quantum approximate optimization algorithm on Ising problems. Preprint at https://arxiv.org/abs/2012.03421 (2020).

  172. Sanders, Y. R. et al. Compilation of fault-tolerant quantum heuristics for combinatorial optimization. PRX Quantum 1, 020312 (2020).

    Article  Google Scholar 

  173. Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008).

    ADS  Article  Google Scholar 

  174. Boixo, S., Ortiz, G. & Somma, R. Fast quantum methods for optimization. Eur. Phys. J. Spec. Top. 224, 35–49 (2015).

    Article  Google Scholar 

  175. Lemieux, J., Heim, B., Poulin, D., Svore, K. & Troyer, M. Efficient quantum walk circuits for Metropolis–Hastings algorithm. Quantum 4, 287 (2020).

    Article  Google Scholar 

  176. Bapst, V. & Semerjian, G. Thermal, quantum and simulated quantum annealing: analytical comparisons for simple models. J. Phys. Conf. Ser. 473, 012011 (2013).

  177. Das, A. & Chakrabarti, B. K. Quantum Annealing and Related Optimization Methods vol. 679 (Springer Science & Business Media, 2005).

  178. Crosson, E. & Harrow, A. W. Simulated quantum annealing can be exponentially faster than classical simulated annealing. In 2016 IEEE 57th Annual Symp. Foundations of Computer Science (FOCS), 714–723 (IEEE, 2016).

  179. Andriyash, E. & Amin, M. H. Can quantum Monte Carlo simulate quantum annealing? Preprint at https://arxiv.org/abs/1703.09277 (2017).

  180. King, A. D., Bernoudy, W., King, J., Berkley, A. J. & Lanting, T. Emulating the coherent Ising machine with a mean-field algorithm. Preprint at https://arxiv.org/abs/1806.08422 (2018).

  181. Tiunov, E. S., Ulanov, A. E. & Lvovsky, A. Annealing by simulating the coherent Ising machine. Opt. Express 27, 10288–10295 (2019).

    ADS  Article  Google Scholar 

  182. Goto, H., Tatsumura, K. & Dixon, A. R. Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems. Sci. Adv. 5, eaav2372 (2019).

    ADS  MATH  Article  Google Scholar 

  183. Tatsumura, K., Dixon, A. R. & Goto, H. FPGA-based simulated bifurcation machine. In 2019 29th Int. Conf. Field Programmable Logic and Applications (FPL), 59–66 (IEEE, 2019).

  184. Goto, H. et al. High-performance combinatorial optimization based on classical mechanics. Sci. Adv. 7, eabe7953 (2021).

    ADS  Article  Google Scholar 

  185. Tatsumura, K., Yamasaki, M. & Goto, H. Scaling out Ising machines using a multi-chip architecture for simulated bifurcation. Nat. Electronics 4, 208–217 (2021).

    Article  Google Scholar 

  186. Orús, R. Tensor networks for complex quantum systems. Nat. Rev. Phys. 1, 538–550 (2019).

    Article  Google Scholar 

  187. Alcazar, J. & Perdomo-Ortiz, A. Enhancing combinatorial optimization with quantum generative models. Preprint at https://arxiv.org/abs/2101.06250 (2021).

  188. Mugel, S. et al. Dynamic portfolio optimization with real datasets using quantum processors and quantum-inspired tensor networks. Preprint at https://arxiv.org/abs/2007.00017 (2020).

  189. Mohseni, N., Navarrete-Benlloch, C., Byrnes, T. & Marquardt, F. Deep recurrent networks predicting the gap evolution in adiabatic quantum computing. Preprint at https://arxiv.org/abs/2109.08492 (2021).

  190. Bojesen, T. A. Policy-guided Monte Carlo: reinforcement-learning Markov chain dynamics. Phys. Rev. E 98, 063303 (2018).

    ADS  Article  Google Scholar 

  191. Huang, L. & Wang, L. Accelerated Monte Carlo simulations with restricted Boltzmann machines. Phys. Rev. B 95, 035105 (2017).

    ADS  Article  Google Scholar 

  192. Bello, I., Pham, H., Le, Q. V., Norouzi, M. & Bengio, S. Neural combinatorial optimization with reinforcement learning. Preprint at https://arxiv.org/abs/1611.09940 (2016).

  193. Dai, H., Khalil, E. B., Zhang, Y., Dilkina, B. & Song, L. Learning combinatorial optimization algorithms over graphs. Preprint at https://arxiv.org/abs/1704.01665 (2017).

  194. Zhou, J. et al. Graph neural networks: a review of methods and applications. AI Open 1, 57–81 (2020).

    Article  Google Scholar 

  195. Dwivedi, V. P., Joshi, C. K., Laurent, T., Bengio, Y. & Bresson, X. Benchmarking graph neural networks. Preprint at https://arxiv.org/abs/2003.00982 (2020).

  196. Schuetz, M. J., Brubaker, J. K. & Katzgraber, H. G. Combinatorial optimization with physics-inspired graph neural networks. Preprint at https://arxiv.org/abs/2107.01188 (2021).

  197. Vinyals, O., Fortunato, M. & Jaitly, N. Pointer networks. Preprint at https://arxiv.org/abs/1506.03134 (2015).

  198. Shor, P. W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41, 303–332 (1999).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  199. Arora, S. & Barak, B. Computational Complexity: A Modern Approach (Cambridge Univ. Press, 2009).

  200. Aaronson, S. BQP and the polynomial hierarchy. In Proc. 42nd ACM Symp. Theory of Computing, 141–150 (ACM, 2010).

  201. Papadimitriou, C. H. & Yannakakis, M. Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43, 425–440 (1991).

    MathSciNet  MATH  Article  Google Scholar 

  202. Goemans, M. X. & Williamson, D. P. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995).

    MathSciNet  MATH  Article  Google Scholar 

  203. Håstad, J. Some optimal inapproximability results. J. ACM 48, 798–859 (2001).

    MathSciNet  MATH  Article  Google Scholar 

  204. Mukai, H., Tomonaga, A. & Tsai, J.-S. Superconducting quantum annealing architecture with LC resonators. J. Phys. Soc. Japan 88, 061011 (2019).

    ADS  Article  Google Scholar 

  205. Onodera, T., Ng, E. & McMahon, P. L. A quantum annealer with fully programmable all-to-all coupling via Floquet engineering. npj Quantum Inf. 6, 1–10 (2020).

    Article  Google Scholar 

  206. Lechner, W., Hauke, P. & Zoller, P. A quantum annealing architecture with all-to-all connectivity from local interactions. Sci. Adv. 1, e1500838 (2015).

    ADS  Article  Google Scholar 

  207. Puri, S., Andersen, C. K., Grimsmo, A. L. & Blais, A. Quantum annealing with all-to-all connected nonlinear oscillators. Nat. Commun. 8, 1–9 (2017).

    Article  Google Scholar 

  208. Kowalsky, M., Albash, T., Hen, I. & Lidar, D. A. 3-regular 3-XORSAT planted solutions benchmark of classical and quantum heuristic optimizers. Preprint at https://arxiv.org/abs/2103.08464 (2021).

  209. Jörg, T., Krzakala, F., Semerjian, G. & Zamponi, F. First-order transitions and the performance of quantum algorithms in random optimization problems. Phys. Rev. Lett. 104, 207206 (2010).

    ADS  Article  Google Scholar 

  210. Aharonov, D. et al. Adiabatic quantum computation is equivalent to standard quantum computation. SIAM Rev. 50, 755–787 (2008).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  211. Mandra, S. & Katzgraber, H. G. A deceptive step towards quantum speedup detection. Quantum Sci. Technol. 3, 04LT01 (2018).

    Article  Google Scholar 

  212. Oshiyama, H. & Ohzeki, M. Benchmark of quantum-inspired heuristic solvers for quadratic unconstrained binary optimization. Preprint at https://arxiv.org/abs/2104.14096 (2021).

  213. Chancellor, N. Modernizing quantum annealing using local searches. New J. Phys. 19, 023024 (2017).

    ADS  Article  Google Scholar 

  214. Bilbro, G. et al. Optimization by mean field annealing. In Advances in Neural Information Processing Systems 1 (NIPS 1988) (ed. Touretzky, D) 91–98 (Morgan Kaufmann, 1989).

  215. Onodera, T. et al. Nonlinear quantum behavior of ultrashort-pulse optical parametric oscillators. Preprint at https://arxiv.org/abs/1811.10583 (2018).

  216. Hamze, F. & de Freitas, N. From fields to trees. Preprint at https://arxiv.org/abs/1207.4149 (2012).

  217. Selby, A. Efficient subgraph-based sampling of Ising-type models with frustration. Preprint at https://arxiv.org/abs/1409.3934 (2014).

  218. Job, J. & Lidar, D. Test-driving 1000 qubits. Quantum Sci. Technol. 3, 030501 (2018).

    ADS  Article  Google Scholar 

  219. Mandra, S., Zhu, Z. & Katzgraber, H. G. Exponentially biased ground-state sampling of quantum annealing machines with transverse-field driving Hamiltonians. Phys. Rev. Lett. 118, 070502 (2017).

    ADS  Article  Google Scholar 

  220. Zhu, Z., Ochoa, A. J. & Katzgraber, H. G. Fair sampling of ground-state configurations of binary optimization problems. Phys. Rev. E 99, 063314 (2019).

    ADS  Article  Google Scholar 

  221. Albash, T. & Lidar, D. A. Adiabatic quantum computation. Rev. Mod. Phys. 90, 015002 (2018).

    ADS  MathSciNet  Article  Google Scholar 

  222. King, J., Yarkoni, S., Nevisi, M. M., Hilton, J. P. & McGeoch, C. C. Benchmarking a quantum annealing processor with the time-to-target metric. Preprint at https://arxiv.org/abs/1508.05087 (2015).

  223. Takesue, H., Inagaki, T., Inaba, K. & Honjo, T. Performance comparison between coherent Ising machines and quantum annealer. NTT R&D Technical Report (NTT Basic Research Laboratories, 2021); https://www.rd.ntt/e/research/JN202103_10945.html

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Acknowledgements

The authors thank S. Aaronson, T. Albash, H. Katzgraber, T. Leleu, S. King, M. Narozniak, S. Vadlamani, T. van Vaerenbergh and D-Wave Systems for discussions and comments on the manuscript. T.B. is supported by the National Natural Science Foundation of China (62071301); NYU-ECNU Institute of Physics at NYU Shanghai; the Joint Physics Research Institute Challenge Grant; the Science and Technology Commission of Shanghai Municipality (19XD1423000,22ZR1444600); the NYU Shanghai Boost Fund; the China Foreign Experts Program (G2021013002L); and the NYU Shanghai Major-Grants Seed Fund. P.L.M. thanks all his collaborators on the topic of Ising machines — especially S. Ganguli, R. Hamerly, T. Leleu, H. Mabuchi, A. Marandi, E. Ng, T. Onodera and Y. Yamamoto — for discussions that have shaped his understanding over the years. P.L.M. acknowledges funding from NSF award CCF-1918549, and NTT Research for their financial and technical support. P.L.M. also acknowledges membership in the CIFAR Quantum Information Science Program as an Azrieli Global Scholar.

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Mohseni, N., McMahon, P.L. & Byrnes, T. Ising machines as hardware solvers of combinatorial optimization problems. Nat Rev Phys 4, 363–379 (2022). https://doi.org/10.1038/s42254-022-00440-8

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