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.
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.
This is a preview of subscription content, access via your institution
Open Access articles citing this article.
Nature Communications Open Access 08 November 2023
Nature Communications Open Access 11 October 2023
Efficient combinatorial optimization by quantum-inspired parallel annealing in analogue memristor crossbar
Nature Communications Open Access 22 September 2023
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 digital issues and online access to articles
$99.00 per year
only $8.25 per issue
Rent or buy this article
Prices vary by article type
Prices may be subject to local taxes which are calculated during checkout
Lucas, A. Ising formulations of many NP problems. Front. Phys. 2, 5 (2014).
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).
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).
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).
Karp, R. M. in Complexity of Computer Computations, 85–103 (Springer, 1972).
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).
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).
Chang, K. & Du, D. C. Efficient algorithms for layer assignment problem. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 6, 67–78 (1987).
Wang, J., Jebara, T. & Chang, S.-F. Semi-supervised learning using greedy Max-cut. J. Mach. Learn. Res. 14, 771–800 (2013).
Collins, T. Graph Cut Matching in Computer Vision (Univ. Edinburgh, 2004).
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).
Turing, A. M. On computable numbers, with an application to the entscheidungsproblem. Proc. Lond. Math. Soc. 2, 230–265 (1937).
Bournez, O. & Pouly, A. in Handbook of Computability and Complexity in Analysis 2018 (eds Brattka, V. & Hertling, P.) 173–226 (Springer, 2018).
Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982).
Buluta, I. & Nori, F. Quantum simulators. Science 326, 108–111 (2009).
Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153 (2014).
Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996).
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).
Labuhn, H. et al. Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models. Nature 534, 667–670 (2016).
Bernien, H. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579–584 (2017).
Keesling, A. et al. Quantum Kibble–Zurek mechanism and critical dynamics on a programmable Rydberg simulator. Nature 568, 207–211 (2019).
Scholl, P. et al. Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms. Nature 595, 233–238 (2021).
Farhi, E. et al. A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292, 472–475 (2001).
Byrnes, T., Yan, K. & Yamamoto, Y. Accelerated optimization problem search using Bose–Einstein condensation. New J. Phys. 13, 113025 (2011).
King, A. D. et al. Observation of topological phenomena in a programmable lattice of 1,800 qubits. Nature 560, 456–460 (2018).
Harris, R. et al. Phase transitions in a programmable quantum spin glass simulator. Science 361, 162–165 (2018).
Vadlamani, S. K., Xiao, T. P. & Yablonovitch, E. Physics successfully implements Lagrange multiplier optimization. Proc. Natl Acad. Sci. USA 117, 26639–26650 (2020).
Geman, S. & Geman, D. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
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).
Hastings, W. K. Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970).
Swendsen, R. H. & Wang, J.-S. Replica Monte Carlo simulation of spin-glasses. Phys. Rev. Lett. 57, 2607–2609 (1986).
Earl, D. J. & Deem, M. W. Parallel tempering: theory, applications, and new perspectives. Phys. Chem. Chem. Phys. 7, 3910–3916 (2005).
Wang, W., Machta, J. & Katzgraber, H. G. Population annealing: theory and application in spin glasses. Phys. Rev. E 92, 063307 (2015).
Zhu, Z., Ochoa, A. J. & Katzgraber, H. G. Efficient cluster algorithm for spin glasses in any space dimension. Phys. Rev. Lett. 115, 077201 (2015).
Kirkpatrick, S., Gelatt, C. D. & Vecchi, M. P. Optimization by simulated annealing. Science 220, 671–680 (1983).
Černý, V. Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. J. Optim. Theory Appl. 45, 41–51 (1985).
Camsari, K. Y., Faria, R., Sutton, B. M. & Datta, S. Stochastic p-bits for invertible logic. Phys. Rev. X 7, 031014 (2017).
Camsari, K. Y., Sutton, B. M. & Datta, S. p-bits for probabilistic spin logic. Appl. Phys. Rev. 6, 011305 (2019).
Borders, W. A. et al. Integer factorization using stochastic magnetic tunnel junctions. Nature 573, 390–393 (2019).
Sutton, B., Camsari, K. Y., Behin-Aein, B. & Datta, S. Intrinsic optimization using stochastic nanomagnets. Sci. Rep. 7, 1–9 (2017).
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).
Arnalds, U. B. et al. A new look on the two-dimensional Ising model: thermal artificial spins. New J. Phys. 18, 023008 (2016).
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).
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).
Mizushima, K., Goto, H. & Sato, R. Large-scale Ising-machines composed of magnetic neurons. Appl. Phys. Lett. 111, 172406 (2017).
Pierangeli, D., Marcucci, G. & Conti, C. Large-scale photonic Ising machine by spatial light modulation. Phys. Rev. Lett. 122, 213902 (2019).
Pierangeli, D., Marcucci, G. & Conti, C. Adiabatic evolution on a spatial-photonic Ising machine. Optica 7, 1535–1543 (2020).
Roques-Carmes, C. et al. Heuristic recurrent algorithms for photonic Ising machines. Nat. Commun. 11, 1–8 (2020).
Cai, F. et al. Power-efficient combinatorial optimization using intrinsic noise in memristor Hopfield neural networks. Nat. Electron. 3, 409–418 (2020).
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).
Behin-Aein, B., Diep, V. & Datta, S. A building block for hardware belief networks. Sci. Rep. 6, 29893 (2016).
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).
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).
Guo, S. Y. et al. A molecular computing approach to solving optimization problems via programmable microdroplet arrays. Matter 4, 1107–1124 (2021).
Byrnes, T., Koyama, S., Yan, K. & Yamamoto, Y. Neural networks using two-component Bose–Einstein condensates. Sci. Rep. 3, 2531 (2013).
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).
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).
Aramon, M. et al. Physics-inspired optimization for quadratic unconstrained problems using a digital annealer. Front. Phys. 7, 48 (2019).
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).
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).
Tsukamoto, S., Takatsu, M., Matsubara, S. & Tamura, H. An accelerator architecture for combinatorial optimization problems. Fujitsu Sci. Tech. J. 53, 8–13 (2017).
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).
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).
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).
Aadit, N. A. et al. Massively parallel probabilistic computing with sparse Ising machines. Preprint at https://arxiv.org/abs/2110.02481 (2021).
Reuther, A. et al. Survey and benchmarking of machine learning accelerators. In 2019 IEEE High Perform. Extreme Comput. Conf. (HPEC), 1–9 (IEEE, 2019).
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).
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).
Merolla, P. A. et al. A million spiking-neuron integrated circuit with a scalable communication network and interface. Science 345, 668–673 (2014).
Skubiszewski, M. An exact hardware implementation of the Boltzmann machine. In Proc. 4th IEEE Symp. Parallel and Distributed Processing, 107–110 (IEEE, 1992).
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).
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).
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).
Le Ly, D. & Chow, P. High-performance reconfigurable hardware architecture for restricted Boltzmann machines. IEEE Trans. Neural Networks 21, 1780–1792 (2010).
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).
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
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).
Okuyama, T., Sonobe, T., Kawarabayashi, K.-i & Yamaoka, M. Binary optimization by momentum annealing. Phys. Rev. E 100, 012111 (2019).
Hopfield, J. J. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl Acad. Sci. USA 79, 2554–2558 (1982).
Hopfield, J. J. & Tank, D. W. ‘Neural’ computation of decisions in optimization problems. Biol. Cybern. 52, 141–152 (1985).
Von Neumann, J. Non-linear capacitance or inductance switching, amplifying, and memory organs. US Patent 2,815,488 (1957).
Wigington, R. A new concept in computing. Proc. IRE 47, 516–523 (1959).
Goto, E. The parametron, a digital computing element which utilizes parametric oscillation. Proc. IRE 47, 1304–1316 (1959).
Csaba, G. & Porod, W. Coupled oscillators for computing: a review and perspective. Appl. Phys. Rev. 7, 011302 (2020).
Raychowdhury, A. et al. Computing with networks of oscillatory dynamical systems. Proc. IEEE 107, 73–89 (2018).
Kuramoto, Y. in International Symposium on Mathematical Problems in Theoretical Physics, 420 (Lecture Notes in Physics vol. 30, Springer, 1975).
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).
Breakspear, M., Heitmann, S. & Daffertshofer, A. Generative models of cortical oscillations: neurobiological implications of the Kuramoto model. Front. Human Neurosci. 4, 190 (2010).
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).
Wu, C. W. Graph coloring via synchronization of coupled oscillators. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 45, 974–978 (1998).
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).
Kalinin, K. P. & Berloff, N. G. Global optimization of spin Hamiltonians with gain-dissipative systems. Sci. Rep. 8, 1–9 (2018).
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).
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).
McGoldrick, B. C., Sun, J. Z. & Liu, L. Ising machine based on electrically coupled spin Hall nano-oscillators. Phys. Rev. Appl. 17, 014006 (2021).
Albertsson, D. I. et al. Ultrafast Ising machines using spin torque nano-oscillators. Appl. Phys. Lett. 118, 112404 (2021).
Chou, J., Bramhavar, S., Ghosh, S. & Herzog, W. Analog coupled oscillator based weighted Ising machine. Sci. Rep. 9, 14786 (2019).
Xiao, T. P. Optoelectronics for Refrigeration and Analog Circuits for Combinatorial Optimization. PhD thesis, Univ. California Berkeley (2019).
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).
Shukla, N. et al. Synchronized charge oscillations in correlated electron systems. Sci. Rep. 4, 4964 (2014).
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).
Dutta, S. et al. An Ising Hamiltonian solver using stochastic phase-transition nano-oscillators. Preprint at https://arxiv.org/abs/2007.12331 (2020).
Zahedinejad, M. et al. Two-dimensional mutually synchronized spin Hall nano-oscillator arrays for neuromorphic computing. Nat. Nanotechnol. 15, 47–52 (2020).
Houshang, A. et al. A spin Hall Ising machine. Preprint at https://arxiv.org/abs/2006.02236 (2020).
Mallick, A. et al. Using synchronized oscillators to compute the maximum independent set. Nat. Commun. 11, 4689 (2020).
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).
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).
Traversa, F. L. & Di Ventra, M. Universal memcomputing machines. IEEE Trans. Neural Netw. Learn. Syst. 26, 2702–2715 (2015).
Di Ventra, M. & Traversa, F. L. Perspective: Memcomputing: leveraging memory and physics to compute efficiently. J. Appl. Phys. 123, 180901 (2018).
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).
Aiken, J. & Traversa, F. L. Memcomputing for accelerated optimization. Preprint at https://arxiv.org/abs/2003.10644 (2020).
Utsunomiya, S., Takata, K. & Yamamoto, Y. Mapping of Ising models onto injection-locked laser systems. Opt. Express 19, 18091–18108 (2011).
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).
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).
McMahon, P. L. et al. A fully programmable 100-spin coherent Ising machine with all-to-all connections. Science 354, 614–617 (2016).
Inagaki, T. et al. A coherent Ising machine for 2000-node optimization problems. Science 354, 603–606 (2016).
Yamamoto, Y. et al. Coherent Ising machines — optical neural networks operating at the quantum limit. npj Quantum Inf. 3, 49 (2017).
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).
Hamerly, R. et al. Experimental investigation of performance differences between coherent Ising machines and a quantum annealer. Sci. Adv. 5, eaau0823 (2019).
Yamamoto, Y., Leleu, T., Ganguli, S. & Mabuchi, H. Coherent Ising machines — quantum optics and neural network perspectives. Appl. Phys. Lett. 117, 160501 (2020).
Honjo, T. et al. 100,000-spin coherent Ising machine. Science Advances 7, eabh0952 (2021).
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).
Kalinin, K. P. & Berloff, N. G. Networks of non-equilibrium condensates for global optimization. New J. Phys. 20, 113023 (2018).
Takata, K. et al. A 16-bit coherent Ising machine for one-dimensional ring and cubic graph problems. Sci. Rep. 6, 34089 (2016).
Inagaki, T. et al. Large-scale Ising spin network based on degenerate optical parametric oscillators. Nat. Photonics 10, 415–419 (2016).
Okawachi, Y. et al. Demonstration of chip-based coupled degenerate optical parametric oscillators for realizing a nanophotonic spin-glass. Nat. Commun. 11, 4119 (2020).
Goto, H. Bifurcation-based adiabatic quantum computation with a nonlinear oscillator network. Sci. Rep. 6, 21686 (2016).
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).
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).
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).
Nixon, M., Ronen, E., Friesem, A. A. & Davidson, N. Observing geometric frustration with thousands of coupled lasers. Phys. Rev. Lett. 110, 184102 (2013).
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).
Lagoudakis, P. G. & Berloff, N. G. A polariton graph simulator. New J. Phys. 19, 125008 (2017).
Berloff, N. G. et al. Realizing the classical XY Hamiltonian in polariton simulators. Nat. Mater. 16, 1120–1126 (2017).
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).
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).
Mahboob, I., Okamoto, H. & Yamaguchi, H. An electromechanical Ising Hamiltonian. Sci. Adv. 2, e1600236 (2016).
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).
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).
Ercsey-Ravasz, M. & Toroczkai, Z. Optimization hardness as transient chaos in an analog approach to constraint satisfaction. Nat. Phys. 7, 966–970 (2011).
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).
Leleu, T. et al. Chaotic amplitude control for neuromorphic Ising machine in silico. Preprint at https://arxiv.org/abs/2009.04084 (2020).
Yin, X. et al. Efficient analog circuits for Boolean satisfiability. IEEE Trans. Very Large Scale Integration (VLSI) Systems 26, 155–167 (2017).
Elser, V., Rankenburg, I. & Thibault, P. Searching with iterated maps. Proc. Natl Acad. Sci. USA 104, 418–423 (2007).
Albash, T. & Lidar, D. A. Adiabatic quantum computation. Rev. Mod. Phys. 90, 015002 (2018).
Das, A. & Chakrabarti, B. K. Colloquium: Quantum annealing and analog quantum computation. Rev. Mod. Phys. 80, 1061–1081 (2008).
Crosson, E. J. & Lidar, D. A. Prospects for quantum enhancement with diabatic quantum annealing. Nat. Rev. Phys. 3, 466 (2021).
Apolloni, B., Carvalho, C. & De Falco, D. Quantum stochastic optimization. Stoch. Process. Their Appl. 33, 233–244 (1989).
Kadowaki, T. & Nishimori, H. Quantum annealing in the transverse Ising model. Phys. Rev. E 58, 5355–5363 (1998).
Farhi, E., Goldstone, J., Gutmann, S. & Sipser, M. Quantum computation by adiabatic evolution. Preprint at https://arxiv.org/abs/quant-ph/0001106 (2000).
Roland, J. & Cerf, N. J. Quantum search by local adiabatic evolution. Phys. Rev. A 65, 042308 (2002).
Amin, M. Effect of local minima on adiabatic quantum optimization. Phys. Rev. Lett. 100, 130503 (2008).
Schaller, G., Mostame, S. & Schützhold, R. General error estimate for adiabatic quantum computing. Phys. Rev. A 73, 062307 (2006).
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).
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).
Muthukrishnan, S., Albash, T. & Lidar, D. A. Tunneling and speedup in quantum optimization for permutation-symmetric problems. Phys. Rev. X 6, 031010 (2016).
Albash, T. & Lidar, D. A. Demonstration of a scaling advantage for a quantum annealer over simulated annealing. Phys. Rev. X 8, 031016 (2018).
Denchev, V. S. et al. What is the computational value of finite-range tunneling? Phys. Rev. X 6, 031015 (2016).
Boixo, S. et al. Computational multiqubit tunnelling in programmable quantum annealers. Nat. Commun. 7, 10327 (2016).
Cerezo, M. et al. Variational quantum algorithms. Nat. Rev. Phys. 3, 625–644 (2021).
Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).
Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm. Preprint at https://arxiv.org/abs/1411.4028 (2014).
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).
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).
Guerreschi, G. G. & Smelyanskiy, M. Practical optimization for hybrid quantum-classical algorithms. Preprint at https://arxiv.org/abs/1701.01450 (2017).
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).
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).
Farhi, E. & Harrow, A. W. Quantum supremacy through the quantum approximate optimization algorithm. Preprint at https://arxiv.org/abs/1602.07674 (2016).
Harrigan, M. P. et al. Quantum approximate optimization of non-planar graph problems on a planar superconducting processor. Nat. Phys. 17, 332–336 (2021).
Qiang, X. et al. Large-scale silicon quantum photonics implementing arbitrary two-qubit processing. Nat. Photonics 12, 534–539 (2018).
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).
Sanders, Y. R. et al. Compilation of fault-tolerant quantum heuristics for combinatorial optimization. PRX Quantum 1, 020312 (2020).
Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008).
Boixo, S., Ortiz, G. & Somma, R. Fast quantum methods for optimization. Eur. Phys. J. Spec. Top. 224, 35–49 (2015).
Lemieux, J., Heim, B., Poulin, D., Svore, K. & Troyer, M. Efficient quantum walk circuits for Metropolis–Hastings algorithm. Quantum 4, 287 (2020).
Bapst, V. & Semerjian, G. Thermal, quantum and simulated quantum annealing: analytical comparisons for simple models. J. Phys. Conf. Ser. 473, 012011 (2013).
Das, A. & Chakrabarti, B. K. Quantum Annealing and Related Optimization Methods vol. 679 (Springer Science & Business Media, 2005).
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).
Andriyash, E. & Amin, M. H. Can quantum Monte Carlo simulate quantum annealing? Preprint at https://arxiv.org/abs/1703.09277 (2017).
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).
Tiunov, E. S., Ulanov, A. E. & Lvovsky, A. Annealing by simulating the coherent Ising machine. Opt. Express 27, 10288–10295 (2019).
Goto, H., Tatsumura, K. & Dixon, A. R. Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems. Sci. Adv. 5, eaav2372 (2019).
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).
Goto, H. et al. High-performance combinatorial optimization based on classical mechanics. Sci. Adv. 7, eabe7953 (2021).
Tatsumura, K., Yamasaki, M. & Goto, H. Scaling out Ising machines using a multi-chip architecture for simulated bifurcation. Nat. Electronics 4, 208–217 (2021).
Orús, R. Tensor networks for complex quantum systems. Nat. Rev. Phys. 1, 538–550 (2019).
Alcazar, J. & Perdomo-Ortiz, A. Enhancing combinatorial optimization with quantum generative models. Preprint at https://arxiv.org/abs/2101.06250 (2021).
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).
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).
Bojesen, T. A. Policy-guided Monte Carlo: reinforcement-learning Markov chain dynamics. Phys. Rev. E 98, 063303 (2018).
Huang, L. & Wang, L. Accelerated Monte Carlo simulations with restricted Boltzmann machines. Phys. Rev. B 95, 035105 (2017).
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).
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).
Zhou, J. et al. Graph neural networks: a review of methods and applications. AI Open 1, 57–81 (2020).
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).
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).
Vinyals, O., Fortunato, M. & Jaitly, N. Pointer networks. Preprint at https://arxiv.org/abs/1506.03134 (2015).
Shor, P. W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41, 303–332 (1999).
Arora, S. & Barak, B. Computational Complexity: A Modern Approach (Cambridge Univ. Press, 2009).
Aaronson, S. BQP and the polynomial hierarchy. In Proc. 42nd ACM Symp. Theory of Computing, 141–150 (ACM, 2010).
Papadimitriou, C. H. & Yannakakis, M. Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43, 425–440 (1991).
Goemans, M. X. & Williamson, D. P. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995).
Håstad, J. Some optimal inapproximability results. J. ACM 48, 798–859 (2001).
Mukai, H., Tomonaga, A. & Tsai, J.-S. Superconducting quantum annealing architecture with LC resonators. J. Phys. Soc. Japan 88, 061011 (2019).
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).
Lechner, W., Hauke, P. & Zoller, P. A quantum annealing architecture with all-to-all connectivity from local interactions. Sci. Adv. 1, e1500838 (2015).
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).
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).
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).
Aharonov, D. et al. Adiabatic quantum computation is equivalent to standard quantum computation. SIAM Rev. 50, 755–787 (2008).
Mandra, S. & Katzgraber, H. G. A deceptive step towards quantum speedup detection. Quantum Sci. Technol. 3, 04LT01 (2018).
Oshiyama, H. & Ohzeki, M. Benchmark of quantum-inspired heuristic solvers for quadratic unconstrained binary optimization. Preprint at https://arxiv.org/abs/2104.14096 (2021).
Chancellor, N. Modernizing quantum annealing using local searches. New J. Phys. 19, 023024 (2017).
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).
Onodera, T. et al. Nonlinear quantum behavior of ultrashort-pulse optical parametric oscillators. Preprint at https://arxiv.org/abs/1811.10583 (2018).
Hamze, F. & de Freitas, N. From fields to trees. Preprint at https://arxiv.org/abs/1207.4149 (2012).
Selby, A. Efficient subgraph-based sampling of Ising-type models with frustration. Preprint at https://arxiv.org/abs/1409.3934 (2014).
Job, J. & Lidar, D. Test-driving 1000 qubits. Quantum Sci. Technol. 3, 030501 (2018).
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).
Zhu, Z., Ochoa, A. J. & Katzgraber, H. G. Fair sampling of ground-state configurations of binary optimization problems. Phys. Rev. E 99, 063314 (2019).
Albash, T. & Lidar, D. A. Adiabatic quantum computation. Rev. Mod. Phys. 90, 015002 (2018).
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).
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
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.
P.L.M. declares an interest in QC Ware Corp., a company producing software for quantum computers, to which he is an advisor. T.B. and N.M. declare no competing interests.
Peer review information
Nature Reviews Physics thanks Natalia Berloff, Masano Yamaoka and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
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
This article is cited by
Nature Reviews Physics (2023)
Efficient combinatorial optimization by quantum-inspired parallel annealing in analogue memristor crossbar
Nature Communications (2023)
Nature Electronics (2023)
Nature Communications (2023)
Designing Ising machines with higher order spin interactions and their application in solving combinatorial optimization
Scientific Reports (2023)