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.
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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.
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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