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A 1,968-node coupled ring oscillator circuit for combinatorial optimization problem solving


Computational architectures that are optimized to solve non-deterministic polynomial-time hard or complete problems are of use in the development of machine learning, logistical planning and pathfinding. A range of quantum-, optical- and spintronic-based approaches have been explored for solving such combinatorial optimization problems, but they remain complicated to build and to scale. Here we report a scalable ring-oscillator-based integrated circuit for optimization problem solving. Our 1,968-node King’s graph ring oscillator array has five levels of coupling strengths and can achieve up to 95% accuracy for randomly generated combinatorial optimization problems. The measured average power consumption of the Ising chip is 0.042 W and it takes less than 50 oscillation cycles to resolve to the ground state. Our device is resilient to environmental and variation effects. By using a multi-phase phase measurement circuit, we also capture the true phase behaviour within a coupled-oscillator integrated circuit.

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Fig. 1: Computation steps and operating principle of the coupled-oscillator-based Ising machine.
Fig. 2: Schematic of a coupled oscillator and 1,968-node Ising chip layout.
Fig. 3: Unit-tile oscillator and support circuits.
Fig. 4: Phase-sampling circuit, measured raw phase maps and post-processing steps.
Fig. 5: Measured results from the 1,968 oscillator chips.

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The problem set and test data presented in this paper are available at

Code availability

The code used to generate the optimal Hamiltonians is available at


  1. Yang, G. Industrial Applications of Combinatorial Optimization (Springer, 1998).

  2. Vangelis, T. P. Applications of Combinatorial Optimization 2nd edn (Wiley, 2014).

  3. Markov, I. L. Limits on fundamental limits to computation. Nature 512, 147–154 (2014).

    Article  Google Scholar 

  4. Markov, I. L. Know your limits. IEEE Des. Test 30, 78–83 (2013).

    Article  Google Scholar 

  5. Hoover, H. J. Greenlaw, R. & Ruzzo, W. L. Limits to Parallel Computation: P-Completeness Theory (Oxford Univ. Press, 1995).

  6. Date, P., Arthur, D. & Pusey-Nazzaro, L. QUBO formulations for training machine learning models. Sci. Rep. 11, 10029 (2021).

    Article  Google Scholar 

  7. Glover, F. Kochenberger, G. & Du, Y. A tutorial on formulating and using QUBO models. Preprint at (2019).

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

  9. Ising, E. Beitrag zur theories des ferromagnetismus. Z. Phys. 31, 253–258 (1925).

    Article  Google Scholar 

  10. Neogy, T. & Roychowdhury, J. Analysis and design of sub-harmonically injection locked oscillators. In 2012 Design, Automation & Test in Europe Conference & Exhibition (DATE) 1209–1214 (IEEE, 2012).

  11. 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).

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  14. 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 (2016).

    Article  Google Scholar 

  15. Hsu, J. How much power will quantum computing need? IEEE Spectrum 5 (2015).

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  18. Honjo, T. et al. 100,000-spin coherent Ising machine. Sci. Adv. 7, eabh0952 (2021).

  19. Wang, T. & Roychowdhury, J. Oscillator-based Ising machine. Preprint at (2017).

  20. Glover, F. Tabu search—part I. ORSA J. Comput. 1, 190–206 (1989).

    Article  Google Scholar 

  21. Glover, F. Tabu search—part II. ORSA J. Comput. 2, 4–32 (1990).

    Article  Google Scholar 

  22. 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).

  23. Takemoto, T. et al. A 2 × 30k-spin multi-chip scalable CMOS annealing processor based on a processing in-memory approach for solving large-scale combinatorial optimization problems. IEEE J. Solid-State Circuits 55, 145–156 (2020).

    Article  Google Scholar 

  24. Takemoto, T. et al. 4.6 A 144Kb annealing system composed of 9×16Kb annealing processor chips with scalable chip-to-chip connections for large-scale combinatorial optimization problems. In 2021 IEEE International Solid-State Circuits Conference (ISSCC) 64–66 (IEEE, 2021).

  25. Bian, Z. et al. The Ising model: teaching an old problem new tricks. D-Wave Systems 2, 1–32 (2010)

  26. Dutta, S. et al. Experimental demonstration of phase transition nano-oscillator based Ising machine. In 2019 IEEE International Electron Devices Meeting (IEDM) 37.8.1–37.8.4 (IEEE, 2019).

  27. Wang, T., Wu, L. & Roychowdhury, J. New computational results and hardware prototypes for oscillator-based Ising machines. In Proc. 56th Annual Design Automation Conference 2019 56, 1–2 (IEEE, 2019).

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I.A. and C.H.K. acknowledges the initial support of the project from the National Science Foundation under ECCS 1739635 and the Semiconductor Research Corporation (SRC) under 2759.007. For the chip testing and data analysis works, W.M. and C.H.K. have been supported in part by the SRC under 3024.001. We would like to thank SRC’s industry liaisons for their technical feedback.

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Authors and Affiliations



W.M., I.A., P.-W.C., S.S.S. and C.H.K. participated in the circuit and architecture design of the integrated circuit. W.M., I.A. and P.-W.C. created the layout for the fabrication of the integrated circuit. W.M. and C.H.K. performed the testing and measurement of the chip. J.M. created the cloud-testing interface and testing automation setup for the chip.

Corresponding author

Correspondence to Chris H. Kim.

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Competing interests

The authors declare the following competing interest: US patent application 17/213,396 (Probabilistic compute engine using coupled ROSCs).

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Nature Electronics thanks Michael Huang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Raw phase map data with post-processed results.

Raw phase maps are shown for uniform, chequerboard, and random pattern graphs. A lightweight post-processing method that emphasizes the relative phase difference between adjacent nodes can eliminate non-ideal effects and produce the expected spin value maps on the right. The weights and spin values are colour coded and overlaid for better visualization. ROSCs are indexed from 1 (upper left corner) to 1,968 (lower right corner), with ROSC 1 being directly above ROSC 42.

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Moy, W., Ahmed, I., Chiu, Pw. et al. A 1,968-node coupled ring oscillator circuit for combinatorial optimization problem solving. Nat Electron 5, 310–317 (2022).

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