Abstract
Combinatorial optimization problems belong to the non-deterministic polynomial time (NP)-hard complexity class, and their computational requirements scale exponentially with problem size. They can be mapped into the problem of finding the ground state of an Ising model, which describes a physical system with converging dynamics. Various platforms, including optical, electronic and quantum approaches, have been explored to accelerate the ground-state search, but improvements in energy efficiencies and computational abilities are still required. Here we report an Ising solver based on a network of electrically coupled phase-transition nano-oscillators (PTNOs) that form a continuous-time dynamical system (CTDS). The bi-stable phases of the injection-locked PTNOs act as artificial Ising spins and the stable points of the CTDS act as the ground-state solution of the problem. We experimentally show that a prototype with eight PTNOs can solve an NP-hard MaxCut problem with high probability of success (96% for 600 annealing cycles). We also show via numerical simulations that our Ising Hamiltonian solver can solve MaxCut problems of 100 nodes with energy efficiency of 1.3 × 107 solutions per second per watt, offering advantages over other approaches including memristor-based Hopfield networks, quantum annealers and photonic Ising solvers.
This is a preview of subscription content, access via your institution
Access options
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
$119.00 per year
only $9.92 per issue
Buy this article
- Purchase on Springer Link
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Code availability
The custom simulation code written in MATLAB and SPICE program for this study are available from the corresponding author upon request.
References
Ising, E. Beitrag zur theorie des ferromagnetismus. Z. Physik 31, 253–258 (1925).
Kochenberger, G. et al. The unconstrained binary quadratic programming problem: a survey. J. Comb. Optim. 28, 58–81 (2014).
Barahona, F. On the computational complexity of ising spin glass models. J. Phys. A: Math. Gen. 15, 3241–3253 (1982).
Lucas, A. Ising formulations of many NP problems. Front. Phys. 2, 5 (2014).
Goemans, M. X. & Williamson, D. P. Improved approximation algorithms for maximum cut and satisflability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995).
Benlic, U. & Hao, J. K. Breakout local search for the max-cut problem. Eng. Appl. Artif. Intell. 26, 1162–1173 (2013).
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).
Kirkpatrick, S., Gelatt, C. D. & Vecchi, M. P. Optimization by simulated annealing. Science 220, 671–680 (1983).
Johnson, M. W. et al. Quantum annealing with manufactured spins. Nature 473, 194–198 (2011).
Boixo, S. et al. Evidence for quantum annealing with more than one hundred qubits. Nat. Phys. 10, 218–224 (2014).
Tsukamoto, S., Takatsu, M., Matsubara, S. & Tamura, H. An accelerator architecture for combinatorial optimization problems. Fujitsu Sci. Tech. J. 53, 8–13 (2017).
Yamaoka, M. et al. 24.3 20k-spin Ising chip for combinational optimization problem with CMOS annealing. In Digest of Technical Papers—2015 IEEE International Solid-State Circuits Conference https://doi.org/10.1109/ISSCC.2015.7063111 (2015).
Takemoto, T., Hayashi, M., Yoshimura, C. & Yamaoka, M. 2.6 A 2 ×30k-spin multichip scalable annealing processor based on a processing-in-memory approach for solving large-scale combinatorial optimization problems. In Digest of Technical Papers—2019 IEEE International Solid-State Circuits Conference https://doi.org/10.1109/ISSCC.2019.8662517 (2019).
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).
Hamerly, R. et al. Experimental investigation of performance differences between coherent Ising machines and a quantum annealer. Sci. Adv. 5, eaau0823 (2019).
Chialvo, D. R. Emergent complex neural dynamics. Nat. Phys. 6, 744–750 (2010).
Wills, T. J., Lever, C., Cacucci, F., Burgess, N. & O’Keefe, J. Attractor dynamics in the hippocampal representation of the local environment. Science 308, 873–876 (2005).
Dutta, S. et al. Spoken vowel classification using synchronization of phase transition nano-oscillators. In Digest of Technical Papers—2019 Symposium on VLSI Technology https://doi.org/10.23919/VLSIT.2019.8776534(2019).
Dutta, S. et al. Programmable coupled oscillators for synchronized locomotion. Nat. Commun. 10, 3299 (2019).
Strogatz, S. H. & Stewart, I. Coupled oscillators and biological synchronization. Sci. Am. 269, 102–109 (1993).
Hopfield, J. J. & Tank, D. W. ‘Neural’ computation of decisions in optimization problems. Biol. Cybern. 52, 141–152 (1985).
Cai, F. et al. Power-efficient combinatorial optimization using intrinsic noise in memristor Hopfield neural networks. Nat. Electron. 3, 409–418 (2020).
Shukla, N. et al. Synchronized charge oscillations in correlated electron systems. Sci. Rep. 4, 1–6 (2014).
Dutta, S. et al. Experimental demonstration of phase transition nano-oscillator based Ising machine. In Technical Digest—2019 International Electron Devices Meeting (IEDM) https://doi.org/10.1109/IEDM19573.2019.8993460 (2019).
Parihar, A., Jerry, M., Datta, S. & Raychowdhury, A. Stochastic IMT (insulator-metal-transition) neurons: an interplay of thermal and threshold noise at bifurcation. Front. Neurosci. 12, 1–8 (2018).
Wiegele, A. Biq Mac Library—a collection of Max-Cut and quadratic 0-1 programming instances of medium size. http://biqmac.uni-klu.ac.at/biqmaclib.pdf (2007).
Booth, M., Reinhardt, S. P. & Roy, A. Partitioning Optimization Problems for Hybrid Classical/Quantum Execution http://www.dwavesys.com/sites/default/files/partitioning_QUBOs_for_quantum_acceleration-2.pdf (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).
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).
Glover, F. Future paths for integer programming and links to artificial intelligence. Comput. Oper. Res. 13, 533–549 (1986).
Wang, T. & Roychowdhury, J. OIM: oscillator-based Ising machines for solving combinatorial optimisation problems. In Unconventional Computation and Natural Computation (Eds. McQuillan, I. & Seki, S.) 11493, 232–256 (Springer International Publishing, 2019)
Chou, J., Bramhavar, S., Ghosh, S. & Herzog, W. Analog coupled oscillator based weighted Ising machine. Sci. Rep. 9, 14786 (2019).
Ahmed, I., Chiu, P.-W. & Kim, C. H. A probabilistic self-annealing compute fabric based on 560 hexagonally coupled ring oscillators for solving combinatorial optimization problems. In 2020 IEEE Symposium on VLSI Circuits 1–2 (2020).
Neukart, F. et al. Traffic flow optimization using a quantum annealer. Front. ICT 4, 29 (2017).
Madan, H. et al. 26.5 terahertz electrically triggered RF switch on epitaxial VO2-on-sapphire (VOS) wafer. In Technical Digest—International Electron Devices Meeting (IEDM) https://doi.org/10.1109/IEDM.2015.7409661 (2015).
Maffezzoni, P., Daniel, L., Shukla, N., Datta, S. & Raychowdhury, A. Modeling and simulation of vanadium dioxide relaxation oscillators. IEEE Trans. Circuits Syst. I. Reg. Papers 62, 2207–2215 (2015).
Simulator, V.S.C. (Cadence Design Systems Inc., 2005).
Acknowledgements
This work was supported in part by ASCENT, one of the six centres in JUMP sponsored by DARPA, NSF and SRC under grant number 2018-JU-2776 (S. Datta); by NSF-SRC SemiSynBio, NSF, under grant numbers CCF-1640081 (S. Datta and Z.T.) and CCF-1644368 (Z.T.); the Nanoelectronics Research Corporation, a wholly owned subsidiary of SRC, through Extremely Energy Efficient Collective Electronics (EXCEL), an SRC-NRI Nanoelectronics Research Initiative, under Research Task ID 2698.004; and by NTT Research (Z.T.).
Author information
Authors and Affiliations
Contributions
S. Dutta and S. Datta conceived the idea. S. Dutta and A.K. performed the measurements, analysed the data and performed the simulations. A.K. fabricated the devices. H.P. and D.G.S. performed the molecular beam epitaxy growth of the VO2 samples. A.R. and Z.T. participated in useful discussions. S. Dutta, A.K., A.R., Z.T. and S. Datta participated in the writing of the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Nature Electronics thanks the anonymous reviewers for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Supplementary Information
Supplementary Discussion, Figs. 1–11 and Table 1.
Rights and permissions
About this article
Cite this article
Dutta, S., Khanna, A., Assoa, A.S. et al. An Ising Hamiltonian solver based on coupled stochastic phase-transition nano-oscillators. Nat Electron 4, 502–512 (2021). https://doi.org/10.1038/s41928-021-00616-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41928-021-00616-7
This article is cited by
-
High-order sensory processing nanocircuit based on coupled VO2 oscillators
Nature Communications (2024)
-
Ferroelectric compute-in-memory annealer for combinatorial optimization problems
Nature Communications (2024)
-
Enhanced convergence in p-bit based simulated annealing with partial deactivation for large-scale combinatorial optimization problems
Scientific Reports (2024)
-
A CMOS-compatible oscillation-based VO2 Ising machine solver
Nature Communications (2024)
-
CMOS-compatible ising machines built using bistable latches coupled through ferroelectric transistor arrays
Scientific Reports (2023)