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An Ising Hamiltonian solver based on coupled stochastic phase-transition nano-oscillators


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.

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Fig. 1: Overview of PTNO-based CTDS as an Ising Hamiltonian solver.
Fig. 2: Creating artificial Ising spin using SHIL.
Fig. 3: Replicating ferromagnetic and antiferromagnetic Ising interactions.
Fig. 4: Experimental demonstration of MaxCut and performance enhancement with annealing.
Fig. 5: Scaling with problem size.
Fig. 6: Performance evaluation of PTNO-based Ising solver.

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.


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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.).

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Authors and Affiliations



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.

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Correspondence to S. Datta.

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Supplementary Discussion, Figs. 1–11 and Table 1.

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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).

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