Abstract
The past decade has seen the emergence of Ising machines targeting hard combinatorial optimization problems by minimizing the Ising Hamiltonian with spins represented by continuous dynamical variables. However, capabilities of these machines at larger scales are yet to be fully explored. We introduce and investigate an almost-linear Ising machine, a machine based on a network of analog spins with piece-wise linear coupling. We show that such networks leverage the computational resource similar to that of the semidefinite positive relaxation of the Ising model. We estimate the expected performance of the almost-linear machine and benchmark it on a set of \(\left\{ 0, 1\right\}\)-weighted graphs. We show that the running time of the investigated machine scales polynomially (linearly with the number of edges in the connectivity graph). As an example of the physical realization of the machine, we present a CMOS-compatible implementation comprising an array of vertices efficiently storing the continuous spins on charged capacitors and communicating externally via analog current.
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Availability of data and materials
The graphs used to obtain Fig. 4 are generic and can be generated using the code referred below. The specific running time values may differ depending on the computer performance and running conditions. However, the estimated scaling, which is the main result of Fig. 4, does not depend on the computer performance as long as the same running conditions are ensured for different graphs.The specific datasets depicted on the manuscript figures are available from the corresponding author on reasonable request.
Code availability
The code of Circut is available at https://www.caam.rice.edu/~zhang/circut/. The code used for simulating the almost-linear Ising machine and generating the supporting data is available at https://github.com/merement/Dice/tree/main/cases/triangular.
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Funding
The work has been supported by the US National Science Foundation (NSF) under Grant No. 1710940 and by the US Air Force Office of Scientific Research (AFOSR) under Grant No. FA9550-16-1-0363.
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All authors contributed to the study conception. AS designed and simulated the custom integrated circuit. ME performed the theoretical analysis and its software validation. Both ME and AS collectively drafted the manuscript. PM supervised the project and revised the manuscript. All authors read and gave approval to the final version of the manuscript.
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Shukla, A., Erementchouk, M. & Mazumder, P. Scalable almost-linear dynamical Ising machines. Nat Comput (2024). https://doi.org/10.1007/s11047-024-09983-4
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DOI: https://doi.org/10.1007/s11047-024-09983-4