Abstract
Ising machines have been attracting attention due to their ability to use mixed discrete/continuous mechanisms to solve difficult combinatorial optimization problems. We present BLIM, a novel Ising machine scheme that uses latches (bistable elements) with controllable gains as Ising spins. We show that networks of coupled latches have a Lyapunov or “energy” function that matches the Ising Hamiltonian in discrete operation, enabling them to function as Ising machines. This result is established in a general coupled-element Ising machine framework that is not limited to BLIM. Operating the latches periodically in analog/continuous mode, during which bistability is removed, helps the system traverse to better minima. CMOS realizations of BLIM have desirable practical features; implementation in other physical domains is an intriguing possibility.
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Notes
- 1.
- 2.
This “energy” is not obviously related to any concept of physical energy, which latches, like all practical electronic elements, consume and dissipate as heat.
- 3.
When not switching, CMOS consumes no power beyond leakage losses.
- 4.
i.e., any small perturbation (e.g., due to noise) from this solution will make the latch settle to one of the other two solutions, at the top left and bottom right.
- 5.
\(\tanh ()\) is merely a convenient analytical choice; any other smoothed step-like function can be used instead.
- 6.
The Lyapunov function is defined in terms of abstract functions \(z(\cdot ;\cdot )\) and \(h(\cdot ,\cdot ;\cdot )\) that are related to the functions \(f(\cdot ;\cdot )\) and \(g(\cdot ,\cdot ;\cdot )\) in the generalized model (7). The relations, captured abstractly as assumptions in Assumption 2, are illustrated concretely for the \(\tanh (\cdot )\) latch model in Sect. 4.
- 7.
This assumption is intrinsic to the Ising model, as already noted in (1).
- 8.
It is easy to show graphically that \(v_+\), \(v_-\) and 0 are the only solutions of \(f(v,\, K) = 0\).
- 9.
The third solution, \(v=0\), is unstable: \({\frac{d f}{d v}}(0) = 24 > 0\).
- 10.
Recall that \(H(\vec \cdot )\) is the Ising Hamiltonian and \(L(\vec \cdot \,; \cdot )\) the Lyapunov function.
- 11.
Each node represents an Ising spin; the weight of the edge between two nodes i and j is \(J_{ij}\).
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Acknowledgements
We thank Tianshi Wang, Nagendra Krishnapura, Yiannis Tsividis and Peter Kinget for discussions that motivated this work. Support from the US National Science Foundation is gratefully acknowledged.
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Roychowdhury, J. (2021). Bistable Latch Ising Machines. In: Kostitsyna, I., Orponen, P. (eds) Unconventional Computation and Natural Computation. UCNC 2021. Lecture Notes in Computer Science(), vol 12984. Springer, Cham. https://doi.org/10.1007/978-3-030-87993-8_9
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