DROID: Discrete-Time Simulation for Ring-Oscillator-Based Ising Design

TL;DR

DROID presents an event-driven discrete-time simulator for CMOS ring-oscillator Ising machines, addressing limitations of continuous-time models by capturing fine-grained timing interactions and transistor nonlinearities through timing-arc lookups and a fine-grained interaction window $W$. By modeling an all-to-all coupled RO Ising hardware with an A2A topology, and using a look_back mechanism to account for unseen predecessor events, DROID achieves ground-state predictions that closely match hardware distributions while delivering speedups from $\sim$10^2 to over $10^3$-fold compared with HSPICE. The approach is validated on silicon-proven A2A hardware and across problem densities, demonstrating that DROID can replicate both timing dynamics and solution distributions (as measured by Earth Mover Distance) with substantial scalability. Overall, the method enables rapid, accurate exploration of large RO-Ising systems, supporting design optimization and hardware-aware problem solving for complex COPs.

Abstract

Many combinatorial problems can be mapped to Ising machines, i.e., networks of coupled oscillators that settle to a minimum-energy ground state, from which the problem solution is inferred. This work proposes DROID, a novel event-driven method for simulating the evolution of a CMOS Ising machine to its ground state. The approach is accurate under general delay-phase relations that include the effects of the transistor nonlinearities and is computationally efficient. On a realistic-size all-to-all coupled ring oscillator array, DROID is nearly four orders of magnitude faster than a traditional HSPICE simulation in predicting the evolution of a coupled oscillator system and is demonstrated to attain a similar distribution of solutions as the hardware.

Figures (13)

• Figure 1: Three five-stage ROs, with a positive and a negative coupling. The green stage is the reference, and odd (even) stages are shown in red (blue).
• Figure 2: (a) Waveforms for nets $a$ and $b$ in two ROs, when uncoupled, and with coupling enabled at $t=0$. (b) Detailed view of the green box in (a), showing the effect of coupling. The period of waveform at $a$ increases while that at $b$ decreases, reducing the phase difference.
• Figure 3: The relative arrival times on nets $a$ (green waveform) and $b$ (blue waveform), top, impact the transition delay on net $c$ (middle). The transition delay as a function of phase difference, and the interaction window $W$ are illustrated at the bottom.
• Figure 4: An illustration of the A2A concept through a small three-RO structure, showing shorting cells $S$ on the diagonal, programmable off-diagonal coupling cells $C$, and enable cells $EN$.
• Figure 5: (a) The results for an example characterization setup, with $J_{ij}=1$, showing the delay shift $f_{J_{ij}}$ as a function of $\phi_{ij}$. (b) HSPICE characterization of $f_{J_{ij}}$ and its tanh approximation as a function of phase difference $\phi_{ij}$.