Bridging the Analog and the Probabilistic Computing Divide: Configuring Oscillator Ising Machines as P-bit Engines

TL;DR

This work addresses the divide between analog oscillator Ising machines (OIMs) and probabilistic p-bit computing by showing how to configure OIMs as p-bit engines through first- and second-harmonic injection, enabling Gibbs/Boltzmann sampling in analog hardware. The authors develop a theoretical framework where oscillator phase dynamics under SHI and FHI map to binary stochastic neurons and networks, with an effective inverse temperature $\beta_{\text{eff}}$ tunable via oscillator quality factor $Q$, SHI slew rate, and sampling times. They demonstrate practical computations, including a 5-node adder and MaxCut, validating Boltzmann statistics (low KL divergence, linear $\log p$ vs energy, and decaying autocorrelation) and showing generalization to the Dynamical Ising Machine (DIM). The results reveal a pathway to hybrid analog-probabilistic computing with potential impact on probabilistic inference and training of stochastic neural models, while outlining design considerations and future extensions to broader analog Ising platforms.

Abstract

Oscillator Ising Machines (OIMs) and probabilistic bit (p-bit)-based computing platforms have emerged as promising paradigms for tackling complex combinatorial optimization problems. Although traditionally viewed as distinct approaches, this work presents a theoretically grounded framework for configuring OIMs as p-bit engines. We demonstrate that this functionality can be enabled through a novel interplay between first- and second harmonic injection to the oscillators. Our work identifies new synergies between the two methods and broadens the scope of applications for OIMs. We further show that the proposed approach can be applied to other analog dynamical systems, such as the Dynamical Ising Machine.

Figures (11)

• Figure 1: Dynamics of a harmonic oscillator under SHI (a) Schematic illustration of the signals required to program an harmonic oscillator as a p-bit. (b) Force field as a function of $\gamma$ ($K_s=0.15$, where $K_s$ is the strength of SHI.). (c) Corresponding energy landscape demonstrating its evolution with the synaptic input, $\gamma$. (d) Specific cuts of the energy landscape at $\gamma=\{0,\,\pm\,0.1,\,\pm\, 0.15,\,\pm\,0.2\}$ ($K_s=0.15$).
• Figure 2: Oscillator-based BSN. Firing probability (symbols) as a function of the synaptic input for: (a) varying levels of noise ($K_n$). (b) different values of $\beta_{\text{eff}}$ derived for $K_n=0.15$. We note that the $\beta_{\text{eff}}$ profile will change for a different $K_n$. The lines in the plot indicate fits using the equation $p=\frac{1+\tanh(\beta_{\text{eff}}V_{\text{inj}})}{2}$ along with the calculated $\beta_{\text{eff}}$; $p:$ firing probability. All fits exhibit $R^2>0.999$
• Figure 3: Full adder Probability histogram measured using the oscillators (orange bars, obtained using $5\times10^{5}$ sweeps) compared with the target Boltzmann distribution (blue bars). Following the convention of Ref. camsari2017stochastic, states are indexed by the decimal value of the binary word $[C_{\mathrm{in}}\;A\;B\;S\;C_{\mathrm{out}}]$. The dominant peaks correspond to the valid entries of the full-adder truth table, as highlighted in the inset. The two distributions show excellent agreement, with a measured KL divergence of $7.68 \times 10^{-4}$. Simulation parameters: $K=0.18$; $K_s(t)=K_{s,\text{max}}(1-e^{-\frac{t}{10^{-2}}})$; $K_n=0.1$.
• Figure 4: Operating OIMs as p-bit platforms. (a) Randomly generated FHI sequence to the oscillators. The application of FHI to the oscillator is accompanied by the suppression of SHI, and vice-versa. (b) Phase response of the oscillators over time. (c) Evolution of the computed graph cut over time/iterations. With stochastic sampling, the system is able to reach the globally optimal solution (MaxCut =39). The red arrows highlight sampling events that improved cut. Simulation parameters: $K=1; K_{s,\text{max}}=2;\: \text{FHI}^0=50;\: K_n \text{ for sampled oscillator}=0.04$; ramping schedule of the SHI signal: $K_s(t) = K_{s,\text{max}} (1 - e^{-\frac{t}{\tau}})$, where $K_{s,\text{max}} = 2$ and $\tau = 10^{-2}$.
• Figure 5: Boltzmann sampling behavior. (a) Plot of $\log$(p) (p: probability) versus energy for varying noise intensities ($K_n = 0.05$, 0.1, 0.15), corresponding to different effective temperatures. The extracted effective inverse temperatures ($\beta$) and the quality of the linear fits ($R^2$) are shown in the inset. (b) Autocorrelation function (ACF) of the system energy as a function of lag, exhibiting a decay toward zero, confirming that the stochastic dynamics effectively decorrelate successive configurations.