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Finite-Time Observability of Oscillatory Instabilities in Synchronous p-bit Dynamics

Naoya Onizawa, Shunsuke Koshita, Takahiro Hanyu

Abstract

Synchronous update schemes in p-bit annealing offer a natural route to massive parallelism, but they can also induce period-2 oscillations that degrade optimization performance. In practical solvers, such oscillations matter only if they become observable within the finite runtime of the device or simulation, yet most existing analyses are formulated in terms of asymptotic stability. As a result, they do not directly address the experimentally relevant question of when oscillatory modes actually appear during finite-duration annealing. Here we develop a finite-time observability framework for synchronous tick-random p-bit dynamics. Starting from a linearized mean-field description, we derive a graph-dependent criterion that predicts whether unstable modes amplify enough within a finite observation window to produce visible signatures in quantities such as the one-step autocorrelation and the energy trace. This shifts the analysis from asymptotic instability to practical detectability and yields a principled estimate of the minimum reduction in synchrony required to suppress oscillations. We validate the framework on G-set benchmark graphs and illustrative graph families. The predicted thresholds capture both the graph dependence of oscillation onset and the finite-time conditions under which oscillations become practically observable. These results provide a graph-aware basis for selecting the update probability in synchronous p-bit annealers without relying on exhaustive instance-by-instance parameter sweeps.

Finite-Time Observability of Oscillatory Instabilities in Synchronous p-bit Dynamics

Abstract

Synchronous update schemes in p-bit annealing offer a natural route to massive parallelism, but they can also induce period-2 oscillations that degrade optimization performance. In practical solvers, such oscillations matter only if they become observable within the finite runtime of the device or simulation, yet most existing analyses are formulated in terms of asymptotic stability. As a result, they do not directly address the experimentally relevant question of when oscillatory modes actually appear during finite-duration annealing. Here we develop a finite-time observability framework for synchronous tick-random p-bit dynamics. Starting from a linearized mean-field description, we derive a graph-dependent criterion that predicts whether unstable modes amplify enough within a finite observation window to produce visible signatures in quantities such as the one-step autocorrelation and the energy trace. This shifts the analysis from asymptotic instability to practical detectability and yields a principled estimate of the minimum reduction in synchrony required to suppress oscillations. We validate the framework on G-set benchmark graphs and illustrative graph families. The predicted thresholds capture both the graph dependence of oscillation onset and the finite-time conditions under which oscillations become practically observable. These results provide a graph-aware basis for selecting the update probability in synchronous p-bit annealers without relying on exhaustive instance-by-instance parameter sweeps.

Paper Structure

This paper contains 35 sections, 1 theorem, 31 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Model and Update Scheme
  4. Ising energy and p-bit dynamics
  5. Tick-random synchronous update
  6. Finite-time observation and oscillatory behavior
  7. Oscillation Phenomenon
  8. Empirical observation
  9. Definition of observable oscillation
  10. Mean-Field Theory and Transient Stability
  11. Mean-field dynamical map
  12. Linearization around fixed points
  13. Finite-time oscillation suppression
  14. Gaussian effective gain approximation
  15. Non-IPR vs IPR-corrected theory.
  16. ...and 20 more sections

Key Result

Theorem 1

Let $\bm{A}$ be the Jacobian of the mean-field map in Eq. (eqS:jacobian). If for a finite horizon $T$, for an observability threshold $R>1$, then no observable oscillation emerges within $T$ steps.

Figures (6)

  • Figure 1: Energy trajectories for G1 at $c=1,2,3$, illustrating suppression of period-2 oscillations as synchrony is relaxed.
  • Figure 2: Energy trajectories for multiple $c$ values (a $c$-sweep) with step 0.4, showing suppression of oscillations as $c$ increases.
  • Figure 3: Normalized second-difference amplitude of the late-time energy trace and normalized optimization performance versus the partial-synchrony control parameter $c$ for representative G-set instances. The blue curves show an energy-based oscillation measure obtained from the late-time energy trace, while the orange curves show the normalized Max-Cut score obtained in the same runs. The dashed line marks the simulation threshold $c^\star$ determined from the autocorrelation criterion $C(1)\ge 0.5$. In each case, increasing $c$ suppresses the large-amplitude period-2 energy oscillations and is accompanied by a rapid recovery of optimization performance, after which the score saturates. The figure therefore links finite-time oscillation suppression directly to a practically relevant performance measure.
  • Figure 4: Theoretical stability boundary $c^*(I_0)$ using Gaussian $\alpha_{\mathrm{eff}}(I_0)$ and a finite-time observability criterion. Each panel contains (top) non-IPR and (bottom) IPR-corrected curves, which emphasize unstable but spatially delocalized modes that couple more strongly to macroscopic observables. Small-$N$ illustrative curves are included for qualitative comparison; quantitative agreement improves for larger illustrative ER graphs (Toy-6/7).
  • Figure S1: Synchronous tick-random p-bit annealing with partial synchrony parameter $c$.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 1: Finite-time oscillation suppression
  • proof