On the solvable-unsolvable transition due to noise-induced chaos in digital memcomputing

TL;DR

This work investigates how numerical and physical noise induce a solvable-unsolvable transition in digital memcomputing machines solving $3$-SAT. The authors map $3$-SAT to continuous-time ODEs with fast variables $v_n\in[-1,1]$ and memory variables $(x_m,y_m)$, and analyze solvability via the clause condition $C_m<0.5$ for all $m$ along with Lyapunov exponents and power spectra. A key finding is the existence of a transiently chaotic regime where the ensemble-averaged mean largest Lyapunov exponent $\langle\overline{\lambda}\rangle$ is positive but instances remain solvable, and spectral features can distinguish regular from chaotic dynamics. The results suggest a noise-induced chaos mechanism behind solvability loss and propose spectral diagnostics to control DMM operation, informing the design of more robust noise-tolerant memcomputing hardware.

Abstract

Digital memcomputing machines (DMMs) have been designed to solve complex combinatorial optimization problems. Since DMMs are fundamentally classical dynamical systems, their ordinary differential equations (ODEs) can be efficiently simulated on modern computers. This provides a unique platform to study their performance under various conditions. An aspect that has received little attention so far is how their performance is affected by the numerical errors in the solution of their ODEs and the physical noise they would be naturally subject to if built in hardware. Here, we analyze these two aspects in detail by varying the integration time step (numerical noise) and adding stochastic perturbations (physical noise) into the equations of DMMs. We are particularly interested in understanding how noise induces a chaotic transition that marks the shift from successful problem-solving to failure in these systems. Our study includes an analysis of power spectra and Lyapunov exponents depending on the noise strength. The results reveal a correlation between the instance solvability and the sign of the ensemble averaged mean largest Lyapunov exponent. Interestingly, we find a regime in which DMMs with positive mean largest Lyapunov exponents still exhibit solvability. Furthermore, the power spectra provide additional information about our system by distinguishing between regular behavior (peaks) and chaotic behavior (broadband spectrum). Therefore, power spectra could be utilized to control whether a DMM operates in the optimal dynamical regime. Overall, we find that the qualitative effects of numerical and physical noise are mostly similar, despite their fundamentally different origin.

Figures (15)

• Figure 1: Numerical noise investigation: Percentage of solved instances and EAMLLE ($\langle\overline{\lambda}\rangle$) vs. integration time step, $\Delta t$. Each point used to make this plot was obtained using an ensemble of 100 solvable 3-SAT instances with clause-to-variable ratio $M/N=7$. The upper bound for each trajectory was set to 20,000 steps. The smallest $\Delta t$ taken is 0.01.
• Figure 2: Numerical noise investigation: Power spectra for a satisfiable instance with $N=1000$ variables and with clause-to-variable ratio $M/N=7$ generated using the integration time step of (a) 0.07, (b) 0.2, (c) 0.3, and (d) 0.4.
• Figure 3: Numerical noise investigation: Power spectra for a satisfiable instance with $N=90$ variables and with clause-to-variable ratio $M/N=4.3$ obtained using the integration time step of (a) 0.05, (b) 0.25, (c) 0.54, and (d) 0.58.
• Figure 4: Physical noise investigation. Main panel: Transition of the percentage of solved cases as a function of noise amplitude $\Gamma$, for a problem instance of $N=1000$ variables at clause-to-variable ratio $M/N=7$. Each point represents 100 realization of the noise. The simulation runs for a maximum of 10,000 steps if convergence to a solution does not occur beforehand . In all cases, the variables $\textnormal{v}_n$'s are initialized to zero, the long term memory $y_{m}$ to 1 and the short term memory $x_{m}$ to $C_m$ which is equal to 1/2 when $\textnormal{v}_n=0$$\forall n$ (midpoint value of the short term memory bounded range [0,1]). The spectrum is then averaged over 100 noise realizations. The MLLE averaged over noise realisation $\langle\overline{\lambda}\rangle_{noise}$ is also displayed. Insets (1), (2), and (3) show representative power spectra of the flow vector field of the fast degrees of freedom of individual instances at different noise levels. Inset (1): Power spectrum of a solved instance at low noise ($\Gamma \approx 0$), where 100% of instances are solved. Inset (2): Power spectra at intermediate noise. The red curve corresponds to the average power spectrum of noise realizations of 85 solved cases, and the black curve to the average of noise realizations of 15 unsolved ones. Inset (3): Average power spectrum of an unsolved instance at high noise over 100 noise realizations ($\Gamma \approx 1.7$ where the solving rate drops to zero).
• Figure 5: Physical noise investigation: Percentage of problem instances solved with respect to the noise strength $\Gamma$ for clause-to-variable ratio $M/N=7$. The solvable-unsolvable transition becomes sharper with increasing number of variables. The time step is set to be inversely proportional to the maximum time derivative of the variables' vector $\textbf{v}$ (see main text for details). For every value of $\Gamma$ and for each size $N$ a batch of 1000 problem instances has been simulated in one run.