Phase analysis of Ising machines and their implications on optimization

TL;DR

This analysis shows that coexistence phase region can be expanded by carefully placing a digitization operation, giving rise to a family of superior Ising machines, as illustrated by the proposed algorithm digCIM.

Abstract

Ising machines, which are dynamical systems designed to operate in a parallel and iterative manner, have emerged as a new paradigm for solving combinatorial optimization problems. Despite computational advantages, the quality of solutions depends heavily on the form of dynamics and tuning of parameters, which are in general set heuristically due to the lack of systematic insights. Here, we focus on optimal Ising machine design by analyzing phase diagrams of spin distributions in the Sherrington-Kirkpatrick model. We find that that the ground state can be achieved in the phase where the spin distribution becomes binary, and optimal solutions are produced where the binary phase and gapless phase coexist. Our analysis shows that such coexistence phase region can be expanded by carefully placing a digitization operation, giving rise to a family of superior Ising machines, as illustrated by the proposed algorithm digCIM.

Figures (4)

• Figure 1: AIM phases as characterized by distinct modes of spin distribution $P(x)$ together with typical AIM dynamics of $x$ vs $t$ ($T=0$). (a) Paramagnet phase. (b) Gapless phase. (c) Gap-opened phase. (d) Binary phase. (e) Gapless-binary coexistence region.
• Figure 2: Phase diagram of CIM in the space of gain $a$ and temperature $T$ with typical analog spin distributions shown in each region, respectively located at $(a, T) = (-1.5, 0.5)$ (paramagnetic phase, defined by $q_1= 0$), $(-1.5, 0.03)$ (gapless spin glass phase, defined by $q_1 > 0$), and $(0.5, 0.03)$ (pseudogap spin glass phase, defined by $q_1 >0$ and $a_{\rm{eff}}>0$). In these distributions, the solid black lines correspond to 1RSB theory, while the grey shadings indicate simulation results. The dashed phase line is obtained in RS. G is the 1RSB gap-opening point at $T = 0$. The paramagnetic phase also consists of a pseudogap region but for clarity it is not shown in the figure (see Sec. 4 of SM).
• Figure 3: Phase diagram and the spin distribution for clipCIM ($B = 1$), simCIM and digCIM in the space of gain $a$ and bound $B$ at zero temperature. In Fig. 3(a), the solid and dashed phase lines are obtained by FRSB and simulation, respectively. Distributions are annotated similarly to those in Fig. \ref{['fig:phase_diagram']}.
• Figure 4: (a) Theoretical dependence of CIM, clipCIM ($B = 1$), simCIM, and digCIM's decoded energy on the gain $a$ at zero temperature. Dotted line shows the simulation results for 10,000 steps, $N = 10,000$ and $T=10^{-5}$, bounded by extremal values in the shaded areas. (b) The same experiments applied to G1 in Gset Ye_2003, which exhibits similar characteristics.