Using continuation methods to analyse the difficulty of problems solved by Ising machines

TL;DR

The paper analyzes how continuation methods can reveal the difficulty of solving Ising-model problems with Ising machines by tracking the bifurcation sequence of the optimal solution. It demonstrates that the problem class (spectral easy, Ising easy, Ising hard) depends on both problem structure and the machine's nonlinear implementation, and that changing the nonlinearity can dramatically alter the bifurcation topology to make hard problems easier. By exploring third-order, fifth-order, and sigmoid nonlinearities, the authors show that introducing additional degrees of freedom in the transfer function can connect the ground-state branch to the origin through saddle-node or cusp points, thereby increasing the fraction of problems solvable by annealing. The work provides a practical framework for designing Ising-machine implementations and shows that continuation analysis can serve as both a diagnostic tool and a solver for Ising networks, albeit with caveats regarding noise in physical devices.

Abstract

Ising machines are dedicated hardware solvers of NP-hard optimization problems. However, they do not always find the most optimal solution. The probability of finding this optimal solution depends on the problem at hand. Using continuation methods, we show that this is closely linked to the bifurcation sequence of the optimal solution. From this bifurcation analysis, we can determine the effectiveness of solution schemes. Moreover, we find that the proper choice of implementation of the Ising machine can drastically change this bifurcation sequence and therefore vastly increase the probability of finding the optimal solution.

Figures (9)

• Figure 1: Spin amplitudes while tracking the ground state using continuation when using equation \ref{['eq:ThirdOrder']} with $\alpha=0$. (a) is the result of a problem with $100$ spins and antiferromagnetic, nearest neighbours interactions, (b) of problem g05_100.2 of the BiqMac library and (c) of problem g05_100.1.
• Figure 2: Continuation of spin amplitude 3 of problem g05_100.1 when solved using the third order nonlinearity ($\alpha=0$). Solid (dashed) lines are used when this spin amplitude is part of a stable (unstable) state.
• Figure 3: Continuation of the spin amplitudes of the first pitchfork branch and the ground state branch of problem g05_100.1 of the BiqMac library using the sigmoid nonlinearity with (a) $\alpha=0.98$ and (b) $\alpha = 0.95$. They gray shaded background indicates the region where the energy of the ground state branch is the ground state energy.
• Figure 4: Continuation of the spin amplitude of all four spins of the first pitchfork branch. This continuation was performed using the sigmoid nonlinearity for fixed $\alpha=0.996$. The lines are solid when the corresponding fixed point is stable and dashed when the corresponding fixed point is unstable. They gray shaded background indicates the region where the energy of the ground state branch is the ground state energy.
• Figure 5: (a) Continuation in the two parameters ($\alpha$ and $\beta$) of the pitchfork bifurcation point (PB) and the two saddle node bifurcations (SN) for the four-spin-system solved with the sigmoid nonlinearity in equation \ref{['eq:tanh']}. The three dashed lines represent the three $\alpha$-values at which the other three plots are obtained. The value of the third spin amplitude as a function of the coupling strength $\beta$ is shown in (b) for $\alpha=0.990$, in (c) for $\alpha=0.994$ and in (d) for $\alpha=0.996$.