How to Incorporate External Fields in Analog Ising Machines

TL;DR

This work addresses how external fields should be incorporated into analog Ising machines (IMs) to solve combinatorial optimization problems. It compares four field-embedding strategies plus a constant-field scaling across three problem classes, using a tanh-based transfer function and linear annealing, to evaluate time-to-solution. The spin-sign method emerges as the most robust and fastest across SK, Beasley, and Max-3-Cut problems, with a notable finding that a fixed scaling factor $\zeta \approx 0.6$ greatly improves Max-3-Cut mappings by mitigating constraint-embedding errors. The findings inform hardware design for analog IMs, suggesting that simple, hardware-friendly spin-sign based field incorporation offers practical performance gains, especially when paired with appropriate field scaling for soft constraints.

Abstract

Ising machines (IMs) are specialized devices designed to efficiently solve combinatorial optimization problems (COPs). They consist of artificial spins that evolve towards a low-energy configuration representing a problem's solution. Most realistic COPs require both spin-spin couplings and external fields. In IMs with analog spins, these interactions scale differently with the continuous spin amplitudes, leading to imbalances that affect performance. Various techniques have been proposed to mitigate this issue, but their performance has not been benchmarked. We address this gap through a numerical analysis. We evaluate the time-to-solution of these methods across three distinct problem classes with up to 500 spins. Our results show that the most effective way to incorporate external fields is through an approach where the spin interactions are proportional to the spin signs, rather than their continuous amplitudes.

Figures (17)

• Figure 1: (a) Example of a 3-spin COP with ground state configuration (-1,1,-1). (b)-(e) Spin evolution under the methods described in \ref{['eq:original external fields', 'equation: aux trick', 'equation: meanAbs trick', 'eq:spin sign method']}, respectively. In (b), the initial direction of the spins (dotted lines) is given by $h_i / (1-\alpha)$, independently of the spin-spin couplings $J_{ij}$. Methods in (c)-(e) are designed to prevent the external fields from dominating the spin-spin couplings. The inset in (e) shows a zoom-in near $\beta=0$.
• Figure 2: Comparison of time-to-solution between different methods to implement external fields (\ref{['eq:original external fields', 'equation: meanAbs trick', 'equation: aux trick', 'eq:spin sign method']}) for SK Hamiltonians with random external fields. Dots in the grey area on the right denote COPs that could be solved by the spin sign method within the allocated compute time of $t_\text{max}=10^4$, but not by the method on the x-axis ($\text{TTS}=\infty$, $\text{SR}=0$). The spin sign method can solve more problems within the allocated time than the other methods, and it generally requires less time to do so.
• Figure 3: Comparison of time-to-solution between different methods to implement external fields for Beasley problems. Dots in the grey area on the right denote COPs that could be solved by the spin sign method within the allocated compute time of $t_{\text{max}}=10^4$, but not by the method on the x-axis ($\text{TTS}=\infty$, $\text{SR}=0$).
• Figure 4: Comparison of time-to-solution between $\zeta=1$ and the optimal $\zeta$ that minimizes the TTS, for each of the methods, for Max-3-Cut problems. Data points in the grey area on the right denote COPs that could not be solved using $\zeta=1$ ($TTS=\infty$, $SR=0$), given the allocated compute time of $t_\text{max}=10^4$.
• Figure 5: Optimal $\zeta$ values (which minimize TTS) as a function of the number of spins. Each dot represents the mean optimal value across 10 Max-3-Cut problems. Shaded areas represent the standard deviation.