Adaptive controllable architecture of analog Ising machine

TL;DR

This work unifies AIM theory by expressing the problem as minimizing a Lyapunov energy $E(\psi)=K(\psi)+\mu R(\psi)$ that aligns with the Ising objective $H(\mathbf{s})=\mathbf{s}^T\mathbf{J}\mathbf{s}+\mathbf{h}^T\mathbf{s}$, and introduces CAIM to adapt the binarization constraint via a control variable $\bm{\mu}(t)$. By combining control Lyapunov functions with a momentum-based optimizer in an asynchronous sampling-feedback loop, CAIM achieves faster convergence and improved accuracy relative to conventional AIMs, as demonstrated on FPGA-controlled LC-oscillator hardware with a 51-node network, yielding about $2\times$ speedup and $7\%$ higher success probability on MaxCut-like tasks. The paper also develops a rigorous equivalence framework and analyzes the fundamental trade-off between correctness and reachability governed by $\mu$, providing predictions for $\widehat{\mu}$ and demonstrating how adaptive control can overcome prior limits. Collectively, these results offer a principled pathway to more scalable and reliable analog Ising solvers with practical impact for combinatorial optimization on near-term hardware.

Abstract

As a quantum-inspired, non-traditional analog solver architecture, the analog Ising machine (AIM) has emerged as a distinctive computational paradigm to address the rapidly growing demand for computational power. However, the mathematical understanding of its principles, as well as the optimization of its solution speed and accuracy, remain unclear. In this work, we for the first time systematically discuss multiple implementations of AIM and establish a unified mathematical formulation. On this basis, by treating the binarization constraint of AIM (such as injection locking) as a Lagrange multiplier in optimization theory and combining it with a Lyapunov analysis from dynamical systems theory, an analytical framework for evaluating solution speed and accuracy is constructed, and further demonstrate that conventional AIMs possess a theoretical performance upper bound. Subsequently, by elevating the binarization constraint to a control variable, we propose the controllable analog Ising machine (CAIM), which integrates control Lyapunov functions and momentum-based optimization algorithms to realize adaptive sampling-feedback control, thereby surpassing the performance limits of conventional AIMs. In a proof-of-concept CAIM demonstration implemented using an FPGA-controlled LC-oscillator Ising machine, CAIM achieves a twofold speedup and a 7\% improvement in accuracy over AIM on a 50-node all-to-all weighted MaxCut problem, validating both the effectiveness and interpretability of the proposed theoretical framework.

Figures (10)

• Figure 1: Proposed CAIM architecture and implementations. (a) A lattice-shaped Ising model and the corresponding oscillator network, where each spin in the Ising model has only two discrete states, while the states of the oscillator network are typically continuous and are usually represented by $0$ and $\pi$ as the two extrema. (b) The CAIM architecture. Compared with a conventional AIM, an additional controller is introduced and independent injection signals can be applied to each oscillator. The controller samples the oscillator states and computes feedback injection signals, thereby enabling precise regulation of the oscillator states, intervening in the intrinsic dynamical evolution of a conventional AIM, and consequently achieving faster and more accurate solution processes. (c) In a conventional AIM, identical injection signals with fixed strength are applied to all oscillators, whereas in a CAIM, different injection signals are applied to different oscillators and their injection strengths are adjusted in real time. (d) A proof-of-concept physical implementation framework of a COIM is presented, in which the controller is realized using an FPGA to perform signal sampling, computation, and feedback.
• Figure 2: Theoretical limitation of AIM due to the trade-off of correctness and reachability. (a) The primal problem in Eqs. \ref{['eq: primal problem 1']}$\sim$\ref{['eq: primal problem 2']} can be regarded as having discrete energy levels, which is equivalent to the limit of regularization strength $\mu\to\infty$. For the relaxed problem, the state space is extended from discrete spin configurations $\bm{s}$ in the sets $\mathcal{S}_i$ (which yield Hamiltonian values $H_i$) to continuous representations $F^{-1}(\bm{s})$, such that the representation energy becomes continuous rather than discrete and spans a finite variation range. If, for some $\mathcal{S}_i$ with $i\neq 0$, the variation range of a certain spin is sufficiently large and its lower bound is the minimum among all variation ranges, then the theoretical solutions of the relaxed problem and the original problem are no longer equivalent. By increasing the regularization strength $\mu$, the variation range of the representation energy can be reduced, and when this range becomes sufficiently small, i.e., smaller than $H_1-H_0$, the solutions of the original and relaxed problems become equivalent. (b) An excessively large regularization strength $\mu$ enhances the attraction near the extrema of the regularization term, thereby forming local minima, and due to the non-increasing energy constraint of the Lyapunov process, once the system is trapped in a local minimum it cannot escape. Moreover, the regularization term typically introduces two extrema for each dimension, such as $\{\pm1\}$ or $\{0,\pi\}$, resulting in $\mathcal{O}(2^n)$ local minimum regions in practice. (c) Fig. (a) discusses solution equivalence from the perspective of optimization theory, while Fig. (b) addresses it from the perspective of dynamical systems; however, the former favors larger $\mu$, whereas the latter prefers smaller $\mu$, leading to a trade-off between correctness and reachability and consequently imposing an upper bound on the practical solution quality of AIMs. (d) Within each time slice of duration $\tau$, the controller first samples the oscillator states, then computes the amplitude according to the sampled states and the control algorithm, and outputs the result at the beginning of the next time slice to adjust the injection signal amplitude, and this process is repeated iteratively.
• Figure 3: Oscillator unit and FPGA workflow design.
• Figure 4: Comparison of COIM and OIM circuit experimental results. (a) An instance of a weighted MaxCut problem (SpinModel) with 50 nodes, where each row and column contains 50 grid points and the color indicates the coupling weight. (b) The real-time evolution of the Ising Hamiltonian during the solving process, where the light-colored curves denote raw energy values and the dark-colored curves represent smoothed results. COIM exhibits faster convergence and yields solutions closer to the ground state. (c) The evolution of different energy components during the solving process for OIM and COIM.
• Figure 5: Differences between COIM and OIM during evolution (Time ($\mu$s), Voltage (V)). Since displaying all 51 oscillators would be redundant, oscillators 0, 1, and 3 are selected as representatives. (a) Oscillation waveforms of COIM oscillators near convergence. (b) The corresponding injection signals, showing that the injection amplitudes in COIM are time-varying and independently controlled for each oscillator. (c) The full-time phase evolution of COIM oscillators, whose rate of change is significantly higher than that of OIM. (d) Oscillation waveforms of OIM oscillators near convergence. (e) Injection signals of OIM, where the curves overlap because the amplitudes are constant and identical across oscillators. (f) The full-time phase evolution of OIM oscillators, which is noticeably slower than that of COIM.

Theorems & Definitions (8)

• Definition 1: Analog Ising machine
• Definition 2: Equivalent solution
• Proposition 1: Condition of equivalent solution
• Theorem 2
• Proposition 3
• Remark 1
• Lemma 4: Convergence of asynchronous momentum
• Remark 2: A special case without delay