Correspondence Between Ising Machines and Neural Networks

TL;DR

This work advances a systematic, provably correct method to run trained neural networks on Ising-type hardware by replacing ground-state computation with high-temperature spin-average computations. It establishes a correspondence between a trained feed-forward network and an Ising Hamiltonian using a temperature gradient, with $J^{(\ell)}=\delta^{\ell-1}W^{(\ell)}$ and $h^{(\ell)}=\delta^{\ell-1}b^{(\ell)}$, and proves that shallow tanh networks are implementable at high temperature while deep binary perceptrons are realizable at zero temperature, under appropriate $\delta$ and $\beta$. The paper provides rigorous mean-field error bounds and constructive proofs (for both the shallow and deep cases), supports the theory with simulated 2-bit multiplication experiments, and discusses practical concerns such as quantization and sparsification for real-world Ising devices. The results point to a scalable path for deploying pretrained neural networks on energy-efficient Ising hardware, with meaningful implications for edge compute and quantum-annealing platforms, while also outlining challenges in extending to large-scale deep architectures. Overall, the work bridges physics-based Ising computation and modern neural networks, offering design principles and proofs of correctness for high-temperature and low-temperature regimes.

Abstract

Computation with the Ising model is central to future computing technologies like quantum annealing, adiabatic quantum computing, and thermodynamic classical computing. Traditionally, computed values have been equated with ground states. This paper generalizes computation with ground states to computation with spin averages, allowing computations to take place at high temperatures. It then introduces a systematic correspondence between Ising devices and neural networks and a simple method to run trained feed-forward neural networks on Ising-type hardware. Finally, a mathematical proof is offered that these implementations are always successful.

Figures (4)

• Figure 1: Heatmap plots of worst case correctness ($\eta$) for two models implementing $2$ bit integer multiplication. Dark blue indicates $+1$ (perfectly correct) while dark red indicates $-1$ (totally incorrect). Any blue area corresponds to a correct circuit. Bright blue and bright red correspond to $\pm0.25$, since the qualitatively important differences in spin averages are concentrated around 0.
• Figure 2: Heatmap plots of spin-wise correctness for each input pattern, plotted at several values of $\delta$ with $\beta=1$ fixed. Model A is above, model B below. The two models behave similarly, exhibiting errors with insufficient temperature gradients (right) and high noise at excessive temperature gradients (left), with an optimal zone of correctness in the center.
• Figure 3: Heatmap plots of spin-wise correctness for each input pattern, plotted at several values of $\beta$ with $\delta=0.07$ fixed. Model A is above, model B below. While both models are correct at intermediate temperatures (center) and noisy at high temperatures (left) at low temperatures (right) model A remains correct while model B develops errors.
• Figure 4: Quantized $2$ bit integer multiplication with integer interaction strengths. Inputs are the left column, outputs right. Node labels are the biases on each spin.

Theorems & Definitions (17)

• Definition 2.1: High-Temperature Solution
• Proposition 2.2: Relationship of High-Temperature and Zero-Temperature Solutions
• proof
• Definition 3.1: Correspondence Principle
• Theorem 5.1
• Definition 5.2: Naive Mean Field System for Adjusted Shallow Networks
• Lemma 5.3: Uniqueness of Mean-Field Solution for Adjusted Shallow Networks
• proof
• Lemma 5.4: Accuracy of Mean-Field Approximation
• proof