Exact Spin Elimination in Ising Hamiltonians and Energy-Based Machine Learning

TL;DR

This work introduces an exact spin-elimination technique for Ising Hamiltonians that removes spins in a single step or via a deterministic sequence while preserving the original ground-state configuration. By replacing each eliminated spin with an appropriate higher-order interaction among its neighbors, the method trades local spin complexity for increased locality or order, enabling efficient reductions on hardware that supports multi-body couplings. The authors demonstrate significant practical benefits across multiple domains, including larger 3-regular Max-Cut instances, qubit-reduced factorization on near-term devices, and enhanced memory recall in Hopfield networks, while preserving ground-state solutions. The approach offers a path toward scalable Ising-based optimization and energy-based learning on next-generation hardware, and points to hardware capable of native multi-body interactions as a key enabler for large-scale applications.

Abstract

We present an exact spin-elimination technique that reduces the dimensionality of both quadratic and k-local Ising Hamiltonians while preserving their original ground-state configurations. By systematically replacing each removed spin with an effective interaction among its neighbors, our method lowers the total spin count without invoking approximations or iterative recalculations. This capability is especially beneficial for hardware-constrained platforms, classical or quantum, that can directly implement multi-body interactions but have limited qubit or spin resources. We demonstrate three key advances enabled by this technique. First, we handle larger instances of benchmark problems such as Max-Cut on cubic graphs without exceeding a 2-local interaction limit. Second, we reduce qubit requirements in QAOA-based integer factorization on near-term quantum devices, thus extending the feasible range of integers to be factorized. Third, we improve memory capacity in Hopfield associative memories and enhance memory retrieval by suppressing spurious attractors, enhancing retrieval performance. Our spin-elimination procedure trades local spin complexity for higher-order couplings or higher node degrees in a single pass, opening new avenues for scaling up combinatorial optimization and energy-based machine learning on near-term hardware. Finally, these results underscore that the next-generation physical spin machines will likely capitalize on k-local spin Hamiltonians to offer an alternative to classical computations.

Figures (6)

• Figure 1: Four gadgets for spin $s_a$ elimination. (a) Left: Spin $s_a$ is coupled to spin $s_b$ with strength $b$ and to spin $s_c$ with strength $c$, while an external field of strength $a$ acts on $s_a$ (indicated by the yellow rectangle). Right: After eliminating $s_a$, spins $s_b$ and $s_c$ acquire an effective coupling of strength $\alpha$ and external fields of strengths $\beta$ and $\gamma$. (b) Left: Spin $s_a$ interacts with spins $s_b$ and $s_c$ through pairwise couplings of strengths $b$ and $c$, as well as a three-body interaction of strength $a$. Right: After eliminating $s_a$, spins $s_b$ and $s_c$ acquire an effective two-body coupling of strength $\alpha$ and external fields (not shown). (c) Left: Spin $s_a$ is coupled to spin $s_b$ with strength $b$, to spin $s_c$ with strength $c$, and to spin $s_d$ with strength $d$. Right: After eliminating $s_a$, the remaining spins $s_b, s_c,$ and $s_d$ acquire effective pairwise couplings $(\alpha, \beta, \gamma)$. (d) Left: Spin $s_a$ interacts with spins $s_b$, $s_c$, and $s_d$ through pairwise couplings of strengths $b$, $c$, and $d$, respectively, along with three-body interactions among any three spins and a four-body interaction involving $s_a, s_b, s_c,$ and $s_d$. Right: After eliminating $s_a$, the remaining spins $s_b, s_c,$ and $s_d$ acquire new two-body pairwise couplings and an effective three-body interaction. The expressions for the updated coupling strengths in cases (a–c) are provided in the main text.
• Figure 2: Spin-elimination strategies for Max-Cut on 3-regular (cubic) graphs. (a) $k$-local hardware: Start with disjoint subgraphs of four nodes (red outlines) and eliminate the central spin. This re-routes the surviving neighbors (magenta edges) to keep degree $\le 4$, removing about one-quarter of the spins per round. Subsequent rounds can remove additional spins in subgraphs of five or more nodes, potentially introducing higher-order couplings as $k$ increases. (b) 2-local hardware: Remove every third node in overlapping four-node subgraphs, boosting the maximum node degree to six but retaining strictly 2-local couplings.
• Figure 3: Degree distribution after spin elimination in Max-Cut problems on 3-regular graphs for different problem sizes $10\le N\le 2^{13}$. Curves show the mean percentage of vertices ending with degree $0, 4, 5,$ or $6$ after 10,000 independent elimination runs on graphs of size $N\le 128$, 1,000 runs for $256\le N \le 1024$, 100 runs for $N\ge 2048$. For every run the raw counts were first converted to a percentage of the original vertex total $N$; the shaded bands depict $\pm 1$ standard deviation of those per-run percentages. Because the spread of a proportion scales like $1/\sqrt{N}$, the bands tighten steadily with increasing system size. On average, over one-third of the nodes can be removed while controlling the increase in node degrees. The spread in the degree distribution is also shown.
• Figure 4: Spin elimination for a cubic Max-Cut instance with $N=20$. Nine out of 20 spins are eliminated in the order $0,1,4,6,7,8,12,13,3$ the upper left to the lower right. Antiferromagnetic couplings appear in black; ferromagnetic couplings in red.
• Figure 5: The distribution of the energy values for $N=8$ in the Möbius Ladder instance. In (a), the full Hamiltonian $H_M^{(0)}$ is enumerated; in (b), three spins are eliminated, giving $H_M^{(3)}$. The ground and first-excited states remain essentially the same, while higher-energy states are pruned. Light orange marks the ground and the first excited states.