Erdos-Turan photonic Ising machines with record-high coupling resolution

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Abstract
Ising machines have emerged as promising platforms for efficiently tackling a wide range of combinatorial optimization problems relevant to resource allocation, statistical inference and deep learning, yet their practical utility is fundamentally constrained by the coarse resolution of spin-spin couplings (Jij). Current implementations, relying on direct modulation of physical parameters, achieve at most 256 discrete coupling levels, which severely hinder the faithfully modeling of arbitrary real-valued interactions in realistic applications. Here we present a novel photonic Ising machine that encodes spins in random lattices while programming couplings in the momentum space of light. By introducing the Sidon set-a mathematical structure ensuring pairwise difference uniqueness - and employing the Erdos-Turan bound, we establish an optical framework in which each spin pair can be assigned a unique Jij. This approach decouples the resolution limit from hardware modulation to the spatial precision in the momentum space of light. Experimentally, we demonstrate a record-high coupling resolution of 7,038 on a simple photonic platform, surpassing previous Ising machines. Our results highlight the power of uniting discrete mathematics with momentum-space photonics, paving the way toward scalable Ising machines capable of faithfully modeling real-world optimization problems.
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