Spatial QUBO: Convolutional Formulation of Large-Scale Binary Optimization with Dense Interactions

2025-06-30

AI Summary

Spatial photonic Ising machines (SPIMs) face fundamental limitations in representing dense, high-rank binary interaction structures—especially for large-scale combinatorial optimization problems with spatially correlated couplings—when operating without optical multiplexing. Method: We propose spQUBO, a novel Quadratic Unconstrained Binary Optimization (QUBO) modeling framework that intrinsically embeds spatial convolutional structure, enabling direct mapping onto SPIM hardware. Contribution/Results: We formally prove that any 2D spQUBO instance admits efficient, multiplexing-free implementation on SPIMs—overcoming the longstanding reliance of photonic solvers on low-rank coupling matrices. By synergistically leveraging fast Fourier transform (FFT) acceleration and SPIM’s native parallelism, spQUBO significantly enhances modeling fidelity and scalability for distance-dependent problems (e.g., VLSI placement, spatial clustering). It extends SPIM applicability from sparse-coupling regimes to high-rank spatially structured problems, establishing a new paradigm for photonic computing in complex combinatorial optimization.

Technology Category

Constraint Satisfaction and Optimization: Distributed CSP/OptimizationSearch and Optimization: Mixed Discrete/Continuous SearchPlanning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingEconomics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

The spatial photonic Ising machine (SPIM) is a promising optical hardware solver for large-scale combinatorial optimization problems with dense interactions. As the SPIM can represent Ising problems with rank-one coupling matrices, multiplexed versions have been proposed to enhance the applicability to higher-rank interactions. However, the multiplexing cost reduces the implementation efficiency, and even without multiplexing, the SPIM is known to represent coupling matrices beyond rank-one. In this paper, to clarify the intrinsic representation power of the original SPIM, we propose spatial QUBO (spQUBO), a formulation of Ising problems with spatially convolutional structures. We prove that any spQUBO reduces to a two-dimensional spQUBO, with the convolutional structure preserved, and that any two-dimensional spQUBO can be efficiently implemented on the SPIM without multiplexing. We further demonstrate its practical applicability to distance-based combinatorial optimization, such as placement problems and clustering problems. These results advance our understanding of the class of optimization problems where SPIMs exhibit superior efficiency and scalability. Furthermore, spQUBO's efficiency is not limited to the SPIM architecture; we show that its convolutional structure allows efficient computation using Fast Fourier Transforms (FFT).

Problem

• Clarify SPIM's intrinsic representation power for Ising problems
• Propose spQUBO for efficient SPIM implementation without multiplexing
• Demonstrate spQUBO's applicability to distance-based optimization problems

Innovation

• Spatial QUBO for convolutional Ising problems
• SPIM implementation without multiplexing
• Efficient computation using Fast Fourier Transforms