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

TL;DR

This work introduces spatial QUBO (spQUBO), a convolutional formulation of Ising/QUBO problems that leverages the spatial geometry of variables to efficiently encode dense, distance-based interactions. It proves that any spQUBO can be reduced to a two-dimensional periodic spQUBO, and that such 2D forms can be implemented on a spatial photonic Ising machine (SPIM) without multiplexing, while preserving the convolutional structure. The authors also show that spQUBO naturally benefits from FFT-based computation, enabling efficient evaluation of the Hamiltonian and facilitating large-scale optimization. They demonstrate practical applicability through distance-based problems like placement and clustering, and compare spQUBO’s spatial-volume scaling against other mapping schemes, highlighting superior efficiency for problems with strong spatial structure. The framework lays a foundation for scalable, dense-interaction optimization on SPIMs and broadens the potential for optical computing in combinatorial optimization.

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).

Figures (7)

• Figure 1: A schematic of the architecture of SPIM.
• Figure 2: A schematic of the discussion about conflicting positions in the two-dimensional case. (a) The black square of size ${\boldsymbol{L}}=(L,L)$ represents the configuration domain where the spins are located. The gray squares represent the spin locations with their indices. For a spin pair $(x_i, x_j)$, their relative position is ${\boldsymbol{r}}\equiv{\boldsymbol{d}}_i-{\boldsymbol{d}}_j$. There exists another spin pair $(x_k, x_l)$ such that the relative position ${\boldsymbol{r}}'\equiv{\boldsymbol{d}}_k-{\boldsymbol{d}}_l$ satisfies ${\boldsymbol{r}}'=(L, 0)-{\boldsymbol{r}}$, where they should have the same coefficient as $J_{kl}=J_{ij}$. (b) The extended configuration domain with padding of size ${\boldsymbol{R}}=(R, R)$. The inner black square of size ${\boldsymbol{L}}$ represents the original configuration domain, and the outer square of size ${\boldsymbol{L}}+{\boldsymbol{R}}={\boldsymbol{L'}}$ represents the expanded domain ${{[{\boldsymbol{L'}})}}$. For the same spin pair $(x_i, x_j)$, the relative position is not greater than the padding size: $|{\boldsymbol{r}}|\le {\boldsymbol{R}}$. If there is a spin pair whose relative position ${\boldsymbol{r}}'$ satisfies the same condition as above ${\boldsymbol{r}}'=(L, 0)-{\boldsymbol{r}}$, either of the spins must fall outside the domain ${{[{\boldsymbol{L}})}}$. Therefore, there is no such conflicting spin pair that must have the same coefficient as $J_{ij}$.
• Figure 3: Reduction of three-dimensional spQUBO to two-dimensional. (a) Example of the three-dimensional spQUBO of with $(L_1, L_2, L_3) = (2, 2, 3)$ and $(R_1, R_2, R_3) = (1,1,1)$. The problem has 4 spins located at ${\boldsymbol{d}}_1=(0,0,0)$, ${\boldsymbol{d}}_2=(1,1,0)$, ${\boldsymbol{d}}_3=(1,0,1)$ and ${\boldsymbol{d}}_4=(0,1,2)$. We assume symmetric couplings; $W_{ij}=W_{ji}$ for all $i, j\in \mathcal{N}$. The darker blue area represents the original configuration domain before applying the padding. (b) The two-dimensional spQUBO obtained by the proposed reduction algorithm with dimension groups $\mathcal{D}_1 = \{1, 3\}$ and $\mathcal{D}_2 = \{2\}$. The new spin positions for the spin at ${\boldsymbol{d}}=(d_1, d_2, d_3)$ are computed as $\tilde{{\boldsymbol{d}}}=(d_1 + d_3(L_1+R_1), d_2)=(d_1 + 3 d_3, d_2)$, and the new spatial shape is $((L_1+R_1)(L_3+R_3), L_2+R_2) = (12, 3)$. The values of the new spatial coupling function $\tilde{f}$ are defined as $\tilde{f}(1,1) = \tilde{f}(-1,-1) = f(1,1,0) = W_{12}$, $\tilde{f}(4,0) = \tilde{f}(-4,0) = f(1,0,1) = W_{13}$, $\tilde{f}(6,1) = \tilde{f}(-6,-1) = f(0,1,2) = W_{14}$, $\tilde{f}(3,-1) = \tilde{f}(-3,1) = f(0,-1,1) = W_{23}$, $\tilde{f}(5,0) = \tilde{f}(-5,0) = f(-1,0,2) = W_{24}$, and $\tilde{f}(2,1) = \tilde{f}(-2,-1) = f(-1,1,1) = W_{34}$, where $f$ is the original coupling function, and they are defined periodically with period $(12, 3)$.
• Figure 4: An example placement problem. (a) The coupling function $f$ and (b) its discrete Fourier transform $\mathcal{F}[{f}]$ of the example placement problem. (c) An approximate solution. The color represents the placement cost at each candidate site. The white crosses represent the placed facilities in the solution. The axes for (a) and (b) represent the integer coordinates in the transformed spQUBO, and those for (c) represent the coordinates in the original problem. The distance-based interaction expressed in Eq. (\ref{['eq:placement_interaction_multidim']}) corresponds to the center blue circle in (a) and converted to the concentric pattern in (b). In the solution shown in (c), the placement is limited to the blue region but distributed within it, where the placement cost is low.
• Figure 5: (a,b) Example of clustering problem of dimension $D=2$. The distribution of the points and clusters (a) at generation and (b) in the obtained approximate solution. The clusters are represented by the shape and color of the points. The spins represented by black triangles denote the spins with invalid output where the exact-one constraint is violated. The axes represent the integer coordinates of the problem. (c-e) The transformed two-dimensional spQUBO for an example of smaller sized problem. The axes represent the integer coordinates of the transformed spQUBO. (c) The spin arrangement, where the yellow dots indicate the locations where spins are mapped. (d) The coupling function $f$ and (e) its discrete Fourier transform $\mathcal{F}[{f}]$. The axes for (a) and (b) represent the coordinates in the original problem, and those for (c-e) represent the integer coordinates in the transformed spQUBO. In the panel (c), the points in the original clustering problem are copied $K$ times corresponding to the possibility of the cluster assignments. The distance-based cost expressed in Eq. (\ref{['eq:clustering_HAg_multidim']}) corresponds to the yellow circle at the center of (d) and the horizontal pattern in (e). On the other hand, the assignment constraints expressed in Eq. (\ref{['eq:clustering_HB_multidim']}) corresponds to horizontally arranged blue dots in (d) and vertical lines in (e).

Theorems & Definitions (20)

• Definition 1: spatial QUBO
• Definition 2: periodic spatial QUBO
• Definition 3: locality
• Theorem 1: Transformation of two-dimensional spQUBOs
• Theorem 2: Transformation of high-dimensional spQUBOs
• Theorem 3
• Theorem 4
• Theorem S1
• Proposition S2
• Proposition S3: Theorem \ref{['thm:higherorder']} in one-dimensional case