Efficient Computation Using Spatial-Photonic Ising Machines: Utilizing Low-Rank and Circulant Matrix Constraints

TL;DR

The paper investigates spatial-photonic Ising machines (SPIMs) for solving Ising problems with low-rank or circulant coupling matrices, addressing NP-hard optimization. It develops and assesses decomposition strategies, notably SVD-based linear combinations of rank-1 Mattis-type matrices and the correlation function method, to broaden representable problem classes. Key contributions include demonstrating practical low-rank approximations in portfolio optimization, introducing the constrained number partitioning (CNP) problem as a hardware-friendly benchmark, and analyzing translation-invariant and circulant graphs on SPIM hardware. The results show that high-quality approximate solutions can be achieved under fixed hardware precision when the coupling rank remains modest, highlighting SPIMs’ potential for energy-efficient optimization in finance and beyond, with the energy function $H = -\sum_{i,j} J_{ij} s_i s_j + \sum_i h_i s_i$ serving as the central objective for photonic realization.

Abstract

We explore the potential of spatial-photonic Ising machines (SPIMs) to address computationally intensive Ising problems that employ low-rank and circulant coupling matrices. Our results indicate that the performance of SPIMs is critically affected by the rank and precision of the coupling matrices. By developing and assessing advanced decomposition techniques, we expand the range of problems SPIMs can solve, overcoming the limitations of traditional Mattis-type matrices. Our approach accommodates a diverse array of coupling matrices, including those with inherently low ranks, applicable to complex NP-complete problems. We explore the practical benefits of low-rank approximation in optimization tasks, particularly in financial optimization, to demonstrate the real-world applications of SPIMs. Finally, we evaluate the computational limitations imposed by SPIM hardware precision and suggest strategies to optimize the performance of these systems within these constraints.

Figures (7)

• Figure 1: (a) The energy of approximate solutions is plotted against the rank of the approximate coupling matrix. The exact interaction matrix represents a random, unweighted, undirected graph with 1000 vertices, all with anti-ferromagnetic couplings, where each edge has an equal probability of having a value of 0 or -1. Each set of parameters uses 100 different random sequences of spin flips to produce the scatter of final energies. The coupling strengths are rounded to the nearest $2^{-8}$ in the approximation. (b) The energy of approximate solutions is plotted against the precision of the approximate coupling matrix. A full-rank ($R=1000$) matrix and two low-rank approximations with ranks $R=256$ and $R=64$ are used for each precision level. (c) and (d) follow the same structure as (a) and (b), but the exact interaction matrix represents a random 3-regular graph, again with all non-zero couplings being anti-ferromagnetic.
• Figure 2: (a) Frequency histogram of eigenvalues obtained from the covariance matrix of S&P 500 stock data. There are only a few dominating eigenvalues, and most eigenvalues are orders of magnitude smaller than the dominant ones. (b) Equal-weighted cardinality-constrained portfolios constructed from the full rank covariance matrix $\mathbf{S}$ (blue), $K = 20$ low rank matrix $\mathbf{S}^{'}$ (orange), and $K = 5$ low rank matrix (green). Here, $\lambda = 0.5$, $\eta = 1$, and $q = 20$. The portfolios were built by minimizing Eq. (\ref{['Portfolio Equation']}) using commercial solver Gurobi.
• Figure 3: (a) The probability of the existence of a perfect solution is plotted against various problem sizes $N$ at fixed values of bias $S$. (b) The colour map shows the probability of the existence of a perfect solution in a random CNP problem instance with $N$ integers and various bias values, and the integers are chosen uniformly and randomly in the range $[1, 2^{12}]$. The probability at each point in the phase space is calculated over 200 random instances. Three phases are identified in the figure, separated by the orange and red dash lines. Region 1, 2, and 3 correspond to the "ordered", "hard", and "perfect" phases proposed in borgs_phase_2003. Region 1 in the graph is not drawn because it is likely to be trivially easy to find the optimum partition in this region for an average problem instance, so it is not meaningful to investigate the probability of a perfect solution's existence in this region.
• Figure 4: Schematic search tree of the complete differencing algorithm.
• Figure 5: (a) The number of configurations the modified complete differencing algorithm must search to determine if a perfect solution exists in a CNP problem instance as a function of size $N$ and bias $S$. (b) Degeneracy of the ground state as a function of size $N$ for CNP problems with different bias ratio parameters $b$. Integers of the CNP instances were drawn uniformly at random from the range $[1, 2^{12}]$.