Encoding arbitrary Ising Hamiltonians on Spatial Photonic Ising Machines

TL;DR

This work introduces and experimentally validate a SPIM instance that enables direct control over the full interaction matrix, allowing the encoding of Ising Hamiltonians with arbitrary couplings and connectivity and paves the way to encoding the full range of NP problems known to be equivalent to Ising models on SPIM devices.

Abstract

Photonic Ising Machines constitute an emergent new paradigm of computation, geared towards tackling combinatorial optimization problems that can be reduced to the problem of finding the ground state of an Ising model. Spatial Photonic Ising Machines have proven to be advantageous for simulating fully connected large-scale spin systems. However, fine control of a general interaction matrix $J$ has so far only been accomplished through eigenvalue decomposition methods that either limit the scalability or increase the execution time of the optimization process. We introduce and experimentally validate a SPIM instance that enables direct control over the full interaction matrix, enabling the encoding of Ising Hamiltonians with arbitrary couplings and connectivity. We demonstrate the conformity of the experimentally measured Ising energy with the theoretically expected values and then proceed to solve both the unweighted and weighted graph partitioning problems, showcasing a systematic convergence to an optimal solution via simulated annealing. Our approach greatly expands the applicability of SPIMs for real-world applications without sacrificing any of the inherent advantages of the system, and paves the way to encoding the full range of NP problems that are known to be equivalent to Ising models, on SPIM devices.

Figures (4)

• Figure 1: (a) Schematic of the experimental setup. (b) Images captured by the camera corresponding to the energy computation according to \ref{['eq:Hamiltonian_pm']} for a given spin configuration. (c,d) The experimental energies versus the theoretical ones for a sparse ferromagnetic system (c), and a sparse spin glass system (d).
• Figure 2: Simulated (a,b,c) and experimental (d,e,f) results for a graph partitioning Hamiltonian, showing the total ($H$) and individual ($H_a$ and $H_b$) energies (a,d), the magnetization (b,e), and the cost (c,f) as a function of the optimization iterations. Inset in (c) depicts the different annealing schedules that were tried (grey) and the one that was used (red). Inset in (f) shows a graphical representation of a toy example of the graph partitioning problem. The dashed lines in (b,c,e,f) represent the solution obtained by the METIS package.
• Figure 3: Multiple runs obtained experimentally for the Unweighted Graph Partitioning Problem (GPP) (a,b) and the Weighted Graph Partitioning Problem (WGPP) (c,d). Both instances have $N=100$ and $p=0.05$. Each column shows the total energy (a,c) and the cost (b,d) for 10 (a,b) and 5 (c,d) different initial spin configurations, showing that the SPIM systematically finds solutions with comparable cost to that obtained by METIS. The insets show all individual runs.
• Figure 4: Optimal cost $C^*$ of the GPP for varying edge probability $p$ from the SPIM experiment, METIS, SPIM simulation and theory. The SPIM experiment points have been averaged over 20 random problem instances. The METIS and SPIM simulation points have been averaged over 50 problem instances. Inset: the improvement due to optimization $\Delta(p)$.