Fully Programmable Spatial Photonic Ising Machine by Focal Plane Division

Abstract

Ising machines are an emerging class of hardware that promises ultrafast and energy-efficient solutions to NP-hard combinatorial optimization problems. Spatial photonic Ising machines (SPIMs) exploit optical computing in free space to accelerate the computation, showcasing parallelism, scalability, and low power consumption. However, current SPIMs can implement only a restricted class of problems. This partial programmability is a critical limitation that hampers their benchmark. Achieving full programmability of the device while preserving its scalability is an open challenge. Here, we report a fully programmable SPIM achieved through a novel operation method based on the division of the focal plane. In our scheme, a general Ising problem is decomposed into a set of Mattis Hamiltonians, whose energies are simultaneously computed optically by measuring the intensity on different regions of the camera sensor. Exploiting this concept, we experimentally demonstrate the computation with high success probability of ground-state solutions of up to 32-spin Ising models on unweighted maximum cut graphs with and without ferromagnetic bias. Simulations of the hardware prove a favorable scaling of the accuracy with the number of spins. Our fully programmable SPIM enables the implementation of many quadratic unconstrained binary optimization problems, further establishing SPIMs as a leading paradigm in non von Neumann hardware.

Figures (4)

• Figure 1: SPIM by focal plane division. (a) Schematic of the SLM screen division. The same spin configuration $\bm{\sigma}$ is encoded in each row. We apply the Gauge encoding to implement each Mattis Hamiltonian $\mathcal{H}_k$ on each row. (b) Intensity detected in the focal plane. By using blazed gratings, we perform a division of the focal plane to separate the signals of different $\mathcal{H}_k$. (c) Sketch of the experimental setup. The spin energy is evaluated by the measured intensities $[I_1,...,I_K]$. The ground-state search operates by updating via digital feedback $\bm{\sigma}$ on the SLM according to the measured energy and a SA algorithm.
• Figure 2: Ground-state accuracy of the SPIM by FPD with $N=16$ for (a) Möbius ladder, (b) Max-Cut and (c) Max-Cut with ferromagnetic bias. The energy histograms show a ground state probability of $95\%$, $50\%$ and $55\%$, respectively. Insets shows the optical energy $F$ and the value of the Ising Hamiltonian during an experimental run.
• Figure 3: Accuracy of the SPIM by FPD for $N=32$ on fully connected biased Max-Cut graphs. Energy distribution of the experimental and SA solutions ($P_{\rm{suc}}=10\%$).
• Figure 4: Scaling analysis of the SPIM accuracy. Energy histograms on $100$-spin (a) Möbius ladder, (b) Max-Cut and (c) biased Max-Cut graphs for the simulated SPIM and SA. The probability of finding low-energy solutions within $a\%$ of the energy minimum is shown by increasing $N$. Shaded areas represent the interquartile range over $100$ different random graphs.