An optical Ising spin glass simulator with tuneable short range couplings

TL;DR

Problem: programmable coupling topologies in optical Ising simulators to address NP-hard ground-state problems. Approach: utilize free-space optics and a thin diffuser to estimate the transmission matrix and realize couplings via $J_{ij} = - \sum_m \Re{\overline{t_{im}} t_{jm}}$; tune the spin interaction length by changing the diffuser–camera distance, enabling regimes from localized to all-to-all interactions. Key findings: experimental and numerical results show controllable spin clustering and, as overlap between regions grows, replica-to-replica fluctuations increase, displaying signatures akin to replica symmetry breaking (RSB). Significance: demonstrates a scalable, algorithm-agnostic optical platform for programmable Ising machines and provides a route to study RSB and cluster interactions, with potential extensions to Hopfield networks.

Abstract

Non-deterministic polynomial-time (NP) problems are ubiquitous in almost every field of study. Recently, all-optical approaches have been explored for solving classic NP problems based on the spin-glass Ising Hamiltonian. However, obtaining programmable spin-couplings in large-scale optical Ising simulators, on the other hand, remains challenging. Here, we demonstrate control of the interaction length between user-defined parts of a fully-connected Ising system. This is achieved by exploiting the knowledge of the transmission matrix of a random medium and by using diffusers of various thickness. Finally, we exploit our spin-coupling control to observe replica-to-replica fluctuations and its analogy to standard replica symmetry breaking.

Figures (4)

• Figure 1: Optical spin-glass simulator.A laser is collimated on a spatial light modulator (SLM) and passes through a thin diffuser to then reach the imaging plane. The surface of the SLM is directly imaged (via a 4f-system) on the surface of the diffuser. The light is focused on two output modes. The inset shows the two rows of the transmission matrix $(t_{im})_{im}$ summed and then reshaped to SLM dimensions. The light reaching the pink (resp. yellow) focus can only come from certain pixels within the pink (resp. yellow) cone on the SLM plane ($\sigma_i$), of which the extent is given by the distance $d$ and/or the diffuser's matrix $t_{im}$, corresponding to solving an Ising Hamiltonian with local couplings.
• Figure 2: Tuning the spin correlation length by optical propagation. a) Schematic representation of the setup in two different distance configurations: $d_1$ in red and $d_2$ in orange, where the colored cones represent where the light reaching one CCD pixel comes from on the SLM. b) Simulation (first col) and experimental (second col) spin maps at three different imaging plane CCD-DIFF distances. The further away the plane, the wider the correlation. c) The amplitudes of the experimental transmission matrices at these same three distances.
• Figure 3: Controlling the interaction between spin clusters.a) Simulation (first columns) and experimental (second columns) spin ground states. b) TM amplitudes obtained summing all rows of the measured TM. Two isolated regions of magnetized spins (when optimizing over to two foci at the CCD plane) as displayed on the SLM (first row). As the distance between CCD and diffuser increases the extent of the magnetized regions also increases, the regions start to overlap and therefore interact (second row).
• Figure 4: Controlling frustration and replica symmetry breaking. a) Two regions of fixed size are increasingly getting closer on the SLM plane. Each point corresponds to ten replicas for one given distance. The graph shows the maximum of the probability distribution of the correlation between replicas $q_{max}$ as a function of the distance between the regions (in px on the CCD image). Replica-to-replica fluctuations vanish when the correlation is close to 1. b) and c) Histograms of the ground-state correlation for different replicas at maximum and minimum distance respectively.