Large-Scale Photonic Ising Machine by Spatial Light Modulation

Quantum and classical physics can be used for mathematical computations that are hard to tackle by conventional electronics. Very recently, optical Ising machines have been demonstrated for computing the minima of spin Hamiltonians, paving the way to new ultrafast hardware for machine learning. However, the proposed systems are either tricky to scale or involve a limited number of spins. We design and experimentally demonstrate a large-scale optical Ising machine based on a simple setup with a spatial light modulator. By encoding the spin variables in a binary phase modulation of the field, we show that light propagation can be tailored to minimize an Ising Hamiltonian with spin couplings set by input amplitude modulation and a feedback scheme. We realize configurations with thousands of spins that settle in the ground state in a low-temperature ferromagneticlike phase with all-to-all and tunable pairwise interactions. Our results open the route to classical and quantum photonic Ising machines that exploit light spatial degrees of freedom for parallel processing of a vast number of spins with programmable couplings.

Figure 1

Ising machine by spatial light modulation. (a) The wave phase in different spatial points gives the spins evolving through optical propagation. (b) An amplitude-modulated laser beam is phase modulated by a reflective SLM and detected by a CCD camera in the far field. (c) A discrete phase mask with binary values ${\varphi }_j=0,\pi$ in the Fourier plane mimics Ising spins ${\sigma }_j=\pm 1$. Inset is an example of the detected intensity when the binary hologram is tailored to generate a squared intensity target $I_T$.

Figure 2

Optically solving the Ising Hamiltonian with all-to-all spin interactions. (a) Unweighted Möbius-Ladder graph with fully connected vertices (results refer to $N=4\times {10}^4$) along with the target intensity $I_T$. (b) Measured evolution of $H$ ($J_{jh}=1$) and $|m|$ for different initial random spin matrices. (c),(d) Observed probability distribution for the (c) energy and (d) magnetization of spin configurations satisfying the interaction constraint $I_T$; red and magenta lines indicate $H$ and $|m|$ of the identified ground-state solutions, respectively. (e) Set of ground-state spin configurations: small-size ferromagnetic clusters with opposite magnetization are visible.

Figure 3

Scaling properties of the ferromagnetic ground state. (a) Observed magnetization varying the spin number. The inset shows the scaling of the machine performance. (b) Spatial spin autocorrelation functions (distance in pixel units) for different $N$. (c) Autocorrelation length as a function of the system size $L$ (dots) and linear fitting behavior (line).

Figure 4

Programming the spin interaction by amplitude modulation. (a) Examples of coupling configurations ($N={10}^4$, top panels) made of random blocks in which the interaction assumes two positive values $({\xi }_j=0,{\xi }_0>0)$. The corresponding spin ground state observed in the red box region is shown in the bottom panel. (b) Measured probability distribution of the correlation $C$ between the ground state and the couplings for the Mattis models. The inset shows a corresponding Möbius-Ladder graph with connected and unconnected nodes.