Wavelength-division multiplexing optical Ising simulator enabling fully programmable spin couplings and external magnetic fields

TL;DR

The work addresses the limitation of spatial photonic Ising machines to fully connected couplings by introducing a wavelength-division multiplexing SPIM that enables fully programmable spin couplings and external magnetic fields. It combines a Cholesky-like, Mattis-type decomposition of the Ising Hamiltonian with a gauge transformation that maps general interactions onto a single phase-only SLM, so the optical center intensity $\tilde{I}$ implements the general Ising energy $H = -J\tilde{I} + H_0$. Experimentally, the authors demonstrate 80-spin systems for $\pm J$ and SK models and a 64-spin $J_1$-$J_2$ model, observing phase transitions among spin-glass, ferromagnetic, paramagnetic, and stripe-antiferromagnetic phases, in agreement with mean-field predictions and susceptibility measurements. The results show high programmability of couplings and fields, enabling large-scale, on-demand SPIM solutions to general combinatorial optimization problems with optical speed and efficiency.

Abstract

Recently, spatial photonic Ising machines (SPIMs) have demonstrated the abilities to compute the Ising Hamiltonian of large-scale spin systems, with the advantages of ultrafast speed and high power efficiency. However, such optical computations have been limited to specific Ising models with fully connected couplings. Here we develop a wavelength-division multiplexing SPIM to enable programmable spin couplings and external magnetic fields as well for general Ising models. We experimentally demonstrate such a wavelength-division multiplexing SPIM with a single spatial light modulator, where the gauge transformation is implemented to eliminate the impact of pixel alignment. To show the programmable capability of general spin coupling interactions, we explore three spin systems: $\pm J$ models, Sherrington-Kirkpatrick models, and only locally connected ${{J}_{1}}\texttt{-}{{J}_{2}}$ models and observe the phase transitions among the spin-glass, the ferromagnetic, the paramagnetic and the stripe-antiferromagnetic phases. These results show that the wavelength-division multiplexing approach has great programmable flexibility of spin couplings and external magnetic fields, which provides the opportunities to solve general combinatorial optimization problems with large-scale and on-demand SPIM.

Figures (6)

• Figure 1: (a) Schematic of the wavelength-division multiplexing SPIM. In this setup, light with different wavelengths is diffracted and focus on a phase-only spatial light modulator (SLM) along the $x$-direction, while the pixels in the $y$-direction are coherently illuminated by incident light of the same wavelength. The spins are encoded with phase modulation on the SLM using Eq. (\ref{['eq:2']}). SC: super-continuum laser; CL: cylindrical lens; FL: Fourier lens; CCD: charge coupled device camera. (b) The Ising model with general interactions and external magnetic fields is transformed into $N$ numbers of Mattis models via gauge transformation, and $H=\sum_{k=1}^{N}{{{H}_{k}}}+{{H}_{0}}$, where ${{H}_{k}}=-J\sum _{i,j=k}^{N+1}{\sigma_{i}^{\prime k}\sigma_{j}^{\prime k}}$ and $J$ and ${{H}_{0}}$ are constants. Here $\sigma_{j}^{\prime k}$ represents the $z$ component of spin $\sigma_{j}$ rotated by an angle ${\alpha}_{j}^{k}$ with respect to the $z$ axis.
• Figure 2: Probability distribution of the Parisi parameter ${{q}}$ as a function of $T$, for $N=80$ spins. (a) Schematic representation for the $\pm J$ model. (b) and (c) present the experimental results for $p=0.7$ and $0.55$, respectively. (d) Schematic for the SK model. (e) and (f) display the results for ${{J}_{0}}=40$ and $8$, respectively, with $\Delta J=\sqrt{80}$ fixed. (g) Schematic for the SK model with a uniform external magnetic field. (h) and (i) present the results for $h=0.2$ and $2$, respectively, with ${{J}_{0}}$ and $\Delta J$ being the same as (f).
• Figure 3: Experimental results for a locally connected ${{J}_{1}}\texttt{-}{{J}_{2}}$ model with cyclic boundary condition. (a) A schematic of the ${{J}_{1}}\texttt{-}{{J}_{2}}$ model, where the thick solid line indicates the nearest neighbor ferromagnetic interaction (${{J}_{1}}>0$) and the blue double line represents the next-nearest-neighbor antiferromagnetic interaction (${{J}_{2}}<0$). Blue and yellow squares denote spins ${\sigma}_{i}=-1$ and $1$, respectively, and the experiment was conducted on an $8\times8$ lattice. (b) Results for the order parameter $(m_x,m_y)$ as a function of $T$, for the parameters ${{J}_{1}}=0.2$ and ${{J}_{2}}=-1$. (c) A spin configuration sampled at $T=70J_1$ representing a PM state. (d-g) Four spin configurations sampled at $T=14.39J_1$ which are adjacent to four striped states, respectively.
• Figure 4: (a) and (b) Probability density distribution of ${{q}}$ for the ${{J}_{1}}\texttt{-}{{J}_{2}}$ models with the fixed nearest neighbor interactions ${{J}_{1}}=1$, while the next-nearest-neighbor interactions are Gaussian distributed with the mean value of ${{J}_{0}}=-1$ and the standard deviation of $\Delta J=0.2$ and $2$, respectively. (c) Experimental measured susceptibility for $\Delta J=0.2$ and $\Delta J=2$.
• Figure S1: (a) and (b) are the checkerboard functions ${{M}_{1}}$ and ${{M}_{2}}$. Here each spin is encoded by ${{N}_{y}} \times {{N}_{x}}$ pixels as shown in the black dashed box. For the $k$-th Mattis-type model, the center of the $j$-th spin is located at $(k{{N}_{x}}W, j{{N}_{y}}W)$, where $W$ represents the pixel width of the SLM.