All-Optical Scalable Spatial Coherent Ising Machine

Networks of optical oscillators simulating coupled Ising spins have been recently proposed as a heuristic platform to solve hard optimization problems. These networks, called coherent Ising machines (CIMs), exploit the fact that the collective nonlinear dynamics of coupled oscillators can drive the system close to the global minimum of the classical Ising Hamiltonian, encoded in the coupling matrix of the network. To date, realizations of large-scale CIMs have been demonstrated using hybrid optical-electronic setups, where optical oscillators simulating different spins are subject to electronic feedback mechanisms emulating their mutual interaction. While the optical evolution ensures an ultrafast computation, the electronic coupling represents a bottleneck that causes the computational time to severely depend on the system size. Here, we propose an all-optical scalable CIM with fully programmable coupling. Our setup consists of an optical parametric amplifier with a spatial light modulator (SLM) within the parametric cavity. The spin variables are encoded in the binary phases of the optical wave front of the signal beam at different spatial points, defined by the pixels of the SLM. We first discuss how different coupling topologies can be achieved by different configurations of the SLM, and then benchmark our setup with a numerical simulation that mimics the dynamics of the proposed machine. In our proposal, both the spin dynamics and the coupling are fully performed in parallel, paving the way towards the realization of size-independent ultrafast optical hardware for large-scale computation purposes.

Figure 1

(a) All-optical spatial CIM. The setup consists of the following: (i) ${\chi }^{(2)}$ NLM pumped by a laser at wavelength ${\lambda }_p$ nonresonant with the cavity mirrors ($M$), which amplifies the signal at ${\lambda }_s=2{\lambda }_p$, (ii) coupling mechanism, (iii) extraction of the signal by a beam splitter (BS) with reflection and transmission coefficients $R_{\mathrm{out}}\mathrm{and}{T}_{\mathrm{out}}=\sqrt{1-R_{\mathrm{out}}^2}$, respectively, and detection ($D$). The coupling is implemented using different configurations of the SLM depending on the implemented graph. (b) Circulant graphs: a first lens (L1) transforms the signal in Fourier space, the SLM multiplies the FT of the field by ${\tilde{Q}}_k$, and a second lens (L2) transforms the modulated field back to real space. (c) General graphs, based on the vector-matrix multiplication scheme. The SLM works in real space, multiplying the incoming field by $Q_{ij}$. The modulated signal is focused along the $y$ axis by a cylindrical lens (CL1), defocused, and rotated by ${90}^{\circ }$ on the $x$-$y$ plane by another cylindrical lens (CL2) with ${45}^{\circ }$ rotated optical axis. The pixels in (b) and (c) depict the OPOs encoded on the $x$-$y$ plane: an amplitude $A_{j,\tau }$ is defined as a single pixel in (b), whereas as a column of $M_y$ pixels in (c).

Figure 2

Histogram of time evolution of the (a) real and (b) imaginary part of the $N=112$ OPO amplitudes (in units of $A_0$, see text), computed using Eq. (3), for short times. As evident, the cavity dynamics enhances the real part of the OPO amplitudes and suppresses their imaginary part.

Figure 3

Ising energy from the OPO phases during the round trips for (ML1),(ML2) Möbius ladder, (K1),(K2) complete graph, (ER1),(ER2) Erdős-Rényi graph, and (BA1),(BA2) Barabási-Albert graph. Upper panels refer to $N=112$, while lower panels to $N=224$. Different lines refer to different initial conditions. Horizontal black dashed lines mark the GS energy found by diagonalizing $\mathbf{J}$ for ML, and the minimal energy found by Metropolis annealing for K, ER, and BA. The insets in the upper panels depict the different connectivities in the four cases (black dots are nodes, and red and green lines depict negative and positive edge weights, respectively), where $N=16$ is used for illustration purposes only.