Nanophotonic Phased Array XY Hamiltonian Solver

Abstract

Solving large-scale computationally hard optimization problems using existing computers has hit a bottleneck. A promising alternative approach uses physics-based phenomena to naturally solve optimization problems wherein the physical phenomena evolves to its minimum energy. In this regard, photonics devices have shown promise as alternative optimization architectures, benefiting from high-speed, high-bandwidth and parallelism in the optical domain. Among photonic devices, programmable spatial light modulators (SLMs) have shown promise in solving large scale Ising model problems to which many computationally hard problems can be mapped. Despite much progress, existing SLMs for solving the Ising model and similar problems suffer from slow update rates and physical bulkiness. Here, we show that using a compact silicon photonic integrated circuit optical phased array (PIC-OPA) we can simulate an XY Hamiltonian, a generalized form of Ising Hamiltonian, where spins can vary continuously. In this nanophotonic XY Hamiltonian solver, the spins are implemented using analog phase shifters in the optical phased array. The far field intensity pattern of the PIC-OPA represents an all-to-all coupled XY Hamiltonian energy and can be optimized with the tunable phase-shifters allowing us to solve an all-to-all coupled XY model. Our results show the utility of PIC-OPAs as compact, low power, and high-speed solvers for nondeterministic polynomial (NP)-hard problems. The scalability of the silicon PIC-OPA and its compatibility with monolithic integration with CMOS electronics further promises the realization of a powerful hybrid photonic/electronic non-Von Neumann compute engine.

Figures (8)

• Figure 1: General view of the photonic hardware used for the XY Hamiltonian solver, and the algorithmic approach for the solver. (a) A diagram showing the XY model solver implemented with a PIC OPA. The phases of the optical phased array are encoded in the light emitted from each emitter pixel (A, A'), and controlled via voltage applied to the resonator phase shifters. In the focal plane (B, B'), the far field pattern is imaged and XY Hamiltonian energy is calculated, then voltages applied to each resonator phase shifter are modified over each optimization iteration to minimize or maximize energy through a feedback process. Phases are extracted using a phase-retrieval algorithm (C, C'). (b) A schematic showing the process of solving the XY model with a PIC OPA. Voltage is applied to the device to tune phases on the antennas (A, A'), which in the far field (B, B') forms a pattern from which the XY energy is calculated and used to feed back to tune the voltages. After the optimization process completes, the phases are extracted from the lowest energy far field pattern (C, C').
• Figure 2: Detailed structure of the PIC-OPA device used for the XY Hamiltonian solver. (a) A 3D rendering of a portion of the 8x8 PIC-OPA chip, with a yellow arrow showing where the laser light enters before branching to reach each of the phased array elements and antennas. The inset shows a scanning electron microscope image of one phased array element consisting an overcoupled ring resonator phase shifter and an antenna. The electronic control of the phase shifter is not shown in this image. (b) A microscope image of the 8 by 8 optical phased array device. An inset shows a camera image of the emission from each antenna, taken in the near field.
• Figure 3: A schematic showing the phase recovery from the experimental data using the Gerchberg-Saxton algorithm. First, (a) an array of random phases are generated. Then, (b) a near field constraint (src) according to the array geometry is multiplied by the random phases. Next, (c) the Fourier transform of the image is taken, and following that, (d) the far field constraint (experimental data or trg) is imposed, and (e) an inverse Fourier transform performed. The images in panels (b), (c), (d), and (e) show the results of the first iteration in this example. These steps are repeated over many iterations until the image after step C resembles the image after step D. In the example shown, (f) shows the reconstructed far field C after 10,000 iterations.
• Figure 4: The energy minimization results for the OPA XY model solver for one optimization run. (a) XY model energy and summed image pixel intensity for each iteration of an optimization for an energy minimum. (b) The simulated energy probability density function for the regular 8x8 OPA, with lines overlaid showing the experimental energies corresponding to the energies marked in (a) shown in different colors. (c) Experimental data showing the far field images corresponding to the energies marked in matching colors in (a) and (b).
• Figure 5: Dependence of XY model energy (calculated from retrieved phases) on the random seed in the Gerchberg-Saxton algorithm. (a) Distributions of solved XY model energies for differently seeded phase retrievals, with selected retrieved phase sets shown as insets marked by Roman numerals. For 100 random seeds, a 10,000 iteration Gerchberg-Saxton phase retrieval is performed using far field data and the histogram of the resulting energies are plotted. (b) Three phase configurations associated with different random seeds. Their corresponding energies are plotted as vertical lines (a) and labeled.