An optical neural chip for implementing complex-valued neural network

Artificial neural networks rely heavily on multiply–accumulate (MAC) operations, creating a computational burden for electronic hardware such as CPUs, GPUs, FPGAs and ASICs. While most networks use real-valued arithmetic, theory suggests complex-valued arithmetic can provide richer representations, faster convergence, improved generalization, and noise-robust memory. Electronic platforms suffer slowdowns when emulating complex numbers as pairs of reals, increasing MAC counts. Optical computing can naturally perform complex arithmetic via interference, with advantages of low power, high bandwidth, parallelism, and large storage. Prior photonic neural demonstrations largely implemented real-valued algorithms, discarding the optical phase and thus underutilizing photonics’ complex degrees of freedom. This work addresses these limitations by proposing and realizing a silicon photonic optical neural chip (ONC) that natively implements complex-valued neural network operations. The study aims to demonstrate feasibility and advantage of complex-valued photonic networks across multiple tasks, comparing against equivalent real-valued implementations on the same platform.

The authors review advances in optical and neurophotonic computing: integrated photonic neural networks, optical reservoir computing, diffractive deep networks, and reprogrammable nanophotonic processors. Prior on-chip neural networks implemented real-valued algorithms despite operating with interference. Photonic accelerators using photonic multiplication also targeted real arithmetic due to photodetection before accumulation. Analog electronics have explored complex-valued networks and reservoir computing with performance benefits. However, general-purpose complex-valued neural networks leveraging optical platforms’ phase and magnitude have been underexplored because existing optical implementations adopt frameworks designed for real-valued digital computation, typically using only intensity and discarding phase. The present work exploits coherent optical detection to access both magnitude and phase, enabling truly complex-valued neural network operations on-chip.

Design and fabrication: The ONC integrates input preparation, complex weight multiplication, and detection on a single silicon photonic chip using a network of Mach–Zehnder interferometers (MZIs) arranged as a multipart interferometer. Each MZI comprises two 50:50 multimode interference (MMI) beam splitters and thermally tuned phase shifters (PSs) implemented with TiN heaters. A coherent 1550 nm laser provides inputs. Input light is split and modulated (magnitude and, when required, phase), with a portion reserved as a reference for coherent detection. The chip ensures stable relative phase and polarization across paths. The demonstrated ONC has 8 modes and 56 PSs; heaters are calibrated with average R² ≈ 0.99. Complex weight matrices are decomposed into MZI phase settings following a unitary/phase decomposition; details provided in supplementary material. Detection: Both intensity and coherent detection are supported on-chip by configuring specific MZIs. Coherent detection mixes signal with the on-chip reference; balanced detection yields currents proportional to cos(φe) and sin(φe) (with a π/2 phase shift), enabling recovery of the signal phase φe and mitigating component noise. Detection choice depends on the activation function: magnitude-only activations use intensity detection; complex activations use coherent detection. Neuron model: A complex-valued neuron computes y = f(Σ w_i x_i + b), with complex weights and bias; inputs may be real or complex. Bias is implemented as a constant input with a trainable complex weight. Weights are realized by configuring on-chip MZIs. For logic tasks, an identity activation is used and weights are updated iteratively using a Hebbian-like rule derived from expected vs. actual outputs. Tasks and benchmarking: Four benchmarks are implemented/simulated: (i) logic gate realization (including XOR) using a single complex-valued neuron with coherent detection; (ii) Iris dataset classification with a single complex layer fed by four real-valued features plus bias; training on 75% of data, testing on 25%; (iii) nonlinear datasets (Circle and Spiral) with two inputs scanned across a grid on-chip to visualize decision boundaries; comparison with a real-valued model; (iv) MNIST handwriting recognition with a complex-valued multilayer perceptron: a 784→4 input compression layer (off-chip), a 4×4 hidden complex layer implemented on ONC, and a 4→10 output layer (off-chip). Training uses TensorFlow with RMSProp (lr=0.01 or 0.005, 12,000 iterations, batch size 100) and complex activation (ModelReLU). Ablation studies vary encoding (complex vs. real magnitude-only) and detection (coherent vs. intensity) to assess contributions of complex-valued weights vs. I/O modalities. Experimental setup and characterization: A 1550 nm tunable laser (≈12 dBm), polarization control, thermal stabilization and temperature control are used to ensure stable interferometry and mitigate thermal cross-talk. Photodetectors interface via TIAs and an ADC. Phase shifter drivers have 16-bit precision. I–V and optical responses are calibrated; optical interference visibility averages ~98.5%. Coherent detection is implemented entirely on-chip to avoid external phase fluctuations and improve stability.

- The ONC natively implements complex-valued arithmetic by encoding and processing both magnitude and phase, overcoming limitations of real-valued optical networks. - Logic gates: A single complex-valued neuron realizes fundamental gates including XOR (not possible with a single real-valued neuron). Training trajectories converge via continuous magnitude and phase modulation. - Iris classification: A single complex-valued layer trained on 75% of the 150-instance dataset and tested on 25% successfully separates the three species. Validation and blind-test plots show good generalization; misclassifications are few (details in figures/supplementary). - Nonlinear datasets (Circle and Spiral): • Simulation: Complex model achieves 100% accuracy on both Circle and Spiral; the real-valued model yields ≈55% (Circle) and ≈89% (Spiral). Complex decision boundaries are nonlinear curves matching dataset geometry, while real-valued boundaries are limited (largely linear/planar). • On-chip experiment: Inputs x1 and x2 scanned over a 21×21 grid (−1 to 1, step 0.1; 441 points). Chip accuracies: 98% (Circle) and 95% (Spiral). Experimental decision boundaries closely match theoretical boundaries, with deviations attributable to input resolution and hardware noise. - Handwriting recognition (MNIST, MLP with 4 hidden complex neurons implemented on-chip): • Complex model: training accuracy ≈93.1%, chip testing accuracy ≈90.5%; faster convergence than real-valued model. • Real-valued counterpart: training ≈84.3%, testing ≈82.0%. • Ablations (hidden N=4): reported testing accuracies show benefits from complex processing and encoding (e.g., “completely complex” testing up to ≈96.0%; real-encoding with complex weights/detection ≈87.0%; real-detection with complex encoding/weights ≈86.5%; both real I/O with complex weights ≈86.5%; completely real model ≈91.0% training/≈82.0% testing in narrative). Across configurations, complex weights consistently outperform purely real-valued networks, and complex input encoding contributes notably to performance gains. • Capacity comparison: A 4×4 complex model (effective capacity 32 real parameters) outperforms an 8×8 real model (capacity 64) in accuracy (e.g., complex 93.1% vs. real 92.3% reported), indicating higher parameter efficiency. Complex models can achieve similar or better performance with smaller chip size (e.g., 12 PSs on a 2-mode complex chip vs. 56 PSs on an 8-mode real chip). - Hardware metrics: ONC with 8 modes and 56 phase shifters; heater calibration R² ≈ 0.99; interference visibility ≈98.5%. - Overall, complex-valued ONC shows higher accuracy, faster convergence, and ability to construct nonlinear decision boundaries versus real-valued counterparts, without increasing hardware complexity (aside from coherent detection measurements).

Implementing complex-valued neural networks directly in optics leverages light’s amplitude and phase, enabling efficient complex arithmetic and richer internal representations. The ONC’s complex neurons can solve tasks (e.g., XOR, nonlinear separations) that are challenging or impossible for single real-valued neurons, and achieve superior performance on nonlinear classification and MNIST with fewer trainable parameters and smaller photonic footprints. Coherent detection allows phase readout and noise mitigation, while intensity detection suffices for magnitude-only activations. The benefits include reduced model size for comparable accuracy, faster convergence, and formation of nonlinear decision boundaries with simple architectures. Beyond classical tasks, the ONC’s interferometric architecture naturally interfaces with non-classical light, offering a path towards quantum optical neural networks and applications such as quantum variational models. Cascadability and noise handling are addressed through electrical interfacing for gain management and by incorporating noise characteristics into training and compensation strategies.

The work demonstrates a single-chip integrated optical neural platform that implements genuine complex-valued neural networks, integrating input encoding, complex weight multiplication, and both intensity and coherent detection. Across logic gates, Iris classification, nonlinear datasets, and MNIST, the complex-valued ONC outperforms real-valued counterparts in accuracy, convergence, and expressive decision boundaries, while requiring fewer effective parameters and comparable hardware complexity. The approach highlights the promise of compact, low-power, high-speed photonic implementations for deep learning and provides a stepping stone towards scalable, cascaded complex-valued photonic networks and quantum optical neural architectures. Future work includes scaling to larger multilayer systems via circuit cascading, further improving noise cascadability and stability, optimizing on-chip nonlinear activations, and integrating non-classical sources and photon-number-resolving detection for quantum-enhanced models.

- Current photonic integration limits the scale of networks achievable compared to electronic deep learning systems; cascading larger networks remains a fabrication and packaging challenge. - Coherent detection introduces additional measurement overhead (approximately twice that of intensity-only), increasing readout time/complexity. - System performance is sensitive to phase stability, photodetector noise, thermal drift, and thermal cross-talk; these factors can degrade decision boundary fidelity and classification accuracy. - Experimental MNIST validations used a limited subset of test instances on-chip (e.g., 200), potentially limiting statistical power; inconsistencies between simulated and on-chip testing accuracies reflect practical hardware constraints and measurement resolution. - Activation functions are constrained by available on-chip operations and detection modes; fully optical nonlinearities with high gain remain challenging.