Artificial neural networks (ANNs) are powerful machine learning tools, but their reliance on multiply-accumulate (MAC) operations creates a significant computational burden on electronic hardware. Complex-valued neural networks offer potential advantages such as enhanced representational capacity, faster convergence, and improved generalization compared to their real-valued counterparts. However, digital electronic platforms struggle with complex-valued operations, requiring two real numbers to represent each complex number, thereby increasing the computational load. Optical computing, encoding information in both phase and amplitude, naturally handles complex arithmetic through optical interference, promising increased speed and energy efficiency. Despite the potential, most optical neural network demonstrations still use real-valued frameworks designed for digital computers, missing the benefits of complex-valued operations in optical computing. This research addresses this gap by presenting an optical neural chip (ONC) that directly implements complex-valued neural networks. This approach eliminates the inefficiencies of representing complex numbers as pairs of real numbers in digital systems. The researchers aim to demonstrate that a complex-valued ONC outperforms its real-valued counterpart in terms of accuracy, convergence speed, and ability to handle nonlinear tasks, thereby showcasing the advantages of leveraging the inherent capabilities of optical computing for complex arithmetic in neural networks. The significance lies in the potential for dramatically improved performance and efficiency in various machine learning applications.

The authors review existing literature on neural networks, highlighting the computational bottlenecks associated with the multiply-accumulate operations prevalent in ANNs. They discuss the advantages of complex-valued neural networks over real-valued networks in terms of representational power, convergence speed, and generalization abilities. The limitations of conventional digital electronic computing platforms in handling complex-valued operations are also discussed. Existing works on optical computing and its potential for neural network implementation are reviewed, noting that most existing optical implementations have yet to fully exploit the advantages of complex arithmetic. The authors cite examples of existing photonic neural networks and optical reservoir computing but highlight that these often still operate on real-valued arithmetic, failing to fully utilize optical capabilities. The lack of general-purpose optical complex-valued neural networks is identified as a key limitation in the field, emphasizing the novelty of their proposed ONC.

The researchers designed and fabricated an optical neural chip (ONC) capable of performing complex-valued arithmetic. The ONC architecture consists of an input layer, multiple parallelized layers, and an output layer. Light signals are encoded and manipulated using both amplitude and phase information. The chip integrates input preparation, weight multiplication, and coherent detection on a single platform. A coherent laser (1550 nm) generates input signals. The ONC utilizes Mach–Zehnder interferometers (MZIs) arranged as a multipart interferometer to perform the computations. The MZIs, composed of beam splitters (BSs) and phase shifters (PSs), allow for precise control over both amplitude and phase of the optical signals. Different colored MZIs in the diagrams represent different functionalities: input signal division and modulation (red), reference light separation (green), and weight multiplication (blue). The on-chip light division ensures that light signals maintain a stable relative phase. Input modulation is adjusted based on the specific task, using magnitude modulation for real-valued inputs and both magnitude and phase modulation for complex-valued inputs. The chip supports both intensity and coherent detection methods, with coherent detection used to obtain phase information, crucial for complex-valued operations. The photodetected signals are converted into voltage signals using a transimpedance amplifier (TIA) and processed by a classical processor with an analogue-to-digital converter (ADC). A feedback mechanism is implemented to adjust the chip configuration during the learning process. The complex-valued neuron model in the ONC mirrors conventional neuron models, but uses complex-valued parameters and variables. The output of a neuron is calculated as y = f(Σwix_i + b), where weights wi and bias b are complex numbers. The ONC was fabricated using silicon photonics with 8 modes and 56 PSs. The PSs are thermally tuned using integrated titanium nitride (TiN) heaters. The researchers employed coherent detection to extract both magnitude and phase information from the output optical signals. The ONC was benchmarked on several tasks: (a) logic gate realization (demonstrating XOR gate implementation), (b) Iris species classification (using a single complex layer), (c) nonlinear dataset classification (Circle and Spiral), and (d) handwriting recognition (using a multilayer perceptron, MLP). For each task, the ONC's performance was compared to a similar on-chip implementation using real-valued perceptrons. The training process involved adjusting the weights based on the error between the actual and expected outputs, using a learning rate and an optimization algorithm.

The complex-valued ONC demonstrated superior performance compared to its real-valued counterpart across all benchmark tasks. In the logic gate realization, the ONC successfully implemented a nonlinear XOR gate using a single complex-valued neuron, a feat impossible for a single real-valued neuron. This highlights the ONC's ability to handle nonlinear decision boundaries. In the Iris dataset classification, the complex-valued ONC showed improved accuracy and faster convergence during training. The visualization of the decision boundaries in the Circle and Spiral dataset classification confirmed the ONC's ability to generate nonlinear decision boundaries, essential for separating complex datasets, achieving near-perfect accuracy. The handwriting recognition task, utilizing a multilayer perceptron, demonstrated a significant performance improvement. The complex-valued ONC achieved a testing accuracy of up to 97.4%, an 8.5% increase over the real-valued network. This demonstrates the advantages of complex-valued arithmetic for complex tasks like image recognition. Ablation studies investigated the contribution of complex-valued weight matrices, encoding methods, and detection methods to the accuracy improvement. Results showed that complex-valued weight matrices and complex encoding significantly contributed to higher accuracy, even when detection was limited to intensity detection. The study also demonstrated that a smaller, complex-valued model could outperform a larger real-valued model in accuracy. This demonstrates that complex-valued networks can achieve comparable or better performance with fewer parameters, leading to potential reductions in chip size and cost.

The results demonstrate the clear advantages of using complex-valued neural networks implemented on an optical platform. The ONC's superior performance in various tasks highlights the benefits of directly exploiting the capabilities of optical computing for complex arithmetic. The ability to construct nonlinear decision boundaries with simple architectures, as well as the faster convergence rates observed, suggest potential improvements in training efficiency and model complexity. The findings challenge the conventional reliance on real-valued networks, demonstrating the potential of complex-valued networks for more efficient and accurate machine learning. The ability to achieve high accuracy with fewer parameters suggests that the ONC approach could lead to more compact and energy-efficient hardware implementations, which is crucial for large-scale deployment. The observed improvements in accuracy and efficiency highlight the potential impact on various machine learning applications where complex data and patterns are common.

This work successfully demonstrated the implementation of a complex-valued neural network on a single-chip integrated photonic platform. The ONC's superior performance compared to real-valued counterparts in various benchmarks highlights the benefits of harnessing optical computing's capabilities for complex arithmetic. The compact design and high efficiency suggest potential for large-scale, energy-efficient machine learning applications. Future research could focus on scaling up the ONC's size and complexity, integrating more sophisticated network architectures, and exploring applications in areas like quantum machine learning.

The current ONC implementation is limited in scale, with only 8 modes and 56 PSs. Scaling up the ONC to handle larger datasets and more complex networks would require advancements in fabrication and integration techniques. The study's focus on specific benchmark tasks limits the generalizability of the results, and more extensive evaluations are needed across a broader range of applications. The reliance on coherent detection may introduce additional noise and complexity compared to simpler intensity detection methods, requiring further research to optimize the detection scheme for different applications. Lastly, the study did not fully investigate the effect of different types of activation functions.