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arXiv:2506.11565 · Tenth International Conference on Information Optics and Photonics

Gradients of unitary optical neural networks using parameter-shift rule

Jinzhe Jiang, Xin Zhang, Li Chen, Yunlong Yu, Hailing Liu

Published Dec 8, 2025• 0 citations
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Abstract

This paper explores the application of the parameter-shift rule (PSR) for computing gradients in unitary optical neural networks (UONNs). While backpropagation has been fundamental to training conventional neural networks, its implementation in optical neural networks faces significant challenges due to the physical constraints of optical systems. We demonstrate how PSR, which calculates gradients by evaluating functions at shifted parameter values, can be effectively adapted for training UONNs constructed from Mach-Zehnder interferometer meshes. We develop a theoretical framework for two distinct scenarios, real-valued outputs and complex-valued outputs of optical networks, which derives exact analytical gradients directly from hardware measurements for phase parameters in optical neural networks. For the first real-valued scenario, the gradient calculation directly conforms to PSR. For the second complex-valued scenario, we improved PSR to obtain the gradient. Then we simulate the training of optical neural networks based on optical PSR and both scenarios complete the classification tasks well. The optical PSR offers a powerful and analytically precise approach for calculating gradients in unitary optical neural networks. This approach offers a promising alternative to traditional in silico training methods and circumvents the limitations of both finite difference approximations and all-optical backpropagation implementations. We present the theoretical framework and practical methodology for applying PSR to optimize phase parameters in optical neural networks, potentially advancing the development of efficient hardware-based training strategies for optical computing systems.

DOI
10.1117/12.3085980

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This paper explores the application of the parameter-shift rule (PSR) for computing gradients in unitary optical neural networks (UONNs). While backpropagation has been fundamental to training conventional neural networks, its implementation in optical neural networks faces significant challenges due to the physical constraints of optical systems. We demonstrate how PSR, which calculates gradients by evaluating functions at shifted parameter values, can be effectively adapted for training UONNs constructed from Mach-Zehnder interferometer meshes. We develop a theoretical framework for two distinct scenarios, real-valued outputs and complex-valued outputs of optical networks, which derives exact analytical gradients directly from hardware measurements for phase parameters in optical neural networks. For the first real-valued scenario, the gradient calculation directly conforms to PSR. For the second complex-valued scenario, we improved PSR to obtain the gradient. Then we simulate the training of optical neural networks based on optical PSR and both scenarios complete the classification tasks well. The optical PSR offers a powerful and analytically precise approach for calculating gradients in unitary optical neural networks. This approach offers a promising alternative to traditional in silico training methods and circumvents the limitations of both finite difference approximations and all-optical backpropagation implementations. We present the theoretical framework and practical methodology for applying PSR to optimize phase parameters in optical neural networks, potentially advancing the development of efficient hardware-based training strategies for optical computing systems.
Deep learning has profoundly reshaped the technology landscape in numerous scientific areas and industrial sectors. This technology advancement is, nevertheless, confronted with severe bottlenecks in digital computing. Optical neural network presents a promising solution due to the ultra-high computing speed and energy efficiency. In this work, we present systematic study of unitary optical neural network (UONN) as an approach towards optical deep learning. Our results show that the UONN can be trained to high accuracy through special unitary gradient descent optimization, and the UONN is robust against physical imperfections and noises, hence it is more suitable for physical implementation than existing ONNs.
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Finding gradients is a crucial step in training machine learning models. For quantum neural networks, computing gradients using the parameter-shift rule requires calculating the cost function twice for each adjustable parameter in the network. When the total number of parameters is large, the quantum circuit must be repeatedly adjusted and executed, leading to significant computational overhead. Here, we propose an approach to compute all gradients using only a single circuit, significantly reducing both the circuit depth and the number of classical registers required. Although the theoretical complexity of gradient calculation and the total number of measurements are not reduced by this approach, a considerable reduction in the number of unique circuit compilations and job submissions is achieved. We experimentally validate our approach on both quantum simulators and IBM’s real quantum hardware, demonstrating that our method significantly reduces circuit compilation time compared to the conventional approach, resulting in a substantial speedup in total runtime.
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