# Gradients of unitary optical neural networks using parameter-shift rule

arXiv: https://arxiv.org/abs/2506.11565 · HTML: https://mlsanity.com/paper/2506.11565/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 22 references that are in the map.
- First posted on arXiv 13 Jun 2025.

## Builds on (most cited first, 4 of 4)

- [Training of photonic neural networks through in situ backpropagation](https://mlsanity.com/paper/1805.09943/) (2018)
- [Spatially Varying Nanophotonic Neural Networks](https://mlsanity.com/paper/2308.03407/) (2023)
- [Training neural networks with end-to-end optical backpropagation](https://mlsanity.com/paper/2308.05226/) (2023)
- [Exact gradients for linear optics with single photons](https://mlsanity.com/paper/2409.16369/) (2024)

## Abstract (arXiv)

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. The method leverages the inherent Fourier series nature of optical interference in these systems to compute exact analytical gradients directly from hardware measurements. 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.