# Exact gradients for linear optics with single photons

arXiv: https://arxiv.org/abs/2409.16369 · HTML: https://mlsanity.com/paper/2409.16369/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 78 references that are in the map.
- First posted on arXiv 24 Sep 2024.

## Builds on (most cited first, 10 of 10)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Stochastic gradient descent for hybrid quantum-classical optimization](https://mlsanity.com/paper/1910.01155/) (2019)
- [Out-of-distribution generalization for learning quantum dynamics](https://mlsanity.com/paper/2204.10268/) (2022)
- [The power and limitations of learning quantum dynamics incoherently](https://mlsanity.com/paper/2303.12834/) (2023)
- [Fock State-enhanced Expressivity of Quantum Machine Learning Models](https://mlsanity.com/paper/2107.05224/) (2021)
- [Matrix optimization on universal unitary photonic devices](https://mlsanity.com/paper/1808.00458/) (2018)
- [Performance of QAOA on Typical Instances of Constraint Satisfaction Problems with Bounded Degree](https://mlsanity.com/paper/1601.01744/) (2016)

## Built on by (most cited first, 3 of 3)

- [Quantum latent distributions in deep generative models](https://mlsanity.com/paper/2508.19857/) (2025)
- [Gradients of unitary optical neural networks using parameter-shift rule](https://mlsanity.com/paper/2506.11565/) (2025)
- [Quantum enhanced ensemble GANs for anomaly detection in continuous biomanufacturing](https://mlsanity.com/paper/2508.21438/) (2025)

## Abstract (arXiv)

Though parameter shift rules have drastically improved gradient estimation methods for several types of quantum circuits, leading to improved performance in downstream tasks, so far they have not been transferable to linear optics with single photons. In this work, we derive an analytical formula for the gradients in these circuits with respect to phaseshifters via a generalized parameter shift rule, where the number of parameter shifts depends linearly on the total number of photons. Experimentally, this enables access to derivatives in photonic systems without the need for finite difference approximations. Building on this, we propose two strategies through which one can reduce the number of shifts in the expression, and hence reduce the overall sample complexity. Numerically, we show that this generalized parameter-shift rule can converge to the minimum of a cost function with fewer parameter update steps than alternative techniques. We anticipate that this method will open up new avenues to solving optimization problems with photonic systems, as well as provide new techniques for the experimental characterization and control of linear optical systems.