# Encryptability As a Coordinate Choice: Depth-One Homomorphic Federated Learning of Quantum Neural Networks

arXiv: https://arxiv.org/abs/2609.30581 · HTML: https://mlsanity.com/paper/2609.30581/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 50 references that are in the map.
- First posted on arXiv 24 Sep 2026.

## Builds on (most cited first, 20 of 20)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Advances and Open Problems in Federated Learning](https://mlsanity.com/paper/1912.04977/) (2019)
- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [Deep Leakage from Gradients](https://mlsanity.com/paper/1906.08935/) (2019)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Inverting Gradients -- How easy is it to break privacy in federated learning?](https://mlsanity.com/paper/2003.14053/) (2020)
- [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)
- [Flower: A Friendly Federated Learning Research Framework](https://mlsanity.com/paper/2007.14390/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group](https://mlsanity.com/paper/1901.08428/) (2019)
- [Federated Quantum Machine Learning](https://mlsanity.com/paper/2103.12010/) (2021)
- [Quantum Federated Learning with Quantum Data](https://mlsanity.com/paper/2106.00005/) (2021)
- [Private quantum computation: An introduction to blind quantum computing and related protocols](https://mlsanity.com/paper/1611.10107/) (2016)
- [Federated Learning with Quantum Computing and Fully Homomorphic Encryption: A Novel Computing Paradigm Shift in Privacy-Preserving ML](https://mlsanity.com/paper/2409.11430/) (2024)
- [Quantum delegated and federated learning via quantum homomorphic encryption](https://mlsanity.com/paper/2409.19359/) (2024)
- [Classical Homomorphic Encryption for Quantum Circuits](https://mlsanity.com/paper/1708.02130/) (2017)
- [Quantum Fully Homomorphic Encryption by Integrating Pauli One-time Pad with Quaternions](https://mlsanity.com/paper/2012.04211/) (2020)
- [RobQFL: Robust Quantum Federated Learning in Adversarial Environment](https://mlsanity.com/paper/2509.04914/) (2025)

## Abstract (arXiv)

Encrypted training relies on keeping server-side updates low-degree. This constraint traditionally excludes models whose weights inhabit a compact Lie group (notably variational quantum circuits, where every trainable weight is an SU(2) rotation). Expressed in Euler angles or discrete alphabets, these updates appear transcendental, historically demanding prohibitive costs: one client--server round per gate, or upwards of 25,000 operations per weight. This penalty is strictly an artefact of coordinates. In the unit-quaternion (spin) chart, group composition is exactly bilinear (degree two, with coefficients in -1,0,+1). Consequently, encrypted rotation updates cost one multiplicative level and federated averaging costs zero in any levelled homomorphic scheme, completely eliminating bootstrapping. This implementation-independent algebraic property is confirmed across two cryptographic backends, introducing only 0.0 and -2.0×10-12 rad of aggregation error. Leveraging this reduction yields a non-interactive protocol for encrypted federated training of hybrid quantum--classical networks. It includes correctness proofs for aggregation and sign handling, plus a compilation lemma proving parameterised entanglers add only constant-factor overhead without altering the depth class. Empirically, a paired five-seed study confirms zero measurable utility tax (Delta=+9×10-6 MSE, p=0.92), and a noise-budget ablation falsifies the hypothesis that encryption noise regularises. These convergence trends replicate across datasets and scale to 20 clients. Finally, hardware validation on a 156-qubit processor achieves 0.9918 fidelity against a 0.99957 unencrypted control.