Encryptability As a Coordinate Choice: Depth-One Homomorphic Federated Learning of Quantum Neural Networks
Abstract
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 (Δ=+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.
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BibTeX
@misc{mordarski2026encryptability,
title = {{Encryptability As a Coordinate Choice: Depth-One Homomorphic Federated Learning of Quantum Neural Networks}},
author = {Marcel Mordarski and Nathan Mani and Arshad Patel and William Knottenbelt and Roberto Bondesan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30581},
eprint = {2609.30581},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30581},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Encryptability As a Coordinate Choice: Depth-One Homomorphic Federated Learning of Quantum Neural Networks
AU - Marcel Mordarski
AU - Nathan Mani
AU - Arshad Patel
AU - William Knottenbelt
AU - Roberto Bondesan
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30581
UR - https://arxiv.org/abs/2609.30581
AB - 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 (Δ=+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.
ER -