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

2026-09-24

AI Summary

This study addresses the excessive homomorphic circuit depth and high communication overhead caused by coordinate-wise selection in encrypted federated learning for quantum neural networks. We propose a low-depth fully homomorphic encryption protocol based on unit quaternion representations, revealing that the prohibitive encryption cost is fundamentally a coordinate artifact. By exploiting the compositional properties of quaternion bilinear groups, rotational updates require only single-layer multiplication, enabling bootstrapping-free aggregation. Integrating this approach with hybrid quantum-classical networks and the Federated Averaging algorithm, the method achieves near-zero aggregation error without utility degradation when scaled to 20 clients. Furthermore, hardware validation on a 156-qubit processor demonstrates an operational fidelity of 0.9918, confirming the practical viability of the proposed framework.

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 $\mathrm{SU(2)}$ 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 $-2.0\times10^{-12}$ 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 ($Δ=+9\times10^{-6}$ 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.

Problem

• Homomorphic Federated Learning
• Quantum Neural Networks
• Encrypted Training
• Compact Lie Group
• Variational Quantum Circuits

Innovation

• Homomorphic Encryption
• Federated Learning
• Quantum Neural Networks
• Unit Quaternion
• Bootstrapping-free