Encrypted federated training of quantum neural networks made efficient
Encryptability As a Coordinate Choice: Depth-One Homomorphic Federated Learning of Quantum Neural Networks
Cryptography and SecurityDistributed, Parallel, and Cluster ComputingMachine Learning
Summary
Training quantum neural networks while keeping data encrypted is usually very slow and costly because of how the math is done. The authors show that by changing the way these quantum operations are represented, the updates become simpler and much cheaper to compute in encrypted form. This new approach lets multiple clients train a shared quantum model securely without interactions that were previously needed. They tested their method on simulated data and actual quantum hardware, showing near-perfect accuracy compared to unencrypted training.
What this means in practice
- •For cloud service providers: Deploy secure and efficient encrypted federated learning of quantum models without costly interaction rounds.
- •For quantum computing engineers: Implement encrypted training protocols for hybrid quantum-classical networks minimizing error and computational overhead.
Authors
Marcel Mordarski, Nathan Mani, Arshad Patel, William Knottenbelt, Roberto Bondesan
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.