Cross quantum homomorphic encryption based on the butterfly network

Quantum homomorphic encryption hides quantum information from unauthorized participants while still allowing them to perform calculations on encrypted quantum information. In this paper, we propose a cross quantum homomorphic encryption scheme based on the butterfly network, in which every receiver can receive the evaluated quantum state with perfect fidelity. The entangled states between senders of the butterfly network are preshared, and the Bell basis measurements are used to encrypt the quantum plaintext. With both classical and quantum channels, the homomorphic evaluation of quantum plaintext can be completed on the butterfly network perfectly and crossly. Our scheme solves a bottleneck problem of cross evaluation at the bottleneck nodes, and improves the evaluation efficiency and total network throughput. In addition, we have designed quantum circuits according to different quantum evaluations with a butterfly network, which provide a basic blueprint for experimental implementation. The security of the scheme is also analyzed.