Universal quantum homomorphic encryption based on $(k, n)$-threshold quantum state sharing

Quantum homomorphic encryption integrates quantum computing with homomorphic encryption, which allows calculations to be performed directly on encrypted data without decryption on the server side. In this paper, we explore distributed quantum homomorphic encryption, focusing on the coordination of multiple evaluators to achieve evaluation tasks, which not only ensures security but also boosts computational power. Notably, we propose a $(k, n)$-threshold universal quantum homomorphic encryption scheme based on quantum state sharing. Each server is capable of executing a universal gate set, including the Clifford gates $\{X,Y,Z,H,S,CNOT\}$ and a non-Clifford T gate. The scheme provides that k evaluation servers chosen from $n$ $(0 < k \leq n)$ cooperate to complete the quantum homomorphic encryption so that the client can get the evaluated plaintext after decryption. Several concrete examples are presented to provide clarity to our solution. We also include security analysis, demonstrating its security against eavesdroppers.