We propose a quantum function secret sharing scheme in which the communication is exclusively classical. In this primitive, a classical dealer distributes a secret quantum circuit $C$ by providing shares to $p$ quantum parties. The parties on an input state $|\psi\rangle$ and a projection $\Pi$, compute values $y_i$ that they then classically communicate back to the dealer, who can then compute $\lVert \Pi C|\psi\rangle\rVert^2$ using only classical resources. Moreover, the shares do not leak much information about the secret circuit $C$. Our protocol for quantum secret sharing uses the {\em Cayley path}, a tool that has been extensively used to support quantum primacy claims. More concretely, the shares of $C$ correspond to randomized version of $C$ which are delegated to the quantum parties, and the reconstruction can be done by extrapolation. Our scheme has two limitations, which we prove to be inherent to our techniques: First, our scheme is only secure against single adversaries, and we show that if two parties collude, then they can break its security. Second, the evaluation done by the parties requires exponential time in the number of gates.

翻译：我们提出了一种量子函数秘密共享方案，其通信过程完全基于经典信道。在该原语中，经典分发者通过向p个量子参与方分配份额来分发秘密量子电路C。各参与方在输入态|\psi\rangle和投影算子Π上计算数值y_i，随后通过经典通信将结果传回分发者，分发者仅需经典资源即可计算\lVert ΠC|\psi\rangle\rVert^2。此外，这些份额几乎不会泄露关于秘密电路C的信息。我们的量子秘密共享协议采用了{\em 凯莱路径}这一工具，该工具已被广泛用于支持量子优越性主张。具体而言，C的份额对应于随机化版本的C并委托给量子参与方，重构过程可通过外推法实现。我们的方案存在两个局限性，我们证明这些局限性是本技术方法固有的：首先，该方案仅能抵抗单一敌手攻击，我们证明若两方合谋即可破坏其安全性。其次，参与方的计算复杂度随门电路数量呈指数级增长。