Quantum function secret sharing

TL;DR

This paper introduces a quantum function secret sharing scheme that uses classical communication and the Cayley path, enabling the distribution and evaluation of quantum circuits with limited security and exponential complexity.

Contribution

It presents a novel quantum secret sharing protocol leveraging the Cayley path, enabling classical communication of quantum circuit shares with specific security and complexity characteristics.

Findings

Shares do not leak much information about the secret circuit

The scheme is secure against single adversaries but not colluding parties

Evaluation requires exponential time in the number of gates

Abstract

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 $\ket{\psi}$ 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.