Quantum Secret Sharing with Grover’s Algorithm

Quantum secret sharing addresses the same problem that classical secret sharing does, but via quantum means. That is, Alice wishes to send a secret to n agents, and any k ≤ n agents can collaborate to recover the secret. However, for any fewer than k agents, collaboration will information-theoretically reveal no information about the secret. Such protocols are called (k,n)threshold schemes, and we focus on (2,2)-threshold quantum schemes in this paper. One technique used in quantum secret sharing is Grover’s algorithm, which solves the unstructured search problem in sub-linear time. This can be utilized by preparing a superposition of two-qubit basis states, then flipping the phase of one of the components. Each agent receives one qubit, and when they collaborate, they can search for the phase error, thus reconstructing Alice’s secret. Hsu’s protocol outlines a (2,2)-threshold scheme, which has been generalized to a (n,n)-threshold scheme for arbitrary n. Tseng’s protocol outlines a similar scheme, but one that doesn’t require that the agents have quantum memory, and it also allows the agents to collaborate using shadows in classical bits. However, more powerful protocols require much more sophistication than Grover’s algorithm, and even with the additional sophistication, none to date have been proven to be unconditionally secure against dishonest agents.