Sumcheck-based delegation of quantum computing to rational server

TL;DR

This paper introduces new one-round rational delegated quantum computing protocols based on sumcheck techniques, allowing flexible gate sets and analyzing reward gaps, including the possibility of constant gaps with entangled servers.

Contribution

It generalizes classical sumcheck protocols to rational quantum settings, enabling flexible gate sets and exploring reward gap amplification with entangled servers.

Findings

Protocols work with any local gate set.

Reward gap can be constant with entangled servers.

Single rational server may be insufficient under certain assumptions.

Abstract

Delegated quantum computing enables a client with weak computational power to delegate quantum computing to a remote quantum server in such a way that the integrity of the server can be efficiently verified by the client. Recently, a new model of delegated quantum computing has been proposed, namely, rational delegated quantum computing. In this model, after the client interacts with the server, the client pays a reward to the server. The rational server sends messages that maximize the expected value of the reward. It is known that the classical client can delegate universal quantum computing to the rational quantum server in one round. In this paper, we propose novel one-round rational delegated quantum computing protocols by generalizing the classical rational sumcheck protocol. The construction of the previous rational protocols depends on gate sets, while our sumcheck technique can be easily realized with any local gate set. Furthermore, as with the previous protocols, our reward function satisfies natural requirements. We also discuss the reward gap. Simply speaking, the reward gap is a minimum loss on the expected value of the server's reward incurred by the server's behavior that makes the client accept an incorrect answer. Although our sumcheck-based protocols have only exponentially small reward gaps as in the previous protocols, we show that a constant reward gap can be achieved if two noncommunicating but entangled rational servers are allowed. We also discuss whether a single rational server is sufficient under the (widely believed) assumption that the learning-with-errors problem is hard for polynomial-time quantum computing. Apart from these results, we show, under a certain condition, the equivalence between $rational$ and $ordinary$ delegated quantum computing protocols. This equivalence then serves as a basis for a reward-gap amplification method.