  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.
