Homomorphic encryption (HE) is an encryption scheme that allows computations to be evaluated on encrypted inputs without knowledge of their raw messages. Recently the first quantum fully homomorphic encryption (FHE) was proposed by Dulek et al. with privacy inherited from classical FHE and thus is computationally secure. On the other hand, Ouyang et al. constructed a quantum HE scheme for Clifford circuits with information-theoretic security (IT-security). It is desired to see whether an information-theoretically-secure (IT-secure) quantum FHE exists. If not, what other nontrivial class of quantum circuits can be homomorphically evaluated with IT-security? We answer the first question in the negative. As for the second one, we propose an IT-secure quantum HE scheme that supports the homomorphic evaluation of a class of quantum circuits, called IQP+ , which is an enlarged class of the instantaneous quantum polynomial-time (IQP) circuits. Our HE scheme is based on a class of concatenated quantum stabilizer codes, whose logical X and Z operators act nontrivially on different sets of qubits.
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