On Statistically-Secure Quantum Homomorphic Encryption

Homomorphic encryption is an encryption scheme that allows computations to be evaluated on encrypted inputs without knowledge of their raw messages. Recently Ouyang et al. constructed a quantum homomorphic encryption (QHE) scheme for Clifford circuits with statistical security (or information-theoretic security (IT-security)). It is desired to see whether an informationtheoretically-secure (ITS) quantum FHE exists. If not, what other nontrivial class of quantum circuits can be homomorphically evaluated with IT-security? We provide a limitation for the first question that an ITS quantum FHE necessarily incurs exponential overhead. As for the second one, we propose two QHE schemes for an enlarged class of the instantaneous quantum polynomial-time (IQP) circuits called IQP. The first scheme TRIV follows directly from the one-time pad. The second scheme IQPP is constructed from a class of concatenated quantum stabilizer codes, whose logical X and Z operators act nontrivially on different sets of qubits.