Quantum Fully Homomorphic Encryption with neither Clifford Gate Decomposition nor Real Representation

We present a novel quantum fully homomorphic encryption (QFHE) scheme, which allows to perform conditional rotations with the control bit in encrypted form. In our scheme, any quantum circuit can be directly evaluated with no need to decompose into Clifford/non-Clifford gates, nor to be transformed into real representation. Our scheme is able to evaluate quantum Fourier transform algorithm faster than previous QFHE schemes in the worst case. The security of our scheme relies on the hardness of the underlying quantum capable FHE scheme, and the latter sets its security on the learning with errors problem and the circular security assumption.