A quantum homomorphic encryption scheme for polynomial-sized circuits

Quantum homomorphic encryption (QHE) is an encryption method that allows quantum computation to be performed on one party's private data with the program provided by another party, without revealing much information about the data nor about the program to the opposite party. It is known that information-theoretically-secure QHE for circuits of unrestricted size would require exponential resources, and efficient computationally-secure QHE schemes for polynomial-sized quantum circuits have been constructed. In this paper we first propose a QHE scheme for a type of circuits of polynomial depth, based on the rebit quantum computation formalism. The scheme keeps the restricted type of data perfectly secure. We then propose a QHE scheme for a larger class of polynomial-depth quantum circuits, which has partial data privacy. We also propose an asymptotically-secure interactive QHE scheme for general polynomial-sized quantum circuits. All these schemes have good circuit privacy. The entanglement and classical communication costs in these schemes are polynomial in the circuit size and (for the last scheme) the security parameter.