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 arbitrary circuits would require exponential resources, and efficient computationally-secure QHE schemes for polynomial-sized quantum circuits have been constructed. In this paper we propose a QHE scheme for a restricted type of circuits. The scheme keeps the restricted type of data perfectly secure, and the privacy of the circuit is optimal. When this scheme is used for general polynomial-sized circuits, the data privacy is partial but the circuit privacy remains optimal in general. The entanglement and communication costs in the two schemes scale linearly with the product of the input size and circuit depth.
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