Quantum one-time pad-based quantum homomorphic encryption schemes for circuits of the non-Clifford gates

Quantum homomorphic encryption can afford secure and convenient delegating computations for the client with weak computing power. The process of the server performing quantum gates on the client’s ciphertext is referred to as the homomorphic evaluation. At present, homomorphic evaluations of the group {H, S, T, controlled-X} for implementing universal quantum computations have been provided. However, except for T gate, there is little research on homomorphic evaluations of other non-Clifford gates, some of which can simplify quantum circuits. In order to optimize homomorphic evaluations of quantum circuits with complex computing functions, three quantum one-time pad-based quantum homomorphic encryption schemes for circuits of the non-Clifford gates are proposed in this paper. Firstly, we give homomorphic evaluations of V gate (the square root of X gate), V† gate, and controlled-Z gate in the Clifford gates. Secondly, for the non-Clifford gates, two methods for the homomorphic evaluation of controlled-V gate (or controlled-V† gate) and four methods for the homomorphic evaluation of Toffoli gate are presented in sequence. Among four homomorphic evaluation methods for Toffoli gate, the method based on Toffoli gate itself consumes the fewest auxiliary qubits and has the lowest evaluation circuit depth. Thirdly, three quantum homomorphic encryption schemes for the single-qubit gates, the double-qubit gates, and the triple-qubit gate are respectively proposed. Finally, inspired by the perfectly secure encryption technology of quantum one-time pad, we prove the security of the proposed schemes, enabling safer and faster completion of the client’s entrusted computations.