This article defines encrypted gate, which is denoted by @math . We present a gate-teleportation-based two-party computation scheme for @math , where one party gives arbitrary quantum state @math as input and obtains the encrypted @math -computing result @math , and the other party obtains the random bits @math . Based on @math , we propose a method to remove the @math -error generated in the homomorphic evaluation of @math -gate. Using this method, we design two non-interactive and perfectly secure QHE schemes named GT and VGT . Both of them are @math -homomorphic and quasi-compact (the decryption complexity depends on the @math -gate complexity). Assume @math -homomorphism, non-interaction and perfect security are necessary property, the quasi-compactness is proved to be bounded by @math , where @math is the total number of @math -gates in the evaluated circuit. VGT is proved to be optimal and has @math -quasi-compactness. According to our QHE schemes, the decryption would be inefficient if the evaluated circuit contains exponential number of @math -gates. Thus our schemes are suitable for homomorphic evaluation of any quantum circuit with low @math -gate complexity, such as any polynomial-size quantum circuit or any quantum circuit with polynomial number of @math -gates.
The QHE scheme EPR proposed by Broadbent and Jeffery @cite_6 makes use of Bell state and quantum measurement. That scheme is constructed by the combination of QOTP and classical FHE, then it is computational secure. Moreover, it is proved to be @math -quasi-compact, where @math is the number of @math -gates in an evaluated circuit. In this article, our schemes also make use of Bell state and quantum measurement. However, our scheme VGT has perfect security and @math -quasi-compactness, so it is better than EPR .
@cite_6
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Fully homomorphic encryption is an encryption method with the property that any computation on the plaintext can be performed by a party having access to the ciphertext only. Here, we formally define and give schemes for quantum homomorphic encryption, which is the encryption of quantum information such that quantum computations can be performed given the ciphertext only. Our schemes allow for arbitrary Clifford group gates, but become inefficient for circuits with large complexity, measured in terms of the non-Clifford portion of the circuit (we use the “ ( 8 )” non-Clifford group gate, also known as the ( T )-gate).