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Suppose some data have been encrypted, can you compute with the data withoutdecrypting them? This problem has been studied as homomorphic encryption andblind computing. We consider this problem in the context of quantum informationprocessing, and present the definitions of quantum homomorphic encryption (QHE)and quantum fully homomorphic encryption (QFHE). Then, based on quantumone-time pad (QOTP), we construct a symmetric QFHE scheme, where the evaluatealgorithm depends on the secret key. This scheme permits any unitarytransformation on any $n$-qubit state that has been encrypted. Compared withclassical homomorphic encryption, the QFHE scheme has perfect security.Finally, we also construct a QOTP-based symmetric QHE scheme, where theevaluate algorithm is independent of the secret key.

Symmetric quantum fully homomorphic encryption with perfect security