  Suppose some data have been encrypted, can you compute with the data without
decrypting them? This problem has been studied as homomorphic encryption and
blind computing. We consider this problem in the context of quantum information
processing, and present the definitions of quantum homomorphic encryption (QHE)
and quantum fully homomorphic encryption (QFHE). Then, based on quantum
one-time pad (QOTP), we construct a symmetric QFHE scheme, where the evaluate
algorithm depends on the secret key. This scheme permits any unitary
transformation on any $n$-qubit state that has been encrypted. Compared with
classical homomorphic encryption, the QFHE scheme has perfect security.
Finally, we also construct a QOTP-based symmetric QHE scheme, where the
evaluate algorithm is independent of the secret key.

Quantum Physics