  Quantum fully homomorphic encryption (QFHE) allows to evaluate quantum
circuits on encrypted data. We present a novel QFHE scheme, which extends Pauli
one-time pad encryption by relying on the quaternion representation of SU(2).
With the scheme, evaluating 1-qubit gates is more efficient, and evaluating
general quantum circuits is polynomially improved in asymptotic complexity.
  Technically, a new encrypted multi-bit control technique is proposed, which
allows to perform any 1-qubit gate whose parameters are given in the encrypted
form. With this technique, we establish a conversion between the new encryption
and previous Pauli one-time pad encryption, bridging our QFHE scheme with
previous ones. Also, this technique is useful for private quantum circuit
evaluation.
  The security of the scheme relies on the hardness of the underlying quantum
capable FHE scheme, and the latter sets its security on the learning with
errors problem and the circular security assumption.
