  We construct a (compact) quantum fully homomorphic encryption (QFHE) scheme
starting from (compact) classical fully homomorphic encryption scheme with
decryption in $\mathsf{NC}^{1}$, together with a dual-mode trapdoor function
family. Compared to previous constructions (Mahadev, FOCS 2018; Brakerski,
CRYPTO 2018) which made non-black-box use of similar underlying primitives, our
construction provides a pathway to instantiations from different assumptions.
Our construction uses the techniques of Dulek, Schaffner and Speelman (CRYPTO
2016) and shows how to make the client in their QFHE scheme classical using
dual-mode trapdoor functions. As an additional contribution, we show a new
instantiation of dual-mode trapdoor functions from group actions.
