Searchable encryption (SE) is a positive way to protect users sensitive data in cloud computing setting, while preserving search ability on the server side, i.e., it allows the server to search encrypted data without leaking information about the plaintext data. In this paper, a multi-client universal circuit-based full-blind quantum computation (FBQC) model is proposed. In order to meet the requirements of multi-client accessing or computing encrypted cloud data, all clients with limited quantum ability outsource the key generation to a trusted key center and upload their encrypted data to the data center. Considering the feasibility of physical implementation, all quantum gates in the circuit are replaced with the combination of π/8 rotation operator set {Rz(π/4), Ry(π/4), CRz(π/4), CRy(π/4), CCRz(π/4), CCRy(π/4)}. In addition, the data center is only allowed to perform one π/8 rotation operator each time, but does not know the structure of the circuit (i.e., quantum computation), so it can guarantee the blindness of computation. Then, through combining this multi-client FBQC model and Grover searching algorithm, we continue to propose a quantum searchable encryption scheme for cloud data. It solves the problem of multi-client access mode under searchable encryption in the cloud environment, and has the ability to resist against some quantum attacks. To better demonstrate our scheme, an example of our scheme to search on encrypted 2-qubit state is given in detail. Furthermore, the security of our scheme is analysed from two aspects: external attacks and internal attacks, and the result indicates that it can resist against such kinds of attacks and also guarantee the blindness of data and computation.