The ability to perform computations on encrypted data is a powerful tool for
protecting privacy. Recently, protocols to achieve this on classical computing
systems have been found. Here we present an efficient solution to the quantum
analogue of this problem that enables arbitrary quantum computations to be
carried out on encrypted quantum data. We prove that an untrusted server can
implement a universal set of quantum gates on encrypted quantum bits (qubits)
without learning any information about the inputs, while the client, knowing
the decryption key, can easily decrypt the results of the computation. We
experimentally demonstrate, using single photons and linear optics, the
encryption and decryption scheme on a set of gates sufficient for arbitrary
quantum computations. Because our protocol requires few extra resources
compared to other schemes it can be easily incorporated into the design of
future quantum servers. These results will play a key role in enabling the
development of secure distributed quantum systems