Abstract. We show that any two-party quantum computation, specified by a unitary which simultaneously acts on the registers of both parties, can be securely implemented against a quantum version of classical semihonest adversaries that we call specious. We first show that no statistically private protocol exists for swapping qubits against specious adversaries. The swap functionality is modeled by a unitary transform that is not sufficient for universal quantum computation. It means that universality is not required in order to obtain impossibility proofs in our model. However, the swap transform can easily be implemented privately provided a classical bit commitment scheme. We provide a simple protocol for the evaluation of any unitary transform represented by a circuit made out of gates in some standard universal set of quantum gates. All gates except one can be implemented securely provided one call to swap made available as an ideal functionality. For each appearance of the remaining gate in the circuit, one call to a classical AND-box is required for privacy. The AND-box can easily be constructed from oblivious transfer. It follows that oblivious transfer is universal for private evaluations of unitaries as well as for classical circuits. Unlike the ideal swap, AND-boxes are classical primitives and cannot be represented by unitary transforms. It follows that, to some extent, this remaining gate is the hard one, like the and gate for classical two-party computation.
Dupuis, Nielsen and Salvail REF gave a protocol for two-party secure quantum computation in the case of specious (a version of quantum semi-honest) adversaries.
Secure two-party quantum evaluation of unitaries against specious adversaries
Advances in Cryptology, LNCS 6223, pages 685--706 (2010)
Computer Science Physics