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We describe how any two-party quantum computation, specified by a unitarywhich simultaneously acts on the registers of both parties, can be privatelyimplemented against a quantum version of classical semi-honest adversaries thatwe call specious. Our construction requires two ideal functionalities togarantee privacy: a private SWAP between registers held by the two parties anda classical private AND-box equivalent to oblivious transfer. If the unitary tobe evaluated is in the Clifford group then only one call to SWAP is requiredfor privacy. On the other hand, any unitary not in the Clifford requires onecall to an AND-box per R-gate in the circuit. Since SWAP is itself in theClifford group, this functionality is universal for the private evaluation ofany unitary in that group. SWAP can be built from a classical bit commitmentscheme or an AND-box but an AND-box cannot be constructed from SWAP. It followsthat unitaries in the Clifford group are to some extent the easy ones. We alsoshow that SWAP cannot be implemented privately in the bare model.

Secure two-party quantum evaluation of unitaries against specious adversaries