The cryptographic task of secure multi-party (classical) computation has received a lot of attention in the last decades. Even in the extreme case where a computation is performed between k mutually distrustful players, and security is required even for the single honest player if all other players are colluding adversaries, secure protocols are known. For quantum computation, on the other hand, protocols allowing arbitrary dishonest majority have only been proven for k=2. In this work, we generalize the approach taken by Dupuis, Nielsen and Salvail (CRYPTO 2012) in the two-party setting to devise a secure, efficient protocol for multi-party quantum computation for any number of players k, and prove security against up to k−1 colluding adversaries. The quantum round complexity of the protocol for computing a quantum circuit with g gates acting on w qubits is O((w+g)k). To achieve efficiency, we develop a novel public verification protocol for the Clifford authentication code, and a testing protocol for magic-state inputs, both using classical multi-party computation.
Advances in Cryptology - EUROCRYPT
Algorithms and Complexity

Dulek, Y.M, Grilo, A, Jeffery, S, Majenz, C, & Schaffner, C. (2020). Secure Multi-party Quantum Computation with a Dishonest Majority. In Proceedings of EUROCRYPR. doi:arXiv:1909.13770v1