Secure multi-party quantum computation protocol for quantum circuits: the exploitation of triply-even quantum error-correcting codes

Secure multi-party quantum computation (MPQC) protocol is a cryptographic primitive allowing error-free distributed quantum computation to a group of $n$ mutually distrustful quantum nodes even when some quantum nodes disobey the instructions of the protocol. Here we suggest a modified MPQC protocol that adopts unconventional quantum error-correcting codes and as a consequence reduces the number of qubits required for the protocol execution. In particular, the replacement of the self-dual Calderbank-Shor-Steane quantum error-correcting codes with triply-even ones permits us to avoid the previously indispensable but resource-intensive procedure of the ``magic'' state verification. Besides, since every extra qubit reduces the credibility of physical devices, our suggestion makes the MPQC protocol more accessible for the near-future technology by reducing the number of necessary qubits per quantum node from $n^2 + \Theta(r)n$, where $r$ is the security parameter, to $n^2 + 3n$.