Secure multi-party quantum computation with few qubits

We consider the task of secure multi-party distributed quantum computation on a quantum network. We propose a protocol based on quantum error correction which reduces the number of necessary qubits. That is, each of the $n$ nodes in our protocol requires an operational workspace of $n^2 + 4n$ qubits, as opposed to previously shown $\Omega\big((n^3+n^2s^2)\log n\big)$ qubits, where $s$ is a security parameter. Additionally, we reduce the communication complexity by a factor of $\mathcal{O}(n^3\log(n))$ qubits per node, as compared to existing protocols. To achieve universal computation, we develop a distributed procedure for verifying magic states, which allows us to apply distributed gate teleportation and which may be of independent interest. We showcase our protocol on a small example for a 7-node network.