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 $n^2+\Theta(s) n$ qubits, as opposed to previously shown $\Omega\big((n^3+n^2s^2)\log n\big)$ qubits, where $s$ is a security parameter. To achieve universal computation, we develop a distributed procedure for verifying magic states based on magic state distillation and statistical testing of randomly selected states. This allows us to apply distributed $T$ gate and which may be of independent interest. We showcase our protocol on a small example for a 7-node network.