We consider the task of secure multiparty 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
n2+4n qubits, as opposed to the previously shown $\mathrm{\ensuremath{\Omega}}\left(({n}^{3}+{n}^{2}{s}^{2})logn\right)$ qubits, where
s is a security parameter. Additionally, we reduce the communication complexity by a factor of
O(n3log(n)) qubits per node 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 in a small example for a seven-node network.