  The parity gate emerged recently as a promising resource for performing
universal quantum computation with fermions using only linear interactions.
Here we analyse the parity gate (P-gate) from a theoretical point of view in
the context of quantum networks. We present several schemes for entanglement
generation with P-gates and show that native networks simplify considerably the
resources required for producing multi-qubit entanglement, like n-GHZ states.
Other applications include a Bell-state analyser and teleportation. We also
show that cluster state fusion can be performed deterministically with parity
measurements. We then extend this analysis to hybrid quantum networks
containing spin and mode qubits. Starting from an easy-to-prepare resource
(spin-mode entanglement of single electrons) we show how to produce a spin
n-GHZ state with linear elements (beam-splitters and local spin-flips) and
charge-parity detectors; this state can be used as a resource in a spin quantum
computer or as a precursor for constructing cluster states. Finally, we
construct a novel spin CZ-gate by using the mode degrees of freedom as
ancillae.
