Entangling spins by measuring charge: A parity-gate toolbox

The parity gate emerged recently as a promising resource for performing universal quantum computation with fermions using only linear interactions. Here we analyze 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 multiqubit entanglement, such as n-Greenberg-Horne-Zellinger (GHZ) states. Other applications include a Bell-state analyzer 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 an alternative spin-controlled-Z gate by using the mode degrees of freedom as ancillae.