Efficient quantum algorithms for $GHZ$ and $W$ states, and implementation on the IBM quantum computer

We propose efficient algorithms with logarithmic step complexities for the generation of entangled $GHZ_N$ and $W_N$ states useful for quantum networks, and we demonstrate an implementation on the IBM quantum computer up to $N=16$. Improved quality is then investigated using full quantum tomography for low-$N$ GHZ and W states. This is completed by parity oscillations and histogram distance for large $N$ GHZ and W states respectively. We are capable to robustly build states with about twice the number of quantum bits which were previously achieved. Finally we attempt quantum error correction on GHZ using recent schemes proposed in the literature, but with the present amount of decoherence they prove detrimental.