# “$W_n$ state” in Bravyi-Kitaev encoding

For a particular simulation, I need the initial (physical) state of the QC to represent the equal superposition of all single-occupancy fermionic second-quantized states. In Jordan-Wigner encoding, I can simply prepare the $$W_n$$ state: \begin{alignedat}{8} &\text{Fermionic state}:\qquad&&(|1000\ldots\rangle+|0100\ldots\rangle+|0010\ldots\rangle+\ldots)/\sqrt{n}\\ &\text{JW encoded state}:\qquad&&(|1000\ldots\rangle+|0100\ldots\rangle+|0010\ldots\rangle+\ldots)/\sqrt{n} = U_{W_n} (|0\rangle^{\otimes n}) \end{alignedat} How do I prepare such a state in the Bravyi-Kitaev encoding?

So far I know how to construct:

1. The BK code matrix $$\beta_n$$ (as defined here). For each fermionic state, it gives the corresponding qubit state.
2. A circuit $$U_{W_n}$$ implementing the $$W_n$$ transformation — which does the job in the case of JW encoding.

My guess is that I should somehow implement a unitary gate which would do sort of a BK transform of the qubits...

More generally, given some superposition of the fermionic states (the one which, in the JW language, can be prepared from $$|0\rangle^{\otimes n}$$ by a known operator $$U$$), how do I construct the corresponding BK-encoded state?