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:
- The BK code matrix $\beta_n$ (as defined here). For each fermionic state, it gives the corresponding qubit state.
- 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?
