Fig. 5: Modular position measurement using an auxiliary GKP state.
From: Factoring an integer with three oscillators and a qubit
The modular value \(x\,\,\,{{{\rm{mod}}}}\,\,N\) of a position-eigenstate \(\left|x\right\rangle\) is obtained by adding x to an auxiliary system in the state \({M}_{N}\left|{\mathsf{GKP}}\right\rangle\), and then measuring the Q-quadrature of that system. Because of the fact that \({M}_{N}\left|{\mathsf{GKP}}\right\rangle \propto {\sum }_{y\in {\mathbb{Z}}}\left|y\cdot N\right\rangle\) is the uniform superposition of position-eigenstates with spacing N, this circuit provides \(w\equiv x\,({{{\rm{mod}}}}\,\,N)\) (and no other information on x). This reasoning has been used to construct syndrome extraction circuits for GKP-codes15.
