Encoding a qubit into a cavity mode in circuit QED using phase estimation

Gottesman, Kitaev, and Preskill have formulated a way of encoding a qubit into an oscillator such that the qubit is protected against small shifts (translations) in phase space. The idea underlying this encoding is that error processes of low rate can be expanded into small shift errors. The qubit space is defined as an eigenspace of two mutually commuting displacement operators Sp and Sq which act as large shifts or translations in phase space. We propose and analyze the approximate creation of these qubit states by coupling the oscillator to a sequence of ancilla qubits. This preparation of the states uses the idea of phase estimation where the phase of the displacement operator, say Sp, is approximately determined. We consider several possible forms of phase estimation. We analyze the performance of repeated and adaptive phase estimation as the simplest and experimentally most viable schemes given a realistic upper limit on the number of photons in the oscillator. We propose a detailed physical implementation of this protocol using the dispersive coupling between a transmon ancilla qubit and a cavity mode in circuit QED. We provide an estimate that in a current experimental setup one can prepare a good code state from a squeezed vacuum state using eight rounds of adaptive phase estimation, lasting in total about 4μs, with 94% (heralded) chance of success.