Selective number-dependent arbitrary Hamiltonian engineering for a cavity

Cavity resonators are promising resources for storing and processing quantum information. Here we investigate a scheme to engineer the Hamiltonian for a photonic cavity using an ancilla qubit. In the strong dispersive coupling limit and number-split regime, one can drive the qubit near selective photon-number-dependent transition frequencies to address the individual photon number states of the cavity. By choosing control driving detunings much larger than the driving strengths, we propose a general approach to engineering a selective number-dependent arbitrary Hamiltonian for the cavity. The engineered Hamiltonian admits various applications including canceling unwanted cavity Kerr effect, creating higher-order nonlinearities for quantum simulations, designing quantum gate operations resilient to noise, and even realizing quantum error correction. Our scheme can be implemented with a coupled microwave cavity and transmon qubit in superconducting circuits systems.