Engineered Kerr Nonlinearities for Precise Quantum Control of Fock States

We present a practical design framework for high-fidelity quantum control in coupled Kerr-nonlinear oscillators, directly addressing the challenge of spectral crowding. We show that systematic spectral degeneracies, which hinder selective addressing, are a direct consequence of rational Kerr-nonlinearity ratios $K₁/K₂$. Our solution is a universal architectural principle: engineer this ratio to be a complex rational value, approximating an incommensurate number to systematically eliminate parasitic resonances. Using a Magnus expansion, we derive a complete effective Hamiltonian, including all Stark-shift corrections, to accurately target transitions. We numerically validate this framework by demonstrating protocols for the deterministic synthesis of NOON states, and high-photon-number Fock states e.g., $n=4$, achieving ideal fidelities exceeding mathcal{F}>99.9\%. The protocols are shown to be robust against environmental decay and thermal effects. This work provides an architectural blueprint for bosonic processors in circuit QED and establishes foundational principles that could inform future designs of multi-mode quantum systems.