Deterministic Generation of Arbitrary Fock States via Resonant Subspace Engineering

Deterministic preparation of high-excitation Fock states is a central challenge in bosonic quantum information, with control complexity that generically explodes as the Hilbert space dimension grows. Here we introduce resonant subspace engineering (RSE), a protocol that analytically confines the infinite-dimensional bosonic dynamics to a two-dimensional invariant subspace spanned by an initial coherent state and the target state. State transfer then reduces to a geodesic rotation on a synthetic Bloch sphere, governed by resonance and phase-matching conditions we derive in closed form. For single Fock states, RSE achieves O\(n^1/4\) scaling in both evolution time and gate depth, showing a fundamental improvement over existing deterministic schemes. The construction generalizes to K-component superpositions via a (K{+}1)-dimensional invariant subspace with full SU(K{+}1) controllability, requiring only 3-5 iterations of operations for superpositions spanning photon numbers 70--100. RSE provides a scalable and analytically transparent framework for large-scale bosonic state engineering and gate synthesis across single- and multimode platforms.