Symmetry-Resolved Parent Hamiltonians for Entangled Bosonic Cat Resources

We derive parent Hamiltonians in terms of oscillator operators for multimode bosonic cat resource states. The construction separates a universal branch Hamiltonian, which confines each mode to the coherent-state support $|\pm\alpha\rangle$, and state-dependent constraint Hamiltonians, which select the desired correlations and symmetry sectors inside the resulting branch manifold. This framework progressively removes degeneracies in the low-energy manifold and yields explicit parent Hamiltonians for GHZ-, cluster-, and W-type cat states. In the large-$|\alpha|$ limit, the bosonic parent Hamiltonians reduce to stabilizer or exchange Hamiltonians acting on an effective logical-qubit basis. The present construction provides a direct bridge between coherent-state bosonic engineering and stabilizer-based quantum information processing.