Holstein Primakoff spin codes for local and collective noise

Quantum error correction is essential for fault-tolerant quantum computation, yet most existing codes rely on local control and stabilizer measurements that are difficult to implement in systems dominated by collective interactions. Inspired by spin-GKP codes in PhysRevA.108.022428, we develop a general framework for Holstein-Primakoff spin codes, which maps continuous-variable bosonic codes onto permutation-symmetric spin ensembles via the Holstein-Primakoff approximation. We show that HP codes are robust to both collective and local-spin noise and propose an explicit measurement-free local error recovery procedure to map local noise into correctable collective-spin errors.