Stabilizer-based verification of QFHE T -gadget state

Quantum fully homomorphic encryption (QFHE) relies on specially prepared gadget states to handle non-Clifford gates, so the correctness of these resource states is essential to the correctness of the overall encrypted computation. In this paper, we study the correctness and verification of QFHE T-gadget-state preparation using the stabilizer formalism. We derive explicit stabilizer generators for the T-gadget state and prove that, for a fixed gadget-state parameter, these generators uniquely characterize the target state up to a global phase. This yields a correctness-equivalence theorem: a prepared gadget state is correct if and only if it satisfies all corresponding stabilizer-generator conditions. Based on this result, we propose a stabilizer-based verification protocol with a cut-and-choose strategy, prove completeness, and establish a batch-level soundness bound based on a worst-case per-gadget detection probability of at least 1/2. Finally, a Qiskit simulation of a toy T-gadget state with an intentionally introduced Pauli error illustrates how a preparation error can be detected through a stabilizer-sign mismatch.