Holomorphic Quantum Error Correction Codes

We develop a holomorphic representation of quantum error correction codes (QECCs) within the Segal--Bargmann space. By encoding qubits into Schwinger boson modes (z a j, z b j) subject to a degree-one homogeneity constraint, we derive closed-form differential operator representations for stabilizers, syndrome extraction, and recovery for fundamental codes (three-, five-, seven-, and nine-qubit codes). Quantum errors are characterized as holomorphic perturbations violating this constraint, while syndrome measurement projects onto eigenspaces of commuting differential operators. Restricting to unit-magnitude variables (z=1) reveals a toroidal space 2n where error syndromes manifest as discrete translations in winding number space 2n, and recovery acts as Hamiltonian flows restoring the winding configuration. In the full Segal--Bargmann space, the code space is a holomorphic submanifold of 2 n-1, with correctable errors as transverse normal directions. Consequently, the Knill--Laflamme condition becomes a Fubini--Study orthogonality condition between the code submanifold and its error-translated images. Topological protection emerges from the U(1) n fiber bundle structure: global phase noise along fibers is unobservable, while base-space errors require active correction. Finally, we establish a path-integral formulation for semiclassical error correction dynamics and show that geometric entanglement via the Segre embedding naturally quantifies code distance. This framework unifies algebraic, geometric, and topological perspectives on fault-tolerant quantum protocols.