Spectral Codes : A Geometric Formalism for QEC

In this talk, I will introduce a novel geometric framework for quantum error correction based on spectral triples in noncommutative geometry. In this formulation, quantum error-correcting codes are described as spectral projections onto the low-energy eigenspaces of Dirac-type operators, where the separation between logical information and local errors is captured geometrically. This approach provides a unified spectral and geometric understanding of key properties such as code distance and error thresholds. Moreover, it accommodates various existing codes, including classical linear codes, stabilizer codes, GKP codes, and topological codes. This geometric perspective also suggests intriguing connections to deformation quantization and holographic quantum error correction, offering promising directions for future research.