Celestial Quantum Error Correction I: Qubits from Noncommutative Klein Space

Quantum gravity in 4D asymptotically flat spacetimes features spontaneous symmetry breaking due to soft radiation hair, intimately tied to the proliferation of IR divergences. A holographic description via a putative 2D CFT is expected free of such redundancies. In this series of two papers, we address this issue by initiating the study of Quantum Error Correction in Celestial CFT (CCFT). In Part I we construct a toy model with finite degrees of freedom by revisiting noncommutative geometry in Kleinian hyperk\"ahler spacetimes. The model obeys a Wick algebra that renormalizes in the radial direction and admits an isometric embedding \`a la Gottesman-Kitaev-Preskill. The code subspace is composed of 2-qubit stabilizer states which are robust under soft spacetime fluctuations. Symmetries of the hyperk\"ahler space become discrete and translate into the Clifford group familiar from quantum computation. The construction is then embedded into the incidence relation of twistor space, paving the way for the CCFT regime addressed in upcoming work.