Generalized Gottesman-Kitaev-Preskill States on a Quantum Torus

Abstract: We introduce a novel formulation of a Generalized Gottesman-Kitaev-Preskill (GKP) state that resolves all of its foundational pathologies, such as infinite energy, non-normalizability, and or- thogonality. We demonstrate that these issues are artifacts of defining the code on an unbounded phase-space. By considering the compact quantum-phase-space intrinsic to systems like coupled quantum harmonic oscillators, we have obtained a Generalized GKP (GGKP) state that is both exact and physically realizable. This is achieved by applying an obtained Quantum Zak Transform (QZT) to Squeezed Coherent States, which reveals that Riemann-Theta functions are the natural representation of these states on the quantum torus phase-space. This framework not only provides a well-behaved quantum state, but also reveals a deep connection between quantum error correc- tion, non-commutative geometry, and the theory of generalized Theta functions. This opens a new avenue for fault-tolerant photonic quantum computing.