Abstract: In recent years, the use of Gottesman-Kitaev-Preskill (GKP) Codes to implement fault-tolerant quantum computation has gained significant traction and evidence for their experimental utility has steadily grown. But what does it even mean for quantum computation with the GKP code to be fault tolerant? In this talk, we discuss the structure of logical Clifford gates for the GKP code and how their understanding leads to a classification of the space of all GKP Codes. For GKP codes in a single mode, we explain their relationship to complex elliptic curves and how smooth implementations of Clifford gates can be understood as homotopically non-trivial loops in their moduli space. This connection establishes a geometric understanding of fault-tolerance for quantum computation via the fiber bundle framework for fault tolerance proposed by Gottesman & Zhang and I argue how this understanding can be extended to other qubit- or qudit based quantum error correcting codes via an embedding provided by the GKP code. As ``universal cover´´ for stabilizer quantum error correction, I speculate how quantum error correction with the GKP can serve to establish a new bridge between real physics and modern developments in mathematics. This talk will be based on joint work with A. Burchards and S. Flammia presented in arxiv:2407.03270 [quant-ph] .
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