GKP Codes: A Rosetta Stone for Quantum Error Correction

Explore the intricacies of Gottesman-Kitaev-Preskill (GKP) Codes in quantum error correction during this hour-long QuICS lecture. Delve into the concept of fault-tolerant quantum computation using GKP codes and their growing experimental utility. Examine the structure of logical Clifford gates for the GKP code and how this understanding leads to a classification of all GKP Codes. Discover the relationship between single-mode GKP codes and complex elliptic curves, and learn how smooth implementations of Clifford gates can be interpreted as homotopically non-trivial loops in their moduli space. Investigate the geometric understanding of fault-tolerance through the fiber bundle framework proposed by Gottesman & Zhang, and consider how this concept extends to other qubit- or qudit-based quantum error correcting codes via GKP code embedding. Contemplate the potential of GKP codes as a "universal cover" for stabilizer quantum error correction and their role in bridging real physics with modern mathematical developments. The lecture is based on joint work with A. Burchards and S. Flammia, as presented in arxiv:2407.03270 [quant-ph].