Geometric aspects of fault tolerance

We highlight three geometric aspects of fault tolerance. First, we construct an operator-algebraic formulation of the fiber bundle framework for fault tolerance, and show that a generalization of the Knill-Laflamme conditions implies a flat connection for this framework. Next, we develop this framework for Gottesman-Kitaev-Preskill (GKP) codes. We show that, for each polarization type, their moduli space embeds smoothly into a Grassmannian of finite-dimensional subspaces of a regularized distribution space. Finally, we explain how GKP codes associated with Jacobians give rise to magic states that can be obtained by code space projections of Gaussian states and identify a $\rm CCZ$-type magic state arising this way.