Fiber Bundle Fault Tolerance of GKP Codes

TL;DR

This paper explores the geometric structure of multi-mode GKP quantum error-correcting codes, revealing how logical Clifford operations correspond to topological features of the code space.

Contribution

It constructs the moduli space of GKP codes as a fiber bundle and proves the Gottesman--Zhang conjecture linking Clifford gates to parallel transport in this space.

Findings

GKP code moduli space is a fiber bundle over symplectically integral lattices.

Logical Clifford gates correspond to non-contractible loops in the moduli space.

Logical identity operations correspond to contractible paths, confirming the conjecture.

Abstract

We investigate multi-mode GKP (Gottesman--Kitaev--Preskill) quantum error-correcting codes from a geometric perspective. First, we construct their moduli space as a quotient of groups and exhibit it as a fiber bundle over the moduli space of symplectically integral lattices. We then establish the Gottesman--Zhang conjecture for logical GKP Clifford operations, showing that all such gates arise from parallel transport with respect to a flat connection on this space. Specifically, non-trivial Clifford operations correspond to topologically non-contractible paths on the space of GKP codes, while logical identity operations correspond to contractible paths.