Fiber Bundle Fault Tolerance of GKP Codes

Fault tolerance is a notion of fundamental importance to the field of quantum information processing. It is one of the central properties a quantum computer must possess in order to enable the achievement of large scale practical quantum computation. While a widely used, general, and intuitive concept, within the literature the term fault tolerant is often applied to specific procedures in an ad-hoc fashion tailored to details of the context or platform under discussion. A more unifying perspective has been proposed by Gottesman and Zhang who conjecture that all types of fault-tolerant gates can be regarded as essentially topological in nature, arising from parallel transport with respect to a flat connection on a space of error correcting codes.

In this talk I will discuss multi-mode GKP (Gottesman--Kitaev--Preskill) quantum error-correcting codes from a geometric perspective, establishing the Gottesman-Zhang conjecture for the case of Clifford operations. First I will show how to construct the space of all GKP codes and how this space relates to the space of symplectic lattices. I will then show that GKP Clifford operations arise from parallel transport with respect to a flat connection on the space of GKP codes. 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.

Pizza and drinks will be served after the seminar in ATL 2117.