Lifting a CSS code via its handlebody realization

TL;DR

This paper introduces a topological method for lifting quantum CSS codes using handlebody realizations, providing an alternative to the Tanner cone-complex approach and applying it to classify lifts of hypergraph-product codes.

Contribution

It demonstrates how handlebody realizations can be used for code lifting, showing equivalence to existing methods and enabling classification of certain code lifts.

Findings

Handlebody realization offers a new perspective for code lifting.

The approach is equivalent to the Tanner cone-complex method.

Classification of lifts of hypergraph-product codes achieved.

Abstract

We present a topological approach to lifting a quantum CSS code. In previous work, we proposed lifting a CSS code by constructing covering spaces over its 2D simplicial complex representation, known as the Tanner cone-complex. This idea was inspired by the work of Freedman and Hastings, which associates CSS codes with handlebodies. In this paper, we show how the handlebody realization of a code can also be used to perform code lifting, and we provide a more detailed discussion of why this is essentially equivalent to the Tanner cone-complex approach. As an application, we classify lifts of hypergraph-product codes via their handlebody realization.