A Sparse $Z_2$ Chain Complex Without a Sparse Lift

TL;DR

The paper constructs a three-degree $ ext{Z}_2$ chain complex from a cellulation of $RP^3 imes[0,1]$ to produce a LDPC quantum code that provably lacks a sparse lift to $ ext{Z}_4$, answering a previously open question. By formulating the lift problem as solving a sparse $ ext{Z}_2$-linear equation $E + oldsymbol{ abla}_Q oldsymbol{ abla}_Z + oldsymbol{ abla}_Z oldsymbol{ abla}_Q=0$ and exhibiting a concrete obstruction, the authors show that no sparse $oldsymbol{ abla}_Z,oldsymbol{ abla}_Q$ exist for their construction, hence no sparse lift to integers. They further discuss the role of non-orientability in obstructing sparse lifts and present a one-dimensional locality result: any $ ext{Z}_2$ chain complex local in 1D admits a local-in-1D lift to $ ext{Z}$ with no torsion and preserved Betti numbers, via a disentangling quantum circuit. The findings illuminate fundamental limits in reverse engineering LDPC quantum codes from sparse integer lifts and connect topological properties to liftability constraints.

Abstract

We construct a sparse $Z_2$ chain complex (with three different degrees, so that it corresponds to a quantum code) which does not admit a sparse lift to the integers, answering a question in Ref. 1.