# Quantum Error Correction on a hexagonal lattice

I have an exercise in QEC, where I have given a hexagonal lattice with periodic boundary conditions (wrapped around a torus), with a qubit at each vertex. I have also given the Stabilizer generators $$X \otimes Y \otimes Z \otimes X \otimes Y \otimes Z$$ and $$Z \otimes Z$$ on the vertical lines of the hexagon. So how is it possible to deduce the logical operator $$X$$ and $$Z$$ on both logical qubits. Or a in a more general way, how can I find by a given lattice and stabilizer generators the logical operations?

In the lecture we have just looked at some QEC like the surface Code (and a version of the surface Code where it is wrapped around the torus) and we have put some lines through the lattice and named it the logical X and Z. If I understood it right we used some stabilizers to change some of the code-qubit values such that it represents the same logical value in the logical qubits. So how can I use this Information to create the logical X and Z operation?

Programmaticaly, you can use stim.Tableau.from_stabilizers([...], allow_underconstrained=True) to solve for the degrees of freedom not covered by the stabilizers (these are typically the logical observables). It won't give you insight into what the observable is, topologically speaking, but it can give you something valid to start from.