Cups and Gates I: Cohomology invariants and logical quantum operations

Understand

We take initial steps towards a general framework for constructing logical gates in general quantum CSS codes.

• Viewing CSS codes as cochain complexes, we observe that cohomology invariants naturally give rise to diagonal logical gates.
• We show that such invariants exist if the quantum code has a structure that relaxes certain properties of a differential graded algebra.
• We show how to equip quantum codes with such a structure by defining cup products on CSS codes.