Quantum Lunch: Ground states and excitations of homological codes
Speaker: Guest Professor Péter Vrana from QMATH
Ground states and excitations of homological codes
Stabilizer codes with a distinguished set of generators of the stabilizer subgroup give rise to a Hamiltonian such that its ground state space is precisely the code space. Small errors can then be seen as low-energy excited states over the ground state. Homological codes form a special class of stabilizer codes, and can be constructed using a chain complex together with a distinguised basis. The main example is Kitaev's toric (or surface) code, which admits elementary excitations which are abelian anyons, and has a ground state degeneracy depending only on the topology of the surface.
We study homological codes arising from more general spaces, which can similarly be seen as quantum systems with local interactions with respect to the underlying geometry. We find that the set of frustration free ground states as well as some excited states can be understood in terms of proper homotopy invariants of the space.