This thesis investigates the far-from-ground-state physics of the surface code, in particular its quantum error correction applications and formulations. We contribute to this field via two lines of research: we study the behaviour of the surface code under coherent errors, which create superpositions of excited states, and we probe topological many body localization (MBL) which protects topological order for all eigenstates.
In the first strand, we develop an interpretation of the error correction threshold for coherent error rotations as a phase transition. For this, we first generalize a numerical method for the simulation of coherent errors in surface codes on square lattices to work with surface codes on general planar graphs. This method is based on a mapping to a free fermion model which allows calculating the expectation values using fermion linear optics. Using this method, we show that the connectivity of the graph can shift the error correcting performance between resilience against *X*- and *Z*-rotations.
Building on this work, we further explore the relationship between coherent errors in surface codes and free fermion models. We develop a formalism to map the surface code under coherent errors to a complex Ising model and from there to a Majorana fermion scattering model. We analyze its conductivity and find that for rotations below the error correction threshold the resulting model is an insulator, and it becomes a metal above the threshold. By estimating the position of this phase transition, we obtain the achievable error correction threshold for coherent errors.
The second line of research is focused on the disordered and perturbed toric code. We implement a recently proposed method that numerically approximates the local integrals of motion that are present in (topological) MBL phases using sets of stabilizers that are dressed by optimized quantum circuits. First, we apply this method to the disordered Kitaev chain as a benchmark. Then, we proceed by adapting it to the toric code. We show how it can be used to distinguish topological and trivial MBL and how it can be combined with exact diagonalization to obtain an approximate phase diagram.ER
