  PhD thesis investigating homological quantum codes derived from curved and
higher dimensional geometries. In the first part we will consider closed
surfaces with constant negative curvature. We show how such surfaces can be
constructed and enumerate all quantum codes derived from them which have less
than 10.000 physical qubits. For codes that are extremal in a certain sense we
perform numerical simulations to determine the value of their threshold.
Furthermore, we give evidence that these codes can be used for more overhead
efficient storage as compared to the surface code by orders of magnitude. We
also show how to read and write the encoded qubits while keeping their
connectivity low. In the second part we consider codes in which qubits are
layed-out according to a four- dimensional geometry. Such codes allow for much
simpler decoding schemes compared to codes which are two-dimensional. In
particular, measurements do not necessarily have to be repeated to obtain
reliable information about the error and the classical hardware performing the
error correction is greatly simplified. We perform numerical simulations to
analyze the performance of these codes using decoders based on local updates.
We also introduce a novel decoder based on techniques from machine learning and
image recognition to decode four-dimensional codes.
