Homological quantum codes beyond the toric code
Nikolas P. Breuckmann · RWTH Publications (RWTH Aachen) · 2017
Computer architectures which exploit quantum mechanical effects can solve computing tasks that are otherwise impossible to perform. A quantum computer operates on a number of small quantum mechanical systems, known as quantum bits, or qubits. Since these systems are realized on the scale of atoms, they are very prone to errors. Errors occur when the environment interacts with the qubits, a process called decoherence. It is widely accepted that it will not be possible to shield qubits completely from the outside world. If one were to perform a quantum computation on the qubits directly, then after a short period of time the information present in the qubits would be lost. To counter decoherence the state of a qubit can be encoded into multiple physical ones. This is called a quantum error correcting code. Performing quantum error correction allows one to extend the life time of the encoded qubit arbitrarily, assuming that the rate of errors remains below a certain threshold value. The use of quantum codes creates an overhead in resources, as for every logical qubit many more physical qubits are needed. The resource overhead for fault-tolerance is problematic, since realizing qubits will be costly, and in the early stages of building quantum computers the number of physical qubits will be limited. The currently favored coding architecture is the toric code and its variant the surface code in which the physical qubits are put on a square grid in which interactions are only between nearest neighbors. In this thesis we will explore quantum codes in which qubits interact as if they were nearest neighbors in more exotic spaces. 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.