PhD thesis: Homological Quantum Codes Beyond the Toric Code

TL;DR

This thesis explores advanced homological quantum codes derived from curved and higher-dimensional geometries, demonstrating their potential for more efficient quantum storage and simplified decoding compared to traditional surface codes.

Contribution

It introduces new quantum codes from negatively curved surfaces and four-dimensional geometries, with analysis of their thresholds, decoding methods, and advantages over existing codes.

Findings

Codes from negatively curved surfaces can outperform surface codes in overhead efficiency.

Four-dimensional codes enable simpler, non-repetitive measurement-based decoding.

A novel machine learning decoder effectively decodes four-dimensional quantum codes.

Abstract

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.