A Quantum Analog of Delsarte's Linear Programming Bounds

TL;DR

This paper generalizes classical linear programming bounds for large codes to quantum metric spaces, advancing quantum error correction theory with new bounds and methods.

Contribution

It extends Delsarte's linear programming bounds to a broad class of finite dimensional quantum metric spaces, unifying previous quantum code bounds.

Findings

Generalized bounds for quantum codes in quantum metric spaces

Unified classical and quantum code bounding methods

Enhanced understanding of quantum error correction limits

Abstract

This thesis presents results in quantum error correction within the context of finite dimensional quantum metric spaces. In classical error correction, a focal problem is the study of large codes of metric spaces. For a class of finite metric spaces that are also metric association schemes, Delsarte introduced a method of using linear programming to compute upper bounds on the size of codes. Within quantum error correction, there is an analogous study of large quantum codes of quantum metric spaces and, in the setting of quantum Hamming space, a quantum analog of Delsarte's method was discovered by Shor and Laflamme and independently by Rains. Later, Bumgardner introduced an analogous method for single-spin codes, or quantum codes related to the Lie algebra $\mathfrak{su}(2)$. The main contribution of this thesis is a generalization of the results of Shor, Laflamme, Rains, and Bumgardner to a class of finite dimensional quantum metric spaces analogous to metric association schemes of the classical case.