UC Davis

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 called 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 $\su(2)$. 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 association schemes of the classical case. This arguably gives a quantum analog of Delsarte's linear programming bounds for association schemes.

In Chapter 1, we first review classical error correction through metric spaces. We then review the mathematical framework of quantum probability, quantum operations, and quantum error correction. In Chapter 2, we review the notion of quantum metrics introduced by Kuperberg and Weaver, which play a role in quantum error correction analogous to metrics in classical error correction. Mathematically motivating examples of quantum metrics arising from the representation theory of Lie algebras are presented. We also present examples of new quantum codes for some of these quantum metrics. In Chapter 3, we present our main result, which is a method of using linear programming to compute upper bounds on the dimension of quantum codes. This method is valid for a class of finite quantum metric spaces that satisfies the conditions of being multiplicity-free and 2-homogeneous. We also present a secondary result that strengthens the bounds when the quantum metric exhibits the property of self-duality. This result is a generalization of Rains' quantum shadow enumerators for binary quantum Hamming space. Lastly, we derive formulas for different families of discrete orthogonal functions needed to compute the linear programming bounds for the quantum metrics presented in Chapter 2.