The locality of quantum codes

Abstract

This thesis concerns itself with the role locality plays in the performance of quantum codes. We first address the difficulty of producing codestates from relatively good codes, specifically we investigate the depth of geometrically local encoding circuits. Previous results struggled to produce bounds allowing for general, non-local classical communications in the circuit. We show that, owed to the entanglement hidden in codestates, the depth of encoding circuits is lower bounded, even when allowing for boundless classical computation. We then move on to address a converse question: given a particular code, what can we say about its geometrical structure? We show that implementing stabiliser codes requires satisfying some connectivity requirements that depend on k and d. We quantify these requirements through the notion of graph expansion. Finally, in the last chapter, we ask the question of the achievability of these requirements. The celebrated BPT bound states upper bound on the parameters that can be achieved for codes local in three dimensions. We provide the first construction that matches this bound.