The realization of quantum computers holds the potential to bring revolutionary advancements in fields such as pharmaceuticals, material development, sensing, logistics, and optimization. However, quantum computers face significant technical barriers in enhancing noise resistance and scaling up.

Consequently, programs must be compiled into sequences of gates under strict constraints on circuit size and inter-chip communication bandwidth. Thus, optimizing the compilation is crucial, as it can dramatically increase execution speed and feasible program size. Nevertheless, achieving optimization or identifying its limitations is a computationally challenging problem since it essentially requires considering all possible compilation methods.

This talk will introduce quantum information theory, which axiomatically handles the quantum phenomena, and explain how compilation optimization problems are reduced to mathematical problems. Additionally, we will present examples of how mathematics, such as number theory and algebraic geometry, is utilized to solve these problems.