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In this talk, we will explore the mathematical problems in two major fields of quantum technology—quantum computing and quantum information. These issues represent breakthrough challenges of the quantum era and reveal mathematics’s profound impact on quantum technology. In quantum computing, we will first introduce the basic concepts of quantum computation, focusing on one of the most representative algorithms—Shor’s Algorithm. We will explain how this algorithm achieves prime factorization on a quantum computer and the challenges encountered in practical implementation. In quantum information, we will introduce the core concept of quantum channels. Quantum channels describe how quantum states are transferred between quantum systems, involving the evolution of quantum states. We will then discuss the “inversion problem” within quantum channels, namely how to mathematically define and solve for the reverse operation of a quantum channel. This issue is important in quantum communication and entanglement studies and relates to deeper mathematical problems such as reversibility and stability.

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