拓樸量子修正碼

Topological Quantum Code

徐溍玢(Chin-Bin Hsu)

指導教授 ： 謝銘倫

國立臺灣大學/理學院/數學研究所/碩士(2022年)

https://doi.org/10.6342/NTU202201694

摘要

眾所皆知，量子電腦的運算能力與速度均優於傳統電腦，因此在電腦科學的發展中佔有舉足輕重的地位。然而，由於粒子的不穩定性，錯誤修正也成為量子演算法中十分關鍵的一環，且與傳統的錯誤修正不同，量子位元的高自由度使得錯誤修正的難度上升。1996年Gottesman以穩定子的方式定義量子修正碼，給出了量子糾錯碼的純數學構造，隨後Kitaev於1997年使用拓樸的概念發明環面碼，開創了拓樸量子修正碼的研究，後來不斷被推廣與優化。本文我們將對量子錯誤修正進行完整的介紹，並著重於拓樸量子修正碼使用的數學工具。

關鍵字

量子科技 ； 錯誤修正 ； 拓樸 ； 圖論 ； 同調代數

並列摘要

Quantum computer is known of being more efficient in computation than classical computer, and hence plays a crucial role in computer science. However, due to the instability of physical particles, error correction becomes an important issue when running quantum algorithms. In contrast to classical error correction, quantum error correction has its difficulty because of the freedom of quantum bits. In 1996, Gottesman invented the stabilizer formalism , which established a purely mathematical construction for quantum codes. In 1997, Kitaev first used topological ideas to construct the toric code, which then generalized to the theory of topological quantum codes. In this article we will introduce various constructions of quantum error correction codes and emphasize mainly on the mathematical idea.

並列關鍵字

Quantum Mechanics ； Error Correction ； Topology ； Graph Theory ； Homological Algebra

參考文獻

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