Studies on topological quantum computation

纠缠熵是拓扑量子计算中的特征量之一。纠缠熵，特别是拓扑纠缠熵，与结点和3-流形的量子不变量密切相关。我们研究了结点和连杆的有色琼斯多项式。对环面结的有色琼斯多项式进行了精确分析，得到了环面结与经典拓扑不变量（如扭量）的显式关系。我们还研究了模形式的傅里叶展开式，得到了它们的渐近特性与复流形熵的关系。

Entanglement entropy is one of the characteristic quantities in the topological quantum computation. The entanglement entropy, especially topological entanglement entropy, is intimately related to the quantum invariants of knots and 3-manifolds. We studied the colored Jones polynomials for knots and links. We exactly analyzed the colored Jones polynomial for torus knots, and we obtained explicit relationship with classical topological invariants such as torsions. We also studied the Fourier expansion of modular forms, and obtained a relationship between their asymptotic behavior and the entropy of complex manifolds.