拓扑量子计算？- 打结和熔化焊剂管
Towards Topological Quantum Computation? - Knotting and Fusing Flux Tubes
摘要：
拓扑量子计算的模型是基于（2+1）-D中任意子（分数统计的准粒子）的编织和融合。物理理论中存在的任意子是由哈密顿量的对称群决定的。在哈密顿量经历全对称群G到有限剩余规范群H的自发对称性破缺的情况下，粒子由子群的量子偶$D(H)$的表示给出。将Drinfeld的量子二重构造应用于有限群H上的函数代数F（H），得到了拟三角Hopf代数D（H）.这项工作的一个主要新贡献是用MAGMA编写的程序，用于计算可以存在于具有任意有限剩余规范群的系统中的粒子（及其性质-包括自旋），以及这些粒子的编织和融合规则。我们显式地计算了两个非阿贝尔群偶的融合规则：$S_3$和$A_5$，并在表中发现了一些有趣的结果、子系统和对称性. SO（3）_4（Chern-Simons理论$SU(2)_4$的限制）及其镜像被发现是8粒子$S_3$量子偶中的3粒子子系统。这些表证明了$S_3$和$A_5$任意子都是马约拉纳的，但这不是所有有限群的情况。在附录中，对SO（3）的其余非交换子群-- $S_4$、$A_4$和$D_4$（无限族$D_n$中的第二个）--的量子双数进行了列表和分析。此外，在量子计算应用中获得任何给定的融合产物的概率被确定并在MAGMA中编程。在整个过程中，提到了与可能的实验的联系。
英文摘要：
Models for topological quantum computation are based on braiding and fusing anyons (quasiparticles of fractional statistics) in (2+1)-D. The anyons that can exist in a physical theory are determined by the symmetry group of the Hamiltonian. In the case that the Hamiltonian undergoes spontaneous symmetry breaking of the full symmetry group G to a finite residual gauge group H, particles are given by representations of the quantum double $D(H)$ of the subgroup. The quasi-triangular Hopf Algebra D(H) is obtained from Drinfeld's quantum double construction applied to the algebra F(H) of functions on the finite group H. A major new contribution of this work is a program written in MAGMA to compute the particles (and their properties - including spin) that can exist in a system with an arbitrary finite residual gauge group, in addition to the braiding and fusion rules for those particles. We compute explicitly the fusion rules for two non-abelian group doubles suggested for universal quantum computation: $S_3$ and $A_5$, and discover some interesting results, subsystems, and symmetries in the tables. SO(3)_4 (the restriction of Chern-Simons theory $SU(2)_4$) and its mirror image are discovered as 3-particle subsystems in the 8-particle $S_3$ quantum double. The tables demonstrate that both $S_3$ and $A_5$ anyons are all Majorana, but this is not the case for all finite groups. In the appendices, the quantum doubles for the remaining nonabelian subgroups of SO(3) - $S_4$, $A_4$, and $D_4$ (the second in the infinite family $D_n$) - are tabulated and analyzed. In addition, the probabilities of obtaining any given fusion product in quantum computation applications are determined and programmed in MAGMA. Throughout, connections to possible experiments are mentioned.
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