图上拓扑演化的量子概率空间
Quantum Probabilistic Spaces on Graphs for Topological Evolutions
摘要：
我们首先考虑任意子粒子在二维表面上的融合规则，然后利用超群来构造纠缠量子马尔可夫链。融合规则导出了一种基于Krein参数和交叉数的关联方案.一种有用的方式是将方案视为编码可能的量子行走（自同构）路径的正则图。我们考虑描述任意子的幺正动力学的辫子B3作为图的自同构子群。由融合（和伴随分裂操作）引起的动力学可以被看作是在生长图上进化的链和在固定图上自同构的编织。在我们的量子概率框架中，如果允许粒子在更长的时间内相干地演化，那么可以为量子模拟编码算法内容的幺正的无限次迭代可以优雅地描述渐近性。我们将定义量子态的玻色-梅斯纳代数，这也是一个冯诺依曼代数以及Frobenius代数建立量子马尔可夫链提供另一个角度的拓扑计算。
英文摘要：
We start with the consideration of fusion rules of anyonic particles evolving on a 2D surface and the a hypergroup comes with it to construct entangled quantum Markov chains. The fusion rules induce an association scheme with Krein parameters and their duals the intersection numbers. One useful way to think of the schemes as regular graphs encoding the paths of possible quantum walks (automorphisms). We consider braid B3 that describes the unitary dynamics of the anyons as the automorphism subgroup of the graphs. The dynamics induced by the fusions (and the adjoint splitting operations) may be viewed as the chain evolving on a growing graph and the braiding as automorphisms on a fixed graph. In our quantum probability framework infinite iterations of the unitaries, which can encode algorithmic content for quantum simulations, can describe asymptotics elegantly if the particles are allowed to evolve coherently for a longer period. We will define quantum states on the Bose-Mesner algebra which is also a von Neumann algebra as well as a Frobenius algebra to build the quantum Markov chains providing another perspective to topological computation.
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