Starting from an association scheme induced by a finite group and the corresponding Bose–Mesner algebra, we construct quantum Markov chains, their entangled versions, using the quantum probabilistic approach. Our constructions are based on the intersection numbers and their duals Krein parameters of the schemes. We make the connection for the first time between the fusion rules of anyonic particles evolving on a 2D surface to the Krein parameters of an association scheme. We consider braid group \(B_3\) 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 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 yet another perspective to topological computation, whereas frameworks such as Unitary Modular Categories can identify and characterize new anyonic systems our framework can build upon them within quantum probabilistic framework that are suitable for asymptotic analysis.

从由有限群和相应的Bose-Mesner代数引起的关联方案开始，我们使用量子概率方法构造量子马尔可夫链，它们的纠缠形式。我们的构造基于方案的交叉点编号及其对偶Kerin参数。我们首次在二维表面上演化的非离子粒子的融合规则与关联方案的Kerin参数之间建立了联系。我们考虑编织组\（B_3 \）该图将Anyon的单位动力学描述为图的自同构子组。融合（和伴随的分裂操作）引起的动力学可以看作是链在增长的图上演化，而编织则看作是在固定图上的自同构。在我们的量子概率框架中，如果允许粒子长时间连贯地演化，则unit的无限迭代可以为量子模拟编码算法内容，可以优雅地描述渐近现象。我们在Bose-Mesner代数（既是von Neumann代数又是Frobenius代数）上定义量子态，以建立量子马尔可夫链，从而为拓扑计算提供了另一个视角，