离散时间量子行走和二叉树上对称行走的框架
Framework for discrete-time quantum walks and a symmetric walk on a binary tree
摘要：
我们制定了一个框架，离散时间量子行走，经典的记忆随机行走的动机。我们提出了一个具体的代表性的经典步行与内存2这是基于。该框架不需要硬币空间，除了酉性之外，它对演化算子没有任何约束，并且是其他方法的统一。作为例子，我们在半无限二叉树上构造了一个对称的离散量子行走。根处的振幅的生成函数以封闭形式计算，作为时间和树中初始水平n的函数，并且我们找到振幅的渐近解和完整的数值解。它表现出一个尖锐的干扰峰和幂律尾，而不是指数衰减的尾巴的广泛峰值分布的经典对称随机游走的二叉树。概率峰值比经典行走的概率峰值大几个数量级（已经在小n处）。量子行走显示了一个多项式算法的加速比n的经典行走，我们推测是2/3的顺序，基于数据的强烈趋势。
英文摘要：
We formulate a framework for discrete-time quantum walks, motivated by classical random walks with memory. We present a specific representation of the classical walk with memory 2 on which this is based. The framework has no need for coin spaces, it imposes no constraints on the evolution operator other than unitarity, and is unifying of other approaches. As an example we construct a symmetric discrete-time quantum walk on the semi-infinite binary tree. The generating function of the amplitude at the root is computed in closed-form, as a function of time and the initial level n in the tree, and we find the asymptotic and a full numerical solution for the amplitude. It exhibits a sharp interference peak and a power law tail, as opposed to the exponentially decaying tail of a broadly peaked distribution of the classical symmetric random walk on a binary tree. The probability peak is orders of magnitude larger than it is for the classical walk (already at small n). The quantum walk shows a polynomial algorithmic speedup in n over the classical walk, which we conjecture to be of the order 2/3, based on strong trends in data.
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