矩阵值Szego多项式与量子随机游动
Matrix valued Szego polynomials and quantum random walks
摘要：
我们考虑整数上的量子随机游走（QRW），这是过去几年在量子计算框架中考虑的一个主题。我们将展示如何CMV矩阵理论提供了一个自然的工具来研究这些过程，并给出类似的结果，卡林和麦格雷戈开发的研究（经典）出生和死亡过程中使用正交多项式的真正的路线。在完美的类比与经典的情况下，研究QRW的一组非负整数可以处理使用标量值（洛朗）多项式和标量值的措施在圆上。在整数上的经典或量子随机游动的情况下，需要考虑这些概念的矩阵值版本。我们展示了我们的工具如何在著名的Hadamard行走的情况下产生结果，但是我们超越了这个平移不变模型来分析使用其他方法难以分析的例子。更确切地说，我们考虑非负整数集合上的QRW.对这些情况的分析导致了在整数上的QRW的情况下不存在的现象，即使一个人将自己限制在一个常数硬币上。这里通过研究行走的递归性质来说明这一点，但同样的方法可以用于其他目的。这里的介绍旨在自成一体，但我们不试图介绍量子随机游动，这是我们引用的文献中调查过的一个主题。对于两个优秀的评论，见[1，18]。另见最近的注释[19]。
英文摘要：
We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation. We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth-and-death processes using orthogonal polynomials on the real line. In perfect analogy with the classical case the study of QRWs on the set of non-negative integers can be handled using scalar valued (Laurent) polynomials and a scalar valued measure on the circle. In the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions. We show how our tools yield results in the well known case of the Hadamard walk, but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods. More precisely we consider QRWs on the set of non- negative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes. The presentation here aims at being selfcontained, but we re- frain from trying to give an introduction to quantum random walks, a subject well surveyed in the literature we quote. For two excellent reviews, see [1, 18]. See also the recent notes [19].
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