Quantum walks on graphs and quantum scattering theory
Quantum walks on graphs and quantum scattering theory
复制标题
图上的量子行走和量子散射理论
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
M. Hillery
中科院分区：
文献类型：
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作者：
E. Feldman;M. Hillery
我们讨论了一般图上的一种特殊的量子行走。我们把两条半无限长的线附加到一个一般的有限图上，我们称之为尾。在尾巴上，行走的粒子在每个时间步前进一个单位，因此它的行为类似于自由传播我们感兴趣的是粒子从一条尾巴开始并通过图传播（其中它的传播不是自由的）需要多少步才能出现在另一条尾巴上。在n步中进行这种行走的概率和这种行走的命中时间可以用图的传输幅度来表示，这是其S矩阵的一个元素。要证明这一点，就必须研究图的透射和反射振幅的解析性质。我们表明，该图可以有绑定状态，不能由一个粒子进入图从它的尾巴之一。定义了量子行走的时间反转不变性，并用于表明如果行走是时间反转不变性，则从不同方向进入图的粒子的传输幅度相同。
We discuss a particular kind of quantum walk on a general graph. We affix two semi-infinite lines to a general finite graph, which we call tails. On the tails, the particle making the walk simply advances one unit at each time step, so that its behavior there is analogous to free propagation We are interested in how many steps it will take the particle, starting on one tail and propagating through the graph (where its propagation is not free), to emerge onto the other tail. The probability to make such a walk in n steps and the hitting time for such a walk can be expressed in terms of the transmission amplitude for the graph, which is one element of its S matrix. Demonstrating this necessitates a study of the analyticity properties of the transmission and reflection amplitudes of a graph. We show that the graph can have bound states that cannot be accessed by a particle entering the graph from one of its tails. Time-reversal invariance of a quantum walk is defined and used to show that the transmission amplitudes for the particle entering the graph from different directions are the same if the walk is time-reversal invariant.