5.1.3 An Example

We now give an example to show how Proposition 5.1.2 can be applied to quantum walks in order to compute their average running time. We consider a quantum walk on an n-circle. It can be seen as a variant of a one-dimensional quantum walk, and it is also a special case of quantum walk on a graph defined in Subsection 2.3.4.

Let Hdsi241_e be the direction space, which is a 2-dimensional Hilbert space with orthonormal basis state |L〉 and |R〉, indicating directions Left and Right, respectively. Assume that the n different positions on the n-circle are labelled by numbers 0,1,…,n − 1. Let Hp be an n-dimensional Hilbert space with orthonormal ...

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