自從Grover 的量子搜尋演算法被提出至今已若干年，其優於傳統搜尋演算法的表現使得量子電腦將為人類的未來提供更多可能性。然而自此之後，在量子搜尋演算法方面的進展卻是停滯不前。對於能否找到更有效率的量子演算法我們抱持著很大的疑問。由於隨機漫步在傳統演算法的領域中受到相當多的重用，於是量子隨機漫步是否能幫助我們建構一些有效率的演算法將是今後研究量子演算法的課題之一。
有一些根據量子隨機漫步所建構的量子搜尋演算法已經被提出來，然
而在此我們主要研究的是Shenvi, Kempe 和Whaley 所提出來的在超立方體(Hypercube)上的量子隨機漫步搜尋演算法。這個方法已經被證明能夠達到與Grover 的搜尋演算法同樣層級的表現。然而，對於量子隨機漫步的性質我們了解得仍不夠清楚。於是我們就量子糾纏的量來對此量子隨機漫步搜尋演算法作分析。經過觀察，意外地發現其量子糾纏的量隨時間的變化趨勢與其得到目標的機率隨時間變化趨勢一致。於是我們以近似的方式作數學上的證明，並對此一特殊結果作解釋。另外，針對量子隨機漫步搜尋演算法，我們也提出一些在直線上作隨機漫步搜尋的嘗試。經由建構不同的錢幣運算子(coin operator)，以及不同空間維度的設計，得到的搜尋效率僅能與傳統搜尋相同，而無法得到Grover 演算法類似的優異表現。於是我們仿效在超立方體上的搜尋演算法，將超立方體與直線做一對應，希望藉由犧牲建構運算子的難易度，來減少所需要的錢幣空間維度(coin dimension)。藉由觀察量子糾纏的量來分析演算法的效率，以及在低空間維度下所嘗試的量子隨機漫步搜尋演算法，希望能為未來量子演算法的領域發揮貢獻。

Quantum random walk search algorithm has shown a quadratic speed up, which is similar
to Grover's algorithm, over classical search algorithms. However, there does not exist any
quantum search algorithm which requires fewer queries than O(pN) when searching on a
database of N items. In this thesis, we analyze the relationship between the entanglement
and the quantum random walk search algorithm. And also, we give some trials for random
walk search on a line. It appears that quantum random walks are useful and °exible tools
for designing quantum algorithms.

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