We present a novel methodological framework for quantum spatial search, generalising the Childs & Goldstone () algorithm via alternating applications of marked-vertex phase shifts and continuous-time quantum walks. We determine closed form expressions for the optimal walk time and phase shift parameters for periodic graphs. These parameters are chosen to rotate the system about subsets of the graph Laplacian eigenstates, amplifying the probability of measuring the marked vertex. The state evolution is asymptotically optimal for any class of periodic graphs having a fixed number of unique eigenvalues. We demonstrate the effectiveness of the algorithm by applying it to obtain search on a variety of graphs. One important class is the n n ³ rook graph, which has N = n ⁴ vertices. On this class of graphs the algorithm performs suboptimally, achieving only overlap after time . Using the new alternating phase-walk framework, we show that overlap is obtained in phase-walk iterations.

我们提出了一种新的量子空间搜索方法论框架，通过标记顶点相移和连续时间量子游走的交替应用来概括 Childs & Goldstone ( ) 算法。我们确定周期图的最佳游走时间和相移参数的封闭形式表达式。选择这些参数以围绕图拉普拉斯本征态的子集旋转系统，从而放大测量标记顶点的概率。对于任何具有固定数量的唯一特征值的周期图，状态演化是渐近最优的。我们通过应用它来获得对各种图的搜索来证明该算法的有效性。一个重要的类是n n ³车图，它有N = n ⁴个顶点。在此类图上，该算法的执行效果欠佳，仅在 time 之后实现重叠。使用新的交替相位游走框架，我们表明在相位游走迭代中获得了重叠。