Span-program-based quantum algorithm for evaluating unbalanced formulas

基于生成程序的公式求值量子算法

Span-program-based quantum algorithm for evaluating formulas

我们给出了一个量子算法，用于在扩展的门集合上评估公式，包括所有的两位和三位二进制门（例如，NAND，3-多数）。该算法是最佳的只读一次公式，每个门的输入在一定意义上平衡。主要的新工具是一个经典的线性代数计算模型之间的对应关系，“跨度程序”，和加权二分图。跨度程序的评估对应于相关联的图的特征值为零的特征向量。因此，量子计算机可以通过将谱估计应用于图来评估跨度程序。例如，评估平衡三元多数公式的经典复杂性是未知的，并且随机化α-β修剪的自然推广已知是次优的。相比之下，我们的算法推广了最佳量子AND-OR公式评估算法，是最佳的平衡三进制多数公式的评估。

Faster quantum algorithm for evaluating game trees

We give an O(sqrt n log n)-query quantum algorithm for evaluating size-n AND-OR formulas. Its running time is poly-logarithmically greater after efficient preprocessing. Unlike previous approaches, the algorithm is based on a quantum walk on a graph that is not a tree. Instead, the algorithm is based on a hybrid of direct-sum span program composition, which generates tree-like graphs, and a novel tensor-product span program composition method, which generates graphs with vertices corresponding to minimal zero-certificates. For comparison, by the general adversary bound, the quantum query compl…

Span programs and quantum query complexity: The general adversary bound is nearly tight for every boolean function

The general adversary bound is a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. We turn this lower bound into an upper bound, by giving a quantum walk algorithm based on the dual SDP that has query complexity at most the general adversary bound, up to a logarithmic factor. In more detail, the proof has two steps, each based on "span programs," a certain linear-algebraic model of computation. First, we give an SDP that outputs for any boolean function a span program computing it that has optimal "witness size." The optimal witness size is shown to coin…